aft Ha y:^ -^ CORNELL UNIVERSITY LIBRARY 3 1924 062 545 276 DATE DUE A m\ M >? MSIt^ APR 1' 1995^ FEB 23 1999 CAVUORO PKINTEntNU.S A The original of tinis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924062545276 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1992. (Cornell Iniuerattg ffiibrarg aitljaca, Wein ^axk BOUGHT WITH THE INCOME O.F THE SAGE ENDOWMENT FUND THE GIFT OV HENRY W. SAGE 1891 HfKIRBBlitXGS Cambridge Tracts in Mathematics and Mathematical Physics General Editors J. G. LEATHEM, M.A. G. H. HARDY, M.A., F.R.S. No. 19 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS by F. S. MACAULAY, M.A., D.Sc. Cambridge University Press C. F. Clay, Manager London: Fetter Lane, E.C. Edinburgh : 100, Princes Street I916 Price 45. 6d. net Cambridge Tracts in Mathematics and Mathematical Physics General Editors J. G. LEATHEM, M.A. G. H. HARDY, M.A., F.R.S. No. 19 The Algebraic Theory of Modular Systems CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager ILnnliDn: FETTER LANE, E.G. eEDinburst: '0°. PRINCES STREET j^eto Borlt; G. P. PUTNAM'S SONS CombaE, ffialtutta anB JHaStas: MACMILLAN AND CO., Ltd SToronto : J. M. DENT AND SONS, Ltd. ffioitso: THE MARUZEN-KABUSHIKI-KAISHA All rights reserved THE ALGEBRAIC THEORY OF MODULAR SYSTEMS by F. S. MACAULAY, M.A., D.Sc. Cambridge : at the University Press 1916 En PREFACE rriHE present state of our knowledge of the properties of Modular -■- Systems is chiefly due to the fundamental theorems and processes of L. Kronecker, M. Noether, D. Hilbert, and E. Lasker, and above all to J. Konig's profound exposition and numerous extensions of Kronecker's theory (p. xiii). Konig's treatise might be regarded as in some measure complete if it were admitted that a problem is finished with when its solution has been reduced to a finite number of feasible operations. If however the operations are too numerous or too involved to be carried out in practice the solution is only a theoretical one ; and its importance then lies not in itself, but in the theorems with which it is associated and to which it leads. Such a theoretical solution must be regarded as a preliminary and not the final stage in the consideration of the problem. In the following presentment of the subject Section I is devoted to the Resultant, the case of n equations being treated in a parallel manner to that of two equations ; Section II contains an account of Kronecker's theory of the Resolvent, following mainly the lines of Konig's exposition ; Section III, on general properties, is closely allied to Lasker's memoir and Dedekind's theory of Ideals ; and Section IV is an extension of Lasker's results founded on the methods originated by Noether. The additions to the theory consist of one or two isolated theorems (especially §§ 50 — 53 and § 79 and its consequences) and the introduction of the Inverse System in Section IV. VI PREFACE The fsubject is full of pitfalls. I have pointed out some mistakes made by others, but have no doubt that I have made new ones. It may be expected that any errors will be discovered and eliminated in due course, since proofs or references are given for all major and most minor statements. I take this opportunity of thanking the Editors for their accept- ance of this tract and the Syndics of the University Press for publishing it. F. S. MACAULAY. London, June, 1916. CONTENTS ART. PAGK Ijjtroddction .... 1 I. THE RESULTANT 2. Resultant of two homogeiieou.s polynomials . -i 6. Eesultant of n homogeneous polynomials . . 7 8. Resultant isobaric and of weight Z ... .11 8. Coefficient of a/'-... «„'''" in /f is ■ii/''^''+'-^" . 11 8. The extraneous factor A involves the coefficients of {F„F,,...,F„.,).,^=oonly 11 9. Resultant is irreducible and invariant . . . . 12 10. The vanishing of the resultant is the necessary and sufficient condition that Fi = ... = F,n — should have a proper solution ... . . . 13 11. The product theorem for the resultant 15 ] 1. U{Fi,...,F^) contains (i^/, . . . , F,^), R is divisible hy R' . 15 12. Solution of equations by means of the resultant . . .15 12. The ?«-resultant resolves into linear homogeneous factors in X,Ui,U2,...,1ti. . . ...... 16 II. THE RESOLVENT 15. Complete resolvent is a member of the module . . 20 15. Complete resolvent is 1 if there is no finite solution . . 21 17. Examples on the resolvent .21 18. The complete M-resolvent i^„ 24 18. (F„)^^u,x,+...+u„xn=Oinod(F„F.„...,F,) . . . 24 19. All the solutions of Fi = F^= ... = Fi. = are obtainable from true linear factors of i^„ 25 20. Any irreducible factor of i^„ having a true linear factor is a homogeneous whole function of x,Ui, ...,u„ . . 26 21. Irreducible spreads of a module . . . . 27 22. Geometrical property of an irreducible spread ... 28 VUl CONTENTS III. GENERAL PROPERTIES OF MODULES AKT. PAGE 23. MIM' = MI{M,M') .... . . .30 23. If M'M" contains M, M' contains MjM" ... 31 24. Associative, commutative, and distributive laws ... 31 25. {M, M')[M, M'] aonta-ma MM' 32 26. MjM' and MKMjM') mutually residual with respect to M . 32 28. MKM„M,,...,M,) = [3flM^,MIM,,...,M/M,] . . 33 28. [Mi,M2,...,Mt]IM=\MJM,M2lM,...,Miliy] . . 33 30. Spread of prime or primary module is irreducible . . 34 31. Prime module is determined by its spread . . 34 32. If M is priaiary some finite power of the corresponding prime module contains M . .... 35 33. A simple module is primary 36 34. There is no higher limit to the number of members that may be required for the basis of a prime module ... 36 34. Space cubic curve has a basis consisting of two members . 37 35. The L. CM. of primary modules with the same spread is a primary module with the same spread .... 37 36. If M is primary MjM' is primary . . .37 37. Hilbert's theorem 38 38. Relations between a module and its equivalent Zf-module . 39 38,42. Condition that an i?-moduleJ/' may be equivalent to J/a:„=i . 39 38. Properties of an Zf-basi.s ... . 40 39. Lasker's theorem . . . . .40 40. Method of resolving a module ... .42 41, 44. Conditions that a module may be unmixed . 43 44 44 . 44 45 nixed . . 47 42. Deductions from Lasker's theorem 42. When MjM' is M and when not 42. No module has a relevant spread at infinity 43. Properties of the modules i/C), i/(») 44. Section of prime module by a plane may be 46. The Hilbert-Netto theorem .48 Unmixed modules 49 48. Module of the principal class is unmixed .... 49 49. Conditions that (/'i, i^'j, ..., i^,.) may be an i?-basis . . 50 50. Any power of module of principal class is unmixed . 51 51, 52. Module with -/-point at every point of J/ . . . .52 CONTENTS IX 52. When a power of a prime module is unmixed 53. Module whose basis is a principal matrix is unmixed Solution or homogeneous linear equations .... Noether's theorem 56. The Lasker-Noether theoi-em IV. THE INVERSE SYSTEM 58. Number of modular equations of an /T-module of the principal class . 59. Any inverse function for degree t can be continued 59. Diagram of dialytic and inverse arrays . 59. The modular equation 1 = . . . . 60, 82. The inverse system has a finite basis .... 61. The system inverse to {F^, F.^, ..., F,,) is that whose Fi derivates vanish identically 62. Modular equations of a residual module 63. Conditions that a system of negative power series may be the inverse system of a module .... 64. Corresponding transformations of module and inverse system 65. Noetherian equations of a module 65. Every Noetherian equation has the derivate 1 = . 65. The Noetherian array 66. Modular equations of simple modules . Properties op simple modules 67. A theorem concerning multiplicity 69. Unique form of a Noetherian equation . 71. A simple module of the principal Noetherian class is a principal system . . .... 72. A module of the principal class of rank n is a principal system .... .... 73. ii^ji' + ii" . . 74. fiV + /V' = W' = W", i^'lieref + r = y-l 75. i/,„ = l+/ii + ;ii2 + ---+Hm • .... 76. H\,-H"v, = II'v^v.- Hr^H,.-H\.^r, where l + r=y-^ 53 54 58 60 61 65 67 67 69 69 70 70 71 71 73 73 75 75 77 77 79 80 81 82 83 83 84 Modular equations op unmixed modules . 85 77. Dialytic array of J/('') . . . ... 86 78. Solution of the dialytic equations of J/(''> . . 88 CONTKNTS AKT. PAGE 79. Unique system of j'-dimensional modular equations of M . 89 79. The 5i-dimonsional equations 89 80. Equations of the simple S"-module determined by the highest terms of the members of an 5-basis of if M . 89 81. If 72 = 1 and if is unmixed, J/ is perfect . . 90 82. If i/('') is a princijaal system so is M . . ■ 90 82. A module of the principal class is a principal system . 90 8.3. i/('') and M are principal systems if the module determined by the terms of highest degree in the members of an S'-basis of i/('') is a principal system ; not conversely . 91 84. Modular equations of an //-module of the principal class . 92 85. Whole basis of system inverse to i/t'') 93 86. Modules mutually residual with respect to an ^-module of the principal class ... ... 94 87. The theorem of residuation . . .96 88. Any module of rank n is perfect .98 88. An unmixed ^'-module of rank n-\ is perfect . . 98 88. An ^-module of the principal class is perfect . . 98 88. A module of the principal class which is not an .ff"-module is not necessarily perfect 98 88. A prime module is not necessarily perfect . . . 98 89. An //-module i/ of rank r is perfect if the module i/aT+2=...=j-.t=o is unmixed . ... 99 90. A perfect module is unmixed .99 90. The L. c. Ji. of a perfect module of rank r and any module in .T,. + i, ...,.r,i only is the same as their product ... 99 91. Value of Hi for a perfect module ...... 99 92. If M, M' are perfect /^-modules of rank c, and if M contains J/', and i/a-,.+, = ...=,TK=o is a principal system, i//i/' is perfect ... . ... 100 Note on the theory of ideals . . 101 DEFINITIONS PAGE Modular sj'stem or module . 1 Member of a module . . 2 Basis of a module . . . . 2 1. Elementary member of {Fi, F^, ■■■, F,,) 3 1. Resultant of (/'i, /;,..., F„) . 4 6. Eeduoed polynomial . . .7 7. Extraneous factoi' . . . lo 8. Leading term of resultant ]0 8. Weight of a coefficient . . .11 8. Isobaric function . ... 11 12. The M-resultant FgM . , .10 12. Multiplicity of a solution . .17 13. Rank and dimensions ... .18 13. Spread of points or solutions . ... 18 14. Reducible and irreducible polynomials . 19 14. Complete resolvent ... . .20 14. Partial resolvent . . . . 20 16. Imbedded solutions . . . .21 18. Complete 2s-resolvent i^,, . . . 24 19. True linear factor of M-resolvent .... 25 21. Irreducible spread ...... 27 21. Order of irreducible spread 27 21. Equations of irreducible spread . ... 28 23. Contained module .... . . 29 23. Least and greatest modules . . . 29 23. Unit module ... . . . 29 23. G.c.M. of J/i,i/2, ..., ifj. . - 29 23. h.c.u. of MuMi,...,Mt . 30 23. Product of J/iji/j, ...,J4 . ... 30 23. P^, 0^, 7-point . . . . 30 23. Residual module . , . 30 29. Prime module . .33 xn DEFINITIONS AKT. PAOB 29. Primary module .... . .33 29. Singular point and spread . . ... 34 33. Characteristic number .... . . 36 33. Simple module ... ... 36 33. iT-modnle ... . .36 38. Equivalent .ff-module . . 39 38. .ff-basis of a module . ... 39 40. Relevant primary modules oi M . . 42 40. Relevant spreads of J/ ... . 42 40. Isolated and imbedded spreads and modules . 42 41. Mixed and unmixed modules . . . 43 43. The modules J/C'), ^1^) 45 47. Module of principal class .... 48 51. Basis consisting of the determinants of a matrix . 52 54. Inverse arrays .... . . 58 56. Noetherian module ... . .61 57. Dialytic array and equations . . 64 57. Inverse array, inverse function . . 64 57. Modular equations . . .64 59. Inverse system 60. ^-derivate of E . 60. Principal system 65. Noetherian equations 69 70 73 65. Underdegree of a polynomial ... .74 68. Multiplicity of a simple module . . .78 68. Multiplicity of a primary module .... 78 68. Primary module of principal Noetherian class . . 78 68. Complete set of remainders 79 68. Simple complete set of remainders .... 79 77. J-- dimensional modular equations 86 77. Regular and extra rows of dialytic array of J/('') 87 77. Regular form of dialytic array of J/(') . . 87 77, 88. Perfect module .... 87 85. Whole basis of sy.stem inverse to J/I'l . . 93 LIST OF REFERENCES References to the following in the text are given by their initial letters, e.g., (L, p. 51). (BN) A. Brill and M. Noether, " Ueber die algebraischen Functionen und ihre Anwenduiig in der Geometrie" {Math. Ann. 7 (1874), p. 269). (D) R. Dedekind, " Sur la Theorie des Nombres entiers algebriques " {Bull. Sc. Math. (1) 11 (1876), p. 278/288 ; (2) 1 (1877), p. 17/41, 69/92, 144/164, 207/248). Also in G. Lejeune Dirichlet's Vorlesnngen ilber Zahlen- theorie (Brunswick, 2nd edition (1871) and 4th edition (1894)). (DW) R. Dedekind and H. Weber, "Theorie der algebraischen Func- tionen einer Veranderlichen " {J. reine angeio. Math. 92 (1882), p. 181). (E) E. B. Elliott, Algebra of Quantics (Clarendon Press, 2nd ed. 1913). (H) D. Hilbert, " Ueber die Theorie der algebraischen Formen " {Math. Ann. 36 (1890), p. 473). (Hi) D. Hilbert, " Ein allgemeines Theorem liber algebraische Formen " {Math. Ann. 42 (1893), p. 320), § 3 of "Ueber die vollen Invariantensysteme." (K) J. Konig, Einleitung in die allgemeine Theorie der algebraischen Grossen (B. G. Teubner, Leipzig, 1903). (Kr) L. Kronecker, (i) " Grundziige einer arithmetischen Theorie der algebraischen Grossen " {J. reine angew. Math. 92 (1882), p. 1). Also expounded with additions in (K) especially in Chap. ix. (ii) " Zur Theorie der Formen hbherer Stufen " {Sitz. Akad. d. Wiss. zu Berlin, 37 (1883), p. 957). (L) E. Lasker, " Zur Theorie der Modulu und Ideale " {Math. Ann. 60 (1905), p. 20). (M) F. S. Macaulay, "On the Resolution of a given Modular System into Primary Systems ■' {Math. Ann. 74 (1913), p. 66). (M]) F. S. Macaulay, " The Theorem of Residuation" {Proc. Lond. Math. Soc. (1) 31 (1900), p. 381). (Mj) F. S. Macaulay, " Some Formulae in Elimination " {Proc. Lond. Math. Soc. (1) 35 (1903), p. 3). XIV LIST OF REFERENCES (M3) F. S. Macaulay, " On a Method of dealing with the Intersections of Plane Curves" [Tram. Am. Math. Soc. 5 (1904), p. 385). (Mo) B. H. Moore, " The decomposition of roodular systems of rank n in n variables" {Bull. Am. Math. Soc. (2) 3 (1S97), p. 372). (N) M. Noether, " Ueber einen Satz aus der Theorie der algebraisohen Functionen" {Math. Ann. 6 (1873), p. 351). (Ne) E. Netto, "Zur Theorie der Elimination" {Acta Math. 7 (1886), p. 101). (S) C. A. Soott, " On a Method for dealing with the Intersections of Plane Curves" {Trans. Am. Math. Soc. 3 (1902), p. 216). (Sa) G. Salmon, " On the order of Restricted Systems of Equations ' {Modern Higher Algebra, Dublin, 4th ed. (1885), Lesson xix). (Wi) Encyklopiidie der Mathematischen Wissenschaften (Teubner, Leipzig, Tell I, Bd. I, Heft 3 (1899), p. 283). G. Landsberg, " Algebraische Gebilde, etc." (Wj) Encyclopedic des Sciences Mathematiques (Gauthier-Villars, Paris, Tome I, Vol. 2, Fasc. 2, 3 (1910, 11), p. 233). J. Hadamard and J. Kiirschak " Proprietes generales des corps, etc." This account of the theory is founded upon that of G. Landsberg b is much fuller both in snbject matter and references. THE ALGEBRAIC THEORY OF MODULAR SYSTEMS Introduction Definition. A modular system is an infinite aggregate of poly- nomials, or whole functions* of n variables XiyX^, ■■■,Xn, defined by the property that if F, Fi, F^ belong to the system Fy^ + F^ and ^i^also belong to the system, where A is any polynomial in a?], ^j; ■••, ^n- Hence if F^, F^, ■■■, i^* belong to a modular system so also does A1F1 + A2F2+ ...+A!:Fk, where A^, A^, ..., A^ are arbitrary poly- nomials. Besides the algebraic or relative theory of modular systems there is a still more diflBcult and varied absolute theory. We shall only consider the latter theory in so far as it is necessary for the former. In the algebraic theory polynomials such as F and aF, where a is a quantity not involving the variables, are not regarded as different polynomials, and any polynomial of degree zero is equivalent to 1. No restriction is placed on the coefficients oi Fi, F^, ..., i^s except in so far as they may involve arbitrary parameters u^, u^, ..., in which case they are restricted to being rational functions of such parameters. The same restriction applies to the coefficients of the arbitrary poly- nomials A-i, Ai, ..., Ay, above. In the absolute theory the coefficients of 2^i, i'''2, ••., -4i, ^2, •■• are restricted to a domain of integrity, generally ordinary integers or whole functions of parameters Mj, u^, ... with integral coefficients; and a polynomial of degree zero other than 1 or a unit is not equivalent to 1. * We use the term wliole function throughout the text (but not in the Note at the end) as equivalent to polynomial and as meaning a wliole rational function. M. 1 2 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS Definitions. A modular system will be called a module (of poly- nomials). Any polynomial F belonging to a module M is called a member (or element) of M. According as we wish to denote that F \& a, member of M in the relative or absolute sense we shall write F = mod M, or F=0 modilf. The notation F= Omodi(f only comes into use in the sequel in connection with the Resultant. A basis of a module M is any set of members i'l, F2, ■•■, Fj^ such that every member of M is of the form JT^Fi + ^2F2+ ■■■ +JC,cFk, where Jii, JTi, ■■■, JT^ are polynomials. Every tnodule of polynomials has a basis consisting of a finite number of members (Hilbert's theorem, §37). The proof of this theorem is from first principles, and its truth will be assumed throughout. The theory of modular systems is very incomplete and offers a wide field for research. The object of the algebraic theory is to dis- cover those general properties of a module which will afford a means of answering the question whether a given polynomial is a member of a given module or not. Such a question in its simpler aspect is of im- portance in Geometry and in its general aspect is of importance in Algebra. The theory resembles Geometry in including a great variety of detached and disconnected theorems. As a branch of Algebra it may be regarded as a generalized theory of the solution of equations in several unknowns, and assumes that any given algebraic equation in one unknown can be completely solved. In order that a polynomial F may be a member of a module J/ whose basis (-F\, F^, ..., F]^ is given it is evident that i'^must vanish for all finite solutions (whether finite or infinite in number) of the equations Fi = F2= ■■.=Fic = 0. These conditions are sufficient if M resolves into what are called prime modules*; otherwise they are not sufficient, and i''must satisfy further conditions, also connected with the solutions, which may be difficult to express concretely. The first step is to find all the solutions of the equations Fi= F, = .■■ = F„ = 0; and this is completely accomplished in the theories of the resultant and resolvent. * Cayley and Salmon constantly assume this. Salmon also discusses particular cases of a number of important and suggestive problems connected with modular systems (Sa). l] THE RESULTANT I. THE RESULTANT 1. The Resultant is defined in the first instance with respect to n homogeneous polynomials Fi, F2, ■■■, Fn in n variables, of degrees 4,4, ..., 4, each polynomial being complete in all its terms with literal co- efficients, all different. The resultant of any n given homogeneous polynomials in 71 variables is the value which the resultant in the general case assumes for the given case. The resultant of n given non- homogeneous polynomials in m — 1 variables is the resultant of the corresponding homogeneous polynomials of the same degrees obtained by introducing a variable Xo of homogeneity. Definitions. An elementary membei- of the module {F^, F^, ..., i^„) is any member of the type w-fi (i = l, 2, ..., «), where t, L=lil2---lni Li=Llli, L2 = Ljlti, ... Ln- Llln- The resultant R oi F^, F^, ... , Fn has already been defined (§1) as the H.c.F. of the determinants of the array of the coefficients of all elementary members of {F^, F^, ■•-, F,i) of degree ^+1. We shall first consider a particular determinant D of the array, viz. that representing (§ 1) the polynomial X(») F, + X('' F,+ ...+ XC-^l Fn of degree / + 1, where X*'' denotes a polynomial in which all terms divisible by x^ or X2 ... or ^i ' are absent, which may be expressed by saying that X*'' is reduced in Xi, x^, ■■■, Xi. The polynomial jr(»> F^ + X(i' F^+...+ X"-" Fn is represented by the bordered array 0,/' + !) ■ • • > "/x''"^'' are all the power products oixj,X2,... ,x„ of degree l+l,a,ndX,,X„ ..., \^ are the coefficients of X(»l, X^'\ ..., X<"-i). That this array has the same number fj. of rows as columns is seen from the fact that one and only one of the elements a-^,a.2, ..., a„* (the coefficients of Xi\xJ\...,xJ"- in F^, F^, .■.,F^ occurs in each row and each column. This is evident as regards the rows. To prove * These are not the same as the aj , a^,..., a„ in the first column of the array. The latter should be represented by some other symbols. 8 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [l that the same is true of the columns, we notice firstly that there is no power product ^'*'^ = Omod(F„F„...,Fn), where w''"^'' is any power product of Xi,Xi,.--,Xn of degree /+1. It can be proved that the factors of D other than R can be divided out of this congruence equation, so that i2 Li in the coefficients of Fi{i=\, 2, ...,n-l). It follows that H, which is a factor of D, is at most of degree Z,, in the coefficients of F„. We shall prove that R is of this degree, and consequently of degree Lt in the coefficients of Fi. Let D' be any other non-vanishing determinant of the array, viz. a)i('+i) 0)/+^) a)/+i> «/ bi ki (^2 ^2 ^2 t'/x *|ll This represents the polynomial ^li^i + 42-^2+ •■■ +-4„i^„, in which i^„ = ^i F, + A.F^ + ... +A^Fn. This gives, by equating coefficients of power products on both sides, 2Aa.= 2aa', 2X6= 2a6', ..., 2XA; = 2a^' as equations for Xj, Xj, ..., X^; and they have a unique solution, since D does not vanish. Let ( j denote the determinant of the substitution corresponding to the solution of the above equations for Xi,X2, ...,X^ as linear functions of a^.ttj, ..., a^. Then if we put 2Xa= 2aa' = X/, 2X6= 2a6' — Xj', ..., 2XX- = 2a^' = X/ we have (0=^. e')=^-"^(oe)=e)--^o^^' by the rule of successive substitutions, or the rule for multiplying determinants. Hence f=e)' Now we can find the solution for Xj, X2, ... , X^^^, or the solution of X^''^F^ + X<'^F, + ...+X<''-'^F„ = A^F, + A,F,+ ... + A„F„, in the following way. First solve the equation F(») F, + F('> F,+ ... + Y^"-'^ Fn-i + X("-^' = An 10 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [l for the unknowns F(»), FW ..., F^-^), X("-'». This equation has a unique solution, since the more particular equation FC) xl' + F(i) x^^'+...+ F"-'') xl'lz'i + X*"-!) = Ar, has a unique solution (for any given polynomial An can be expressed in one and only one way in the form on the left) and shows that the number of the coefficients of F'"', F''', ..., FC"'', X'^'^' is equal to the number of equations they have to satisfy. Substituting the value thus found for jr("~i) in the equation X(°)i?'i + XWi?', + ... + X^''-'^ Fn = A,F, + A,F,+ ... + A,,Fn, it becomes XCi F, + X« F„.+ ...+ X'"-^' Fn-^ = {A, + F«) Fn)F + ... + (4„_, + F("-=) Fn) Fn-,, where F'"', F^', ..., F'""^' have been found. Next solve the equation ^(0) F^ + ^(1) i?', + . . . + zc-'i Fn-2 + X("-^) = ^ n-i + F("-^> i'";, , which has a unique solution for j?'l»», Z^'\ ...,Z^''-'\ X^'^-'l "We can proceed in this way till X'"', X'^', ...,X("-'', i.e. K, K, ■••, V. have all been found. In this method of solving the unknowns on the left are associated with Fi, F2, ... , Fn-i only and not with .F„. Hence ( ) is a rational function of the coefficients oi Fi, F2, ... , Fn whose denominator is independent of the coefficients of F,,, and the same is therefore true of -^ = ( j . Hence every determinant D' of the array has a factor in common with D which is of degree Ln in the coefficients of Fn- The resultant B, which is the h.c.f. of all the determinants D', is therefore of degree Lj in the coefficients oi Fi(i= 1,2, ... , n). If we put D = AB, A is called the extraneous factor oi D. We have proved that A is independent of the coefficients of Fn ; and it is proved at the end of § 8 that A depends only on the coefficients of (Fj, F2, ..., i^,j-i)a^,=0. 8. Properties of the Resultant. Since D has a term ar ...a/" (§ 6) B has a term a,^' a/" ... a,,"^". This is called the leading term of B. Since D vanishes when Ci, C2, ..■,€„, all vanish (§ 6) the same is true of B; ioT D =AB and A is independent of Cj, Cn, ... , c,i. The extraneous factor A of D is a minor of D, viz. the minor obtained by omitting all the columns of D cwresponding to power l] THE RESULTANT 11 products reduced in n-1 of the variables and the rows which contain the elements a^,ai, ...,a„ in the omitted columns {M-z, p. 14). Thus D/A, where A is this minor of D, is an explicit expression for R. Each coefficient a of F^,, F^, ■■■, Fn is said to have a certain numerical weight, equal to the index of the power of one particular variable (say *■„) in the term of which a is the coefficient. In the case of non-homogeneous polynomials the variable chosen is generally the variable x^ of homogeneity. Also the weight of a^ is defined as p times the weight of a, and the weight of a^tV ... as the sum of the weights of a*", 6', c"", .... A whole function of the coefficients is said to be isobaric when all its terms are of the same weight. The resultant is isobaric and of loeight L. Assign to Xi,X2, ■•.,Xn the weights 0,0, ...,0,1. Then the coefficients of Ft,, B\,...,Fn have the same weights as the power products of which they are the coefficients. The «th row of the determinant D represents the poly- nomial cDii^ = aiU)i"+^^ + 6ia)2^^+^U...-i-^il fc] = -Z/2 ^2 ~ ■ ■ • ^ -^ ji-1 'n-1 ~ n '2 — ^7i-l ~ -^71 • Finally let aj, a^, ..., a„, Cj.Co, ■-., c„_i denote the same coefficients of i^i, Fi, ..., i^„ as in § 6. We assume Rn irreducible and have to prove that R is irreducible. F, is of weight i„ in the coefficients of Fi'>\F^''\ ...,i^J^/and each coefficient is a homogeneous polynomial in x^-^, Xn of degree equal to its weight in reference to x„. Hence F,= Ax'^^l^ + Bx^_\^ Xn+ ..., where J , .S, ... are whole functions of the coefficients of i^i,i''2, ■■■,Fn-i of the same dimensions as Rn- When Xn = 0, F„ becomes the resultant of (i^.i"), i^2'°', • - , i^„-i),„.o' viz- ^» ^f -1 ; ^ence A = Rn. Also the whole coefficient of a/"' a.^^'' ...a^""" in F^ is ^'fTj"'' where a'„_i is the coefficient of i^o'"-' in i?'^™^ (§ 8), viz. Hence B has a term i'„_i a/" . . . af^g' *»-i ' ^ ^' ^'^^ cannot be divisible by Rn, since Rn does not involve b. Hence we find that F, = Rnxl\+Bx^^:l\n+-- where B is neither zero nor divisible by K„. Now if R' vanishes one of the solutions of Fn = for Xn-i : «„ will be the same as in one of the solutions of Fi= ... = Fn-i = (§ 10), and will therefore be a solution of -P„ = 0; i.e. R' = requires R„ = 0, and Rfi is divisible by each irreducible factor ot R'. But (§ 3) i?„ = i?J"a/" + 5'«/"~^+ ..., where^' =(- if^hB'" mod Bn, so that B' is neither zero nor divisible by Rn- Hence, as in § 3, R^ has an irreducible factor of the form Rj^'a,? + ..-, and has no other l] THE RESULTANT 13 factor involving the coefficients of Fi, F^, •.., F^-i- This must therefore be a factor of R'. Again R' is what R becomes when all the coefficients of Fn other than those of F^ are put equal to zero. Hence R has an irreducible factor of the form R^a^ + ■ ■ • , where q>p. The remaining factor of R is independent of the coefficients of i^i,i^2, •••, Fn-i, and therefore also of the coefficients of Fn when m > 2. Hence R is irreducible. It easily follows that R is invariant for a homogeneous linear substitution whose determinant ( ,) does not vanish. Suppose that -S = and that this is the only relation existing between the coefficients ot Fj, F,, ..., F^. Then not more than one relation can exist between the coefficients of F^', i'V, • ■ • , Fn, the poly- nomials into which Fi, F^, ■■■,Fn transform. Since R = there are less than /u, linearly independent members of {F^, F^, ■■•, F^) of degree l+l and therefore less than /x linearly independent members of (Fi, F^', ■.■,Fn') of degree l+l, and the only single relation between the coefficients of Fi,F^, ■.■,Fn which will admit this is R' = 0. Hence R = Q requires R' = 0, and R' is divisible by R. The remaining factor of R' is independent of the coefficients of Fi, F2, ■■■,Fn, and can be shown to be f , j . A proof that R is invariaut without assuming it irreducible is given in (E, p. 17). 10. The necessary and sufficient condition that the equations Fi = Fi= ...= Fn=0 may ham a proper solution is the vanishing of R. In the general case, when the coefficients are letters, ARx,!-^^ = mod (F„F„ ...,F,). Put a;„ = 1 and change* ct to Ci — Fi(i = l,2,...,n); then A does not change, being independent of Ci.Ca, ■■■,Cn (§8); but R changes to R - AiFi - A2F2- ■■■ - AnFn , and this must vanish ; hence R=Omod(F,F„...,Fr:)^,=i. Hence R vanishes if the equations Fi = F2- ■■■ = Fn^O have a solution in which «,j=l, i.e. if they have a proper solution. To prove that ^=0 is a sufficient condition, we shall assume that ^=0 is the only relation existing between the coefficients of Fi, Fa, ■■■, Fn. There are then less than /j- linearly independent members of (Fi, F2, ...,Fn) of degree 1+ 1. Hence the coefficients Zp^,p^,...,pn of * Called the Kronecker substitution. 14< THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [l the general member of degree 1+ i must satisfy an identical linear relation 2Cp,.p.2 p„SpuP, P„ = t). P1+P2+ ■■■ +Pn=l+'i- The coefficients of the general member of degree / also satisfy one and only one identical linear relation, whether R vanishes or not. To prove this it has to be shown that the number iVof linearly independent members of (i^i.i^s, ■■•,F,^ of degree Z is 1 less than the number p of power products of degree I. If no relation exists between the ■coefficients of F^, F^, ■■■ ,Fn the equation X(»)i^i + A'Wi^2 + . . . + X<''-i)i?; = A^F, + A,F,+ ...+ AnFn .can always be solved by the method of § 7, where A^,, A^, ■■■, A^ are arbitrary given polynomials. Hence N is not greater than the number of coefficients in Xi«', X^'>, ... , XC-^', or in XC'^i'' + X'^'^x.h +■■■+ XC-^'a;,,'", "viz. p - 1, since, when this expression is of degree I, every power product except a;i'~^X2°~^...a;^~^ occurs once and only once in it. Hence N^p-1. Any particularity in i^i, i^2, ■■•,Fn can only affect the value of N by idiminishing it. Hence for the remainder of the proof it will be sufficient to show that iV= p — 1 in a particular example in which R = 0. Let F,= (cc,-a;,)xy\ F,= (x, - x,) xy\ . . . , F„ = {x„-x^) x,y\ Then ^ = since the equations Fi = F2= ... = Fn = have the proper solution a'i= «2= • • • = a:,, = 1. Let x^^x^'' . . . x,?" be any power product of ■degree /. \i pi~^li change x-^'x^"^ to ;?! ' ~ ' x^'^ where jOi + />2 = ^i - 1 + ^2 1 this is equivalent to changing x-^^x.^'' ...x,^'^ to x^'xf'' ... x^"" + A-^Fx- Again if q^ ^ Z2 change x^'xf' to x^ '~^x-f' ; and if g2 < k proceed to the first pr> Ir and change ^r*'"' «^+"\' to xj'^'^x'!^^^ . If we continue this process, going round the cycle Xt,, x^, . . . , Xj^ as many times as is necessary, the power product x^'xf^ ... x^ will eventually become changed to «i '~ a'2^ ... x^ ■ Hence these two power products are congruent mod {F-i, F^, ..., F,^, while neither of them is a member of (i^i, F^, ...,Fn), since they do not vanish when Xi = ...=Xn=l. Hence N=p- 1. Let -F'= 223,,52,...,t,„a;i*'a;2'' ■•• ^n'" be the general member of {Fi, F2, ..-jFn) of degree I; then XtF is a member of degree ^ + 1 in which the coefficient of x^'x^'' ... xj'" is Zp,,p^ m-i.-.pk- Hence ^''PuPi,--,Pn^Pi.--,Vi-h--,Pn =0 (« = Ij 2, . . . , ?j), or 2c,,.gj,„.,gi+i,...,,j„55,,j,,...,g„ = {{ = 1, 2, . . . , u). l] THE RESULTANT 15 These n equations in Zq^,q.^ g„ are therefore equivalent to one only; and the continued ratio C5,+i,,j,...,g„ :c3,,,5,+i ,„ : ... ■Cq„q^,...,q^+\ is the same for all sets of values of ^i, g'2, •••, Sk whose sum is I. Equating to ^^■.a^■....•.a,^, it follows that Cp^^p^ p„ is proportional to a^'a^' ...a^" {p-^+p^+ ...+Pn=l + \). Hence it follows that (oj, oj, ..., u„) is a solution of the equations Fi = F2= ...=Fn = 0. 11. The Product Theorem. If Fn is the product of two polynomials Fn, Fn, the resultant B of Fi, F^, ..., Fn is the product of the resultants E , R" of Fi, F^, ..., Fn and Fi,Fi,..., Fn- For in the general case R' and R" are irreducible, and if either vanishes R vanishes. Hence R is divisible by R'R". Also it can be easily verified that the leading terms of R and R'R" are identical. Hence iZ = ^'^". This result can easily be extended to the case in which any or all oi Fi, Fi, ■■■, Fn resolve into two or more factors. If Fx,F^, ■■.,Fn are all members of the module (Fi, F^', ■■■,Fn') the resultant R of F^, F2, ---jFn is divisible by the resultant R' of F^, F;, ..., Fn. For a R'=0 then ^ = 0. 12. Solution of Equations by means of the Resultant. The method of the resultant for solving equations can only be applied in what is called the principal case, that is, the case in which the number r of the equations is not greater than the number n of the unknowns, and the resultant Fu of the equations with respect to Xi,W2, ..., Xt-\ (after a linear substitution of the unknowns) does not vanish identically. When jPo vanishes identically the method of the resultant fails, but the equations can be solved by the method of the resolvent, due to -Kronecker, as explained later. The method of the resolvent is also applicable to any number of equations whether greater or less than the number of unknowns. Homogenemis Equations. Let the equations \ieFi = F^ = ...=Fr=0 of degrees /j, 4, •■•> Ir, where r ^ n. We assume that their resultant Fa with respect to i^i, 0^2, ..., x,-..^ does not vanish. We regard x^, X2, ..., Xr as the unknowns, the solutions being functions of iTr+i, •••, ^u- But instead of solving for one of the unknowns x^ , x^, ..., Xr we solve for a linear combination of them, viz. for x=UiXi + U2X2+ ... + u^Xr* where Mi, u^, •-., le,- are undetermined quantities. Let F^ stand for X - UiX-i_- U2X2- ... -UrX,.. Then we regard Fi = F2= ■■■ = Fr=Fn = * Called the Liouville substitution. 16 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [l as the given system of equations with w^, ..., Xr, x a,s unknowns, and their resultant Fo'"' with respect to a;,, x^, ..., Xr gives the equation i^„<''l = Ofora;. Definition, i^o'"' is called the u-resultant of {F^, F^, ..., Fr). Applying the reasoning of § 9 it is seen that Fo is the resultant (with respect to .«!, ..., x,—i, Xo) of i^i, F.^, ...,Fr when a;^, a-v+i, ••-,;»» are changed to x,.x„, x,.+iXo, ■■■, x^Xa, and is a homogeneous polynomial in Xr, Xr+i, ■■■, Xn of degree L=lil2 ■■■ Ir, viz. where Br+i is the resultant of (Fi, F^, ..., i^r)a:^^i=...=a^,=0! and does not vanish; for a homogeneous substitution beforehand between Xr, Xr+i, ■■■, Xn ouly would bc carried through to Fo. Similarly i^o'"' is the resultant (with respect to Xi, ..., Xr, Xo) of Fi, F2, ■■•, Fr, Fu when x, Xr+i, ■■■, Xn are changed to xxo, Xr+iXo,--., x^Xo, and is a homogeneous polynomial Jt'r+iX^+ ... in x, 37^+1, ■••, if^n where R'r+^ is the resultant of (F^, F2, ■■■, Fr, i^„).iY^j=,.,=.r„=o- It is easily seen* that -R',.+i = .Sr+i- Hence i^o*"' =i?r+i x^+ ..., where Br+i + 0. To each solution Xr=Xrt of Fo = corresponds a solution (x^, x^i, ..., Xrt) of the equations Fi = F2= ••■ =Fr = for Xi, x^, ..., Xr (§ 10). There are therefore L solutions altogether, and they are all finite, since i?r+i + 0. Similarly to each of the L solutions x = Xi of i*!)'"' = there corre- sponds a solution {x-^i, x^i, • • ■ , aj^i, ;?;») of jPj = . . . = jfV = -^m = ; and as regards (xu, ocu, ..., a-Vi) the L solutions must be the same as those obtained by solving i^o= 0. Hence it follows that where x^, x^, ..., Xri are independent of Mj, u^, ■••, m,-- Hence i?'„("' = iZ,.+in(a;-Mia;ii- ...-Mr«ri) (« = 1, 2, ..., L). Thus i^o'"' is a pi-oduct of L factors ivhich are linear in x, Ui, lu, ..., Ur, and the coefficients of Ui, u-i, ..., m,. in each factor supply a solution of the equaMons Fi= F2= ... = Fr = 0. Also the number of solutions is either L = lj2...lr or infinite, the latter being the case when F^ vanishes identically. * By introducing a as coefficient of x in F„ it is seen that iJ',.+i is divisible by a^ by considering weight with respect to x. Also the whole coefficient of a'' in iJ'^^i is JJr+i (§ 8) . Hence iJ^+i = a' i^r+i = -Rr+i • l] THE RESULTANT 17 If Du is the determinant for {F^, F^, . .., Fr, FJ, regarding x^, x^, ..., Xr, Xo as the variables, like the i> of § 6, we have D,, = AFo^"K The ex- traneous factor J. depends onlyon the coefficients of (.Pi, F^, ..., Fr)x„=o, that is, oi {Fi, Fa, ..., i^,.)aY^, = ...=x„=o. Hence A is a pure cortstaoit, independent of Xr^.l, ■■■,Xn andof Ui, u^, ■■■, u,., and we may take D„= as the equation for x. Definition. The number of times a linear factor x - u^x^ — ... - M^^rs is repeated in i''o'"' or D^ is called the multiplicity of the solution .(«H, «2i, ••■! '"KrO- This term has a definite geometrical interpretation; it is the number of solutions or points, in the general case distinct, which come into coincidence with a particular solution or point in the particular example considered. In the case of n homogeneous equations in n unknowns such that i2=t=0, the complete solution consists of the non-proper solution (0, 0, . . ., 0) with multiplicity L = lili ... In- Non-homogeneous Equations. In the case of non-homogeneous equations a linear substitution beforehand affects only x-i, x^, ..., «■„ and not the variable x^ of homogeneity. Hence it is possible for i^r + i to vanish identically, while Fq and i^o'"' do not, no matter what the original substitution may be. In this case there is a diminution in tlie number of finite solutions for x, but not in the number of linear factors of i^o'"'- To a factor Ui Xii + U2Xii + ... + u,. a-,.; of i^o'"' not involving x corresponds what is called an infinite solution of i^i = i^2 = • ■ • = -fr = in the ratio ^j; ta?,; : ■•■ :»h- Infinite solutions are however non- existent in the theory of modular systems (§ 42). An extreme case is that in which i^o'"' does not vanish identically, but is independent of x, when all the L solutions are at infinity. It may happen that a system of non-homogeneous equations has only a finite number of finite solutions while the resultant F^ vanishes identically. In such a case the method of the resultant fails to give the solutions. Example. The equations Xi = x^ + Xi x^ = Xs + XiX.2 = have the finite solution Xi = x^ = a-j = ; but the resultant vanishes identically because the corresponding homogeneous equations are satisfied by Xo = iri = 0, a system of two independent equations only. 18 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ll II. THE RESOLVENT 13. We shall follow, with some material deviations, Kunig's exposition of Kronecker's method of solving equations by means of the resolvent. The equations are in general supposed to be non- homogeneous; and homogeneous equations are regarded as a particular case. Thus a homogeneous equation in n variables represents a cone of w — 1 dimensions with its vertex at the origin. Homogeneous co- ordinates are excluded. The problem is to find all the solutions of any given system of equations Fi = F2= ■■■ = JPf^O in n unknowns x-^, x^, ..., Wn- The unknowns are supposed if necessary to have been subjected to a homo- geneous linear substitution beforehand, the object being to make the equations and their solutions of a general character, and to prevent any inconvenient result happening (such as an equation or polynomial being irregular* in any of the variables) which could have been avoided by a linear substitution at the beginning. In theoretical reasoning this preliminary homogeneous substitution is always to be understood ; but is seldom necessary in dealing with a particular example. The solutions we shall seek are (i) those, if any, which exist for «i when a:,, x^, ..., Xn have arbitrary values; (ii) those which exist for Xi, X2, not included in (i), when X3, ..., Xn have arbitrary values; (iii) those which exist for Xy, x^, x-,,, not included in (i) or (ii), when X4, .., a",! have arbitrary values; and so on. A set of solutions for' Xi, x.i; ..., Xr when a-v+i; ■■•,■^1! have arbitrary values is said to be of rank r, and the spread of the points whose coordinates are the solutions is of rank r and dimensions n - r. If there are solutions of rank r and no solutions of rank < r the system of equations Fi= F^.^ ■■■ = Fk=0 and the module (i^i, F2, ••■, Fi^ are both said to be of rank r. 14. The polynomials F^, F^, ■•■, F,,, and also all their factors are regular in x^. Hence their common factor I) can be found by the ordinary process of finding the h.c.f. of F^, F^, ■■■, Fi- treated as polynomials in a single variable ^1. If Z* does not involve the variables we take it to be 1. If it does involve the variables the solutions of D = treated as an equation for x-i give the first set of solutions of the equations Fi = F2= ... = F^ = mentioned above. * A polynomial of degree I is said to be regular or irregular in xi aocordin" as the term Xi'- is present in it or not. II] THE RESOLVENT 19 In the algebruc theory of modules we regard any algebraic equation in one unknown, whether the coefficients involve parameters or not, as completely soluble, i.e. we regard any given non-linear polynomial in one variable as reducible. A polynomial in two or more variables is called reducible if it is the product of two polynomials both of which involve the variables. A polynomial which is not reducible is called (absolutely) irreducible. Any given polynomial is either irreducible or uniquely expressible as a product of irreducible factors, leaving factors of degree zero out of account. It is assumed that the irreducible factors of any given polynomial are known. Thus the polynomial D above may be supposed to be expressed in its irreducible factors in Xi, Xi, ..., Xn, and to each irreducible factor corresponds an irreducible or non-degenerate spread. Put Fi = D'f>i (i = l, 2, ..., k). Then^i,2, •■•, 4>k have no common factor involving the variables, and the same is true of the two polynomials Xj^i + X,^2+ ... +\i<^t and /xji^i + /i2<^3+ ■•• +M;;<^t, where the X's and /a's are arbitrary quantities. Regarding them as two polynomials in a single variable Xi we calculate their resultant, and arrange it in the form P,F,<'^ + P,F/^ + ... + P,c,F,,^'\ where Pi, p2, ■•■, pk, are different power products of the X's and /n's, and i^i'^', i^a'^'i •••, -^Ai''' are polynomials in X2, x^, ..., Xn not involving the X's and yu's. Each T'V is regular in X2; for [any homogeneous linear substitution beforehand oi x^, Xs, ■■■, Xn among themselves only would be carried through to the i^/^'. Find the h.c.f. i)i'> of F^''\ F^'\ ..., F^}'^ treated as polynomials in a single variable x.„ and put i^/" = D'" /'' (« = 1, 2, . . . , k,). Then find the resultant of X,<^jW + A2<^2'" + - + K'^u}'^ and p.^'^ + f^,,^'^ + ... + /a,„ 2^'\ ■■ ■ , i'^\ F,^\ FA .. • . F^A with H.C.F. DC^), A <#./', - ., <^*n and so on. 2—2 20 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ll Now any solution of jPi = i^2 = • • ■ = -Pi = is a solution of i> = or of i='f>2= • • • = <^*: = 0. And any solution of , = <^2 = . . . = <^^ = is a solu- tion of i^,*'> = J5;<'» = . . . = i^i,('' = 0, since ^piFfi> = mod {^Xt^u ^t^ii), and therefore a solution of Z»W = or of <^iP) = c^^'"' = ■ • • = <^fe''' = 0- Hence any solution of i^i = jPa = . . . = i^t = is a solution of I)=0 or of i)(i) = or of <^i''' - <^2''' = • • • = <^t,''' = 0. Proceeding in a ■ similar way we find that any solution of Fi= ...=Fj:=0 is a solution of DD^'^ ... i)'"-^) = 0, since (^i'""'', ^2'""", ••• , <^1^;^^' are polynomials in a single variable av at most and have no common factor. Gonveosely if ^i ^4. ••■ 1 ^n is any solution of i)'-' = the resultant of 'S,Xi4>l''^^ and 2/Xj<^;('' with respect to x.. vanishes when Xs= is, and 2Xj<^i('' = 2/A,i(^i''' = have a solution x., = L when .r3 = f3; i.e. the equations ^1''' = . . . = <^jt,<'* = 0, and therefore also the equations i?'j(i) = . . . = i^'j. W = 0, have a solution f,, m, ^t, •••, •■«'.>; and, by the same reasoning, the equations Fi = F, = ... =i^,, = have a solution ^1, Si, ^3, «4, ••• , ^n- Similarly to any solution of BD^'^ ... ZX""'' = 0, say a solution $,, a-,+i,...,Xn of Z)''-'' = 0, there corresponds a solution ^1, I2, •••, h,Xi+i, ..., «,i of the equations i'"'i = i^2 = ...=F,, = 0. Hence from the solutions of the single equation DB^^^ ... i)i"~'' = we can get all the solutions of the system Fi = F2= ... = Fi; = 0, since all the solutions of the latter satisfy the former. Definitions. DD^^^ ... i)'"-'' is called the complete {total) resolvent of the equations F^ = i^2= • • • =F^ = and of the module (F^.F^,..., F^). Z)'*"'' is called the complete partial resolvent of rank i, and any whole factor of 2>''''* is called a partial resolvent of rank i. 15 . The complete resolvent is a member of the module (Fi ,F.2,..., F,,). For SftZ^'' = mod {%Ki>i, 1i^i<^:)= A1Xii + B^ix,4>,, where A, B axe. whole functions of x^, x^, ..., x,i, Aj, ..., A,;, /xj, ..., ^,,. Hence by equating coefficients of the power products pi on both sides, we have Z',W = 0mod(<^„ = mod {F„ F,, ... , F^)* or Z)i)W<^/'> = mod (F„ F„ ... , F,). Similarly i)i)('> . • . XX""".^/"-^' = mod (F,,F,,..., F,,) ; and since the <^i'""'' include one variable only (or none at all) and have ♦ Not D-FjCt = Omod (Fi.Fj, ..., F^) because any common factor of i'^.i?,, ..., Fj not involving the variables is not included in D and is left out of account. Il] THE RESOLVENT 21 no common factor, we can choose polynomials at in the single variable so that 2ai<^i'"-" = 1. Hence Z)Z»W ... i>("-i> = mod (Fi,F^,..., F,). If the equations i^i = i^o = . . . = Fk = have no finite solution the complete resolvent is equal to 1 ; consequently 1 is a member of (Fi, F^, ... ,F^, and every polynomial is a member. 16. We have seen that to every solution Xi = ^i of Z>'*"'' = there corresponds a solution ^i, i-i,---, ii, Xi+i,..., Xn of the equations Fx = F^= ... = -ffc = 0. It may happen that there is an earlier complete partial resolvent i>'^'-'' which vanishes when Xj=ij, ... , Xi = ii. In such a case the solution ^i, ... , |j, iTj+i, ..., x^ of Fi= ■..--=F^=^Q corresponding to a solution of Z)*'"'* = is included in the solutions corresponding to Z)'-'"') = 0, and may be neglected if we are seeking merely the complete solution of Fi = F2= ■■. = Fj: = 0. Such a solu- tion is called an imbedded solution. All solutions corresponding to an irreducible factor of Z>''"^' will be imbedded if one of them is imbedded. 17. Examples on the Resolvent. Geometrically the re- solvent enables us to resolve the whole spread represented by any given set of algebraic equations into definite irreducible spreads (§ 21). It has been supposed that the complete resolvent also supplies a definite answer to certain other questions. The following examples disprove this to some extent. Example i. Find the resolvent of n homogeneous equations Fi = F2= ■•. = Fn = of the same degree / and having no proper solution. Since there are no solutions of rank <"-'>. The first derived set of polynomials Z'/'', F.fi\ ..., F„y^ are homo- geneous and of degree P, the 2nd set i^/', FJ^\ ...are homogeneous and of degree l\ and the (n - l)th set i^jC-^', F^'^-'^\ ... are homogeneous and of degree f . This last set involve only one variable a-„, and .iJi-i therefore have the common factor «■„ , which is therefore the required complete resolvent. We should arrive at a similar result if we changed Xi to Xi + ai{i=l, 2, ..., n) beforehand, thus making the polynomials non- homogeneous. The complete resolvent would then be {xn + a,,)^ The resultant would be (.r„ + a,j)' . The diiference in the two results is explained by the fact that the resultant is obtained by a process 22 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [H applying uniformly to all the variables, and the resolvent by a process applied to the variables in succession. Example ii. Konig (K, p. 219) defines a module or system of equations as being simple or mixed according as only one or more than one of the complete partial resolvents D, Z>''', ... , Z)(""'l differs from unity. Kronecker (Kr, p. 31) says that the system of equations Fi=t\= ■.■=F,;=0\s irreducible in this case ; and the Ency. des Sc. Math. (Ws, p. 352) repeats Konig's definition. We give two examples to show that this definition is a valueless one. If u, V, IV are three linear functions of three or more variables, any polynomial which contains the spread of u = v-0 is of the form Au + Bv; if it also contains the spread of u = iv = 0, B must vanish when t< = w = 0, hence B must be of the form Cu + Dw, and Au + Bv of the form A'u+ B'vic ; if it also contains the spread of v = w = (), A' must be of the form C'v+D'w, and A'u + B'vw of the form G'uv + D'uiv + B'vw. Hence a polynomial which contains all three spreads is a member of the module {vii\ wu, iiv), and also any member of the module contains the three spreads. This module, although composite, is not mixed in any proper sense of the word. Besides having partial resolvents of rank 2 corresponding to the three spreads the module has a partial resolvent of rank 3 corre- sponding to its singular spread u = v = w = Q. This last partial resolvent does not correspond to any property of the module which is not included in the properties corresponding to its partial resolvents of rank 2 ; in other words the partial resolvent of rank 3 is purely redundant. The resolvent Z>'''Z>''^' can be found as follows : Suppose u = aa + aiXi + aoX.2+ ..., v = bo + 1^X1 + h^x., + ..., w = Ca + CiXi + C2X.,+ .... Then the resultant of X-^vw + Kwu + ^-^uv and ii-ivw + it-^wu + ii-iUxi with respect to x-^, apart from a constant factor, is {ciV- biw) {a-iW -Cjm) {hiU- coiv) ' CiV - b-,w a., w — c,u hiU — a, v ) X ■{- ^ - + ^ — + ' (i its four irreducible factors corresponding to the spreads v = w=0, w = u = 0, It =v=0, (A.,fi3 - A3112) u = (Aj^i- Ai^j) V ■■= (Ai^j - A.,/x,j) w. Hence .0'^' = (cj v - b, w) («! iv — Ci u) {b^ u — a^ v) ; and ^1''' = (CiV— b^w), 2''' = (a^w - c^u), cffi^ = (b^u - aiv), Il] THE RESOLVENT 23 from which we obtain Example iii. Compare and find the resolvents of the two modules The resolvent of ill/' will be found by obtaining the resultant with respect to x-^ of the two equations KXi + X^X-^Xi + ^sXiXi + XiXi + As {Xi' + x} + X-yX^X^ = 0, and MiiJ^i^ + ii--i,x-i,x.f + /^.i^ia' + /^s (^/ + ^J^s^ + XiX^x^j = 0. This resultant is the same as that of the first equation and ('■^iMs) ^i' + (^2/^5) x^Xi + (Xgftj) a^i A'2^ + (X4/X5) xi = except for a factor Xj'. The roots of the last equation are t^iX^, 0.2X2, a-^Xn. Hence the resultant, apart from a constant factor, is n {(Xja^ + X^a? + \^a + A4) x^ + Xj (a^ + 1 + ax^) xi\, {a = a^, a^, a^) or iC,* n { (Xja''* + Xjtt^ + Xgtt + X4) X2 + X5 (a^ + 1 + aXs)}. Hence the complete resolvent is x^^, since no values of x^, x^ inde- pendent of the X's and /a's will make the remaining product of factors of the above resultant vanish. The complete resolvent of M, worked in the same way, is also x^; i.e. M and M' have the same complete resolvent, although they are not the same module. M, but not M', contains the two modules M" = (a?3 — 1, Xy +X1X2 + x?, Xi'^Xi + x-ixi), i.e. every member of iJf is a member of M" and of M'". Thus kC/2, — ^\ \"^\ ' ^\ ^2 * *^2 J "~ V'^^l **'2 *^1 ^2 /) tt/g -^^ tt/nl itj ~r U/] U/2 "^ tt'2 J ^ V***! ^2 3C\SC^ J J it J "T tt/2 "'" tt'1^^2^3 ~~ \^1 """ '^1^2 ' ^2 / ^1 **- 2 I ^3 J* The module 31 is what is called the l.c.m. of M', M", M"'. The two modules M ", M'" have Xi = X2=X3—1 =0 and Xi=X2 = X3+l =X) for their spreads, which are imbedded in the spread a;, = a^j = of the first component of ilf, viz. M'. M'\s then properly speaking a mixed module although this is not indicated by its complete resolvent xi. It has two imbedded spreads, the points (0, 0, + 1). The complete resolvent should have the factors 24 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ll a^3 ± 1 to indicate these, but it has no such factors. The complete resolvent may indicate imbedded modules which do not exist as in Ex. ii, or it may give no indication of them when they do exist as in Ex. iii. Example iv. It is stated in the Encyk. der Math. Wiss. (Wi, p. 305) and repeated in (Wj, p. 354) that if only one complete partial resolvent i)'''* differs from 1, and Z"''' has no repeated factor, the module is the product of the prime modules corresponding to the irreducible factors of X*''''. The absurdity of this statement is shown by applying it to the module (m, vw), where u, v, iv are the same as in Ex. ii. The complete resolvent is .D'^' = (biu - aiv) (cjM — Oiw), and the product of the prime modules (u, v), (u, w) corresponding to its two factors is (tr, uv, uiv, viv) 4= (u, vw). 18. The i/-resolvent. The solutions of i^i = i^, -... = i^^ = are obtained in the most useful way by introducing a general unknown ;» standing for UiXi + 112X2 + ... + z«„a;,j, where Ui, u^, ■■■, u„ are undeter- mined coefficients. This is done by putting X Wg X2 ~ • . • ^nXii in the system of equations Fi = E2= ■■■ = Fk = 0. We thus get a new system /i=/2= ... =/t = in x, x„, x^, ..., Xn, where f, = u,'^F, ( ^~^^^°-;^-"~' -'-'^",^-., .... ^„) (i= 1, 2, ..., /!;), the multiplier u^h being introduced to make /,• integTal in Mj. There is evidently a one-one correspondence between the solutions of the two systems, viz. to the solution fi, ^.,, ..., |„ of i^i =F.= ... =Fk = there corresponds the solution i, £., ■■■, tu ot fi=/2= ■■■ =ft = 0, and vice versa, where f = «,i*''"'' = to which corresponds a solution ^, 4i ■•■, ir, a^r+i, ■•-, i>^n of fi=fi=-- =fi: = (g 14) and a solution ^i, ^2, ..., $,., «•,.+,, .. , Xn of i''i = i^2= ... = i^j; = 0, where ^ = «tili + ... + ««,.4 + ««,-+i a^^+i + ...+«„«„. The linear factors of F,, expressed in the above form supply all the solutions of /i=/2= ... =A = 0, viz. f, 4, •••, i,-, «•,.+!, •••, ;»«,. and all the solutions of Fi = Fi= ■.■=Fk == 0, viz. f 1 , 4 , • ■ • , ^r , ^r+i , ■■■, a;„ , of the several ranks r = 1, 2, ..., « ; but it is only when li, 4, •••, ir are independent of Wi, 2 {x,-, a;,.+i, . . . , a;„) = ^ n (a*, - Xri) - for x^, whose roots are the d values of x,- corresponding to given arbitrary values of «r+i, •■•, Xn- The coefficient of u^u^~^ gives an equation where <^' is r— and (^j , or <^ S ^- — , is a polynomial in ^,., ^,.+1 , ..., Xn. * Also these coefficients are members ot IF-i, F^, ... , FA if IB,,) is a member of (Fi,F^, ...,F^), as it will be proved to be when (Fj, J^ji ••■ ^^k} is a prime module (§ 31). 28 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ll Similarly we have Xi<^' — -2 = 0, ■■■, x,.-i4>'— ^,._i = 0. The equations are called more particularly the equations of the spread, the first giving the different values of ir,. as functions of Xr+i, ..., x^, and the others giving a-'i, x.^, ..., x,—i as rational functions of x,., Xr+i, ■■■, «•„. If Xr, Xr+\, ■■■, Xn have such values that ^ = ^' = then <^i, <^2, ..., <^r-i all vanish and the expressions above for x^, x«, ..., x,.-! become inde- terminate. In such a case the values of Xi, x.2, ..., x,—i may be found by taking other equations from {Rv)x=u,x,+...+u,xXn f^r them. 22. Geometrical property of an irreducible spread. An algebraic spread in general is one which is determined by any finite system of algebraic equations, and consists of all points whose coordinates satisfy the equations and no other points. Such a spread has already been shown to consist of a finite number of irreducible spreads each of which is determined by a finite system of equations. The characteristic property of an irreducible spread is that any alge- braic spread which contains a part of it, of the same dimensions as the irreducible spread, contains the whole of it. Let i^i = i^2 = • • ■ = -ft = be the equations determining any algebraic spread, and F-l = Fi = ... = F'^' = the equations determining an irre- ducible spread. The spread they have in common is determined by the combined system of equations Fi= F2= ... = Ft = Fi = ... =F\' = 0, and is contained in the irreducible spread and has the same or less dimensions. If it is of the same dimensions as the irreducible spread the complete w-resolvent of Fi= ... = Fk = F^' = ... = F\' = will have an irreducible factor RJ' of the same rank as the irreducible factor R^' of the complete 2<-resolvent of Fi =F2= ■■.=F\' = Q corresponding to the spread of the same. Also all the roots of R^' = regarded as an equation for x are roots of ^,/ = 0. Hence Ru is divisible by R^", and since they are both irreducible they must be identical. Hence the spread of Fi= ... =Fk = F^= ...=F\' = contains the whole of the spread of Fi = F.^= ...= F'yj = 0, and the spread of i^j = i^, = • ■ ■ = F;; = contains the same. This proves the property stated above. Ill] GENERAL PROPERTIES OF MODULES 29 III. GENERAL PROPERTIES OF MODULES 23. Several arithmetical terms are- used in connection with modules suggesting an analogy between the properties of polynomials and the properties of natural numbers. Two modules have a g.c.m., an L.C.M., a product, and a residual (integral quotient) ; but no sum or difference. Also a prime module answers to a prime number and a primary module to a power of a prime number. Such terms must not be used for making deductions by analogy. Definitions. Any member F of a module M is said to contain M. Also the module (F) contains M. It is immaterial in this statement as in many others whether we regard i^ as a polynomial or a module. The term contains is used as an extension and generalisation of the phrase is divisible by. More generally a module M is said to contain another M' if every member of M contains M' ; and this will be the case if every member of the ba-sis of M contains M'. Thus {Fi, F^, ■•■, F^) contains {Fi, F2, ■■-, Fk+i), and a module becomes less by adding new members to it. If M contains M' and M' contains M we say that M, M' are the same module, or 31= M'. If itf contains M' the spread of M contains the spread of 31', but the converse is not true in general. If in a given finite or infinite set of modules there is one which is contained in every other one, that one is called the least module of the set ; or if there is one which contains every other one, that one is called the greatest module of the set. Two modules cannot be com- pared as to greater or less unless one contains the pther. There is a module which is contained in all modules, the unit module (1). Also (0) may be conceived of as a module which contains all modules ; but it seldom comes into consideration and will not be mentioned again. These two modules are called non-proper modules, and all others are pro^jer modules. In general by a module a proper module is to be understood. The G.C.M. of k given modules 31^, 31., ■■■, 31^^ is the greatest of all modules 31 contained in 31, and ilfj... and 3It, and is denoted by (3fi, 3Ii, ..., 3It^. In order that 31 may be contained in each of 3Ii, 3^2, ••■, 3Ih, or that each of 3Ii, M^, ..., ilf^ may contain 31, it is 30 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill necessary and sufficient that all the members of the bases of M^, M«, ...,Mk should coutaia M ; hence the module whose basis consists of all these members contains all the modules M, and is at the same time one of the modules M. It is therefore the greatest of all the modules M and the g.c.m. of Mi, M^, ..., 31^. The notation {M^^M^, ..., 3ft) agrees with the notation {Fi,F«, ..., i'''*), since the latter is the G.c.m. of Fi, F2, ■■■, Fic regarded as modules. The L.c.M. of il/i, M2, ..., 3Jk is the least of all modules iKf contain- ing Ml and M^ ... and 31^, and is denoted by [3Ii, M-,, ■■■, Mt]. Its members consist of all polynomials which contain 31^ and M^ ■■■ and 3It; for the basis of any module 31 containing Mi and M^ •■• and 31^ must consist of a certain number of such polynomials, and the whole aggregate of such polynomials constitutes a module 3/ which is the least of all the modules 31. The product of 3Ii, 31^, ..., 3'Ik is the module whose basis consists of all products FiF^--- F^, where Fi is any member of the basis of 3i.i{i = l,2,...,k). The product is denoted hy Mi3f„... M^, and is evidently a definite module independent of what bases may be chosen for Ml, 3I2, ...,3I^. The product 3IiM,...3I,. contains the L.c.M. [3Ii,M,,...,Mt]. The product of y modules each of which is the same module M is denoted by M'' and is called a power of 31. If P is the point (ffli, a,, ..., a„) the module {xi-ai, x^-a^, ■■■,a:,,-a,i) is denoted by P. If is the origin the module is {xi,,a'.,, ...,.r„), and O^ is a module having for basis all power products of .Tj, x.., ...,.t„ of degree y. A polynomial F, or module 31, which contains P'^ is said to have a y-point at P The residual (L, p. 49) of a given module 31' with respect to another 31 is the least module whose product with 31' contains it/ and is denoted by 3Ij3r. Its members consist of every polynomial whose product with each member separately of the basis of 31' is a member of M; for the basis of any module whose product with 31' contains 31 must consist of a certain number of such polynomials, and the whole aggregate of such polynomials constitutes the least such module. In the case of the natural numbers the residual of ?n' with respect to m is the least number whose product with m contains m, and is the quotient of m by the g.c.m. of m and m. It is the same to some extent with modules, viz. 31/31' = M/(M, M'); for if 31/31 = 3r' then 31" is the least module such that 31' 31' contains 31, and is therefore the least module such that {31, M') M" contains 31, i.e. 31" = 31/(31, 31'). Ill] GENERAL PROPERTIES OF MODULES 31 Nevertheless MI{M, M') is not called the quotient of M by {M, M') because it is not true in general that the product of {M,M') and MI{M, M') is M. If M, M', M" are three modules such that M'M" contains M it is clear that M' contains MjM" and M" contains 31/ M'. Since M3I' contains M, 31 contains 31/31'. The module 31/31' is a module con- tained in 31 having a special relation to 31 independently of what 31' may be (§ 26 (i)). There is a least module which can be substituted for M.' without changing 31/31', viz. 3I/{3I/M'), § 26 (ii). This module is contained in {31, 31'), for {31, 31') can be substituted for 31' without changing 31/31', but is in general different from {31, 31'). 24. Comment on the definitions. The non-proper unit module (1) has no spread. Conversely a module which has no spread is the module (1), since the complete resolvent is .1 and is a member of the module. The unit module is of importance from the fact that it often comes at the end of a series of modules derived by some process from a given module. {3Ii, 31-2, ■■■,3Ik) and [311,31^, ..., 31^] obey the associative law [J/j, 3I2, 3I3] = [[ilfi, 31^], 313] = [3fi, [3'1„ 3Is]], and the commutative l&w {3Iu 3f^) = {3I2, 31,). Also (it/i.il/^, ■■■,3Ik) obeys the distributive law M {3£„ 31^)^{3I3I„ 3131,) ; but [3I„ 3L_, ..., MJ does not. Example. As an example of the last statement we have (*'] , X2) [(^1 , ^2 ), (Xi X2)} — {Xi, X2) (Xi X-i , Xi X.2 ) = [Xi X2) {Xi , X.{) , while [(«i , x^ {x^-, xi), («, , x^ {x-, x^'] = {x, x^) {x, , x^). Given the bases of 3Ii,3f«,...,3fi,,3I,3I' we know at once a basis for {M-,, 31^, . . . , 31,^ and for 31, 31^ . . . M,, ; but it may be extremely difficult to find a basis for [3Ii,3l2, ■■■ , 3ft] or for 31/31'. Hilbert (H, pp. 492-4, 517) has given a process for finding a basis of [itfi, ifa, •■•j-ATs]; and the same process can be applied for finding a basis for 31/31'. This process is chiefly of theoretical value in so far as it has any value. We can have (i) M3I' = 3I3I", or 3I/3f' = 31/31", without 3I'=3I"; (ii) 31/31'= 3£" without 3I/3I"= 31' ; (iii) 31/ 31' =31" and 31/31" = 31' without 31 =31' 31"; and (iv) 31 ^31' 31" without 31/31' = 31" or 31/31" = 31'. Examples, (i) {x^,x^{x-„x.^-={x-,,x.^{x^-,.v.i), {x-,, X2) /{Xi, X2) = {Xi, X2)' /{Xi , X2) ; 32 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill (ii) (x,, ^2)7(«l^ «2') = (x„ x,), while (.rj, x^fKx,, x,) = («,, x.f ; (iii) (x^\ xf)l{x, , x^) = {xi , x^y and (x,\ x^^)l{x, , x^f = {x^ , x,), while (xi', xf) =t= {x-, , x^ {x^ , x^y ; (i v) (Xi , x.y = {xi\ Xi' X2 , xi) {Xi\ Xi xi, xi), ■while {xi,xifl{xi,x^Xi,xi) and {xi,xyi{xi,XiXi,xi) are both equal to {x^, xif. 25. The product of the g.C.m. and L. c. m. of two modules contains the product of the modules. Let 3I={F„F„...,Fk) and M' = {F,',Fi, ...,F\.) be the two modules and let Fz, be any member of the basis of their l.c.m. Then, since i^x = mod M, FIFl - mod MM' ; and since i^z, = mod M', FiFz = 0T^od3I3f' ; i.e. the product of any member of the basis of (M, M') with any member of the basis of [il/, M'] contains MM', or \m, M') [M, M'] contains MM'. When M,M' have no point in common {M,M') = {\) and con- sequently [M,M'} contains MM', i.e. [M,M'] = MM'. This case is proved by Kouig (K, p. 356) ; although it is to be noticed that {M, M') cannot be (1) in the case of modules of homogeneous poly- nomials. Thus the l.c.m. of any finite number' of simple modules (§ 33) is the same as their product (Mo). 26. The modules MjM' and Mj{MIM') are mutually residual with respect to M, i.e. each is the residual of the other ivitk respect to M. Let MjM' = M" and MI{M/M') = M"'; then we have 31'"= Ml 31", and we have to prove that 31" = 31131'". Let 31131'" = 31". Now M'3I" contains 31; therefore 31' contains 311 M" or 31'". Also 3T'M"' contains 31 ; therefore M" contains 3'IjM'" or 3T\ Again, since 3T contains 31'" (proved) and 3I"'3I" contains 31, 3T3I'" contains M, i.e. M" contains MI3I' or 31". But M" contains M" (proved). Hence 31" = M" = Ml3r'. Two results follow from this : (i) MIM' is a module contained in 31 of a particular type ; for 3I/3T and its residual with respect to 3f are mutually residual with respect to M, and this is not true in general of any module contained in Tlf and the residual module (Ex. ii, § 24). (ii) The least module ivhich can be substituted for 31' without changing M/3r is 3I/(3I/3f). Let 31'" be any module such that MI3P'' = 31/31' ; then the product of 31'" and 31131' contains 31, and 31" contains 311(31/31'). Aho' 31/(31/31') is one of the modules 31'" ; Ill] GENERAL PROPERTIES OF MODULES 33 for if M/(M/M') = M'" then MjM"' = MIM',.hy the theorem. Hence M/{M/M') is the least of the modules 31'" which can be substituted for M' without changing MjM'. 27. If M', M" are mutually residual with respect to any module they are mutually residual with respect to M'M". Suppose M', M" are mutually residual with respect to M. Then Jf'Jf" contains M; and if M'M"jM' = M"', M'M'" contains M'M" which contains M ; hence M'" contains MjM' or M". Also M" contains M'M"/M' or M'". Hence M" = M'" --= M'M"IM'. Similarly M' = M'M"IM" (cf. statement iv, § 24). Any module M with respect to which M', M" are mutually residual contains \_M', M"] and is contained in M'M". 28. If M, Ml, M^, ..., M-k are any modules, then M{{M„M„ ...,M^)^{MIM„MIM,, ..., il//i»fj, and [Ml, M,, ...,M^]IM= [M^jM, M^jM, ..., M^/M]. For M/(Mi,3L, .-..Mk) contains M/Mi and therefore contains [3I/Mi, M13I^, ..., M/M,.]. Also M,[3f/3I, ..., 3I/M^] contains Mi X M/3Ii which contains M; hence (M-i, ■■■,3Q [31/ M^, ..., 31/ M^] contains 31, and [31/31, ■■■, M/Mt] contains 31/ (M^, ...,M^). This proves the first part. Again [31, ..., 31^/ M contains Mij3I and therefore contains [3IijM, ...,3IkjM]. Also M[3IijM, ...,3I„/3r\ contains Mi and therefore contains [M-y, ..., 3Ii^; hence [M-i/ M, ■ ■ ■ , M^/ M] contains [Ml, ...,Mk]/M. This proves the second part. 29. Prime and Primary Modules. Definitions. A prime module is defined by the property that no product of two modules contains it without one of them containing it. A primary module is defined by the property that no product of two modules contains it without one of them containing it or both containing its spread. Hence if one does not contain the spread the other contains the module. Primary modules will be understood to include prime modules. Lasker introduced and defined the term primary (L, p. 51), though not in the same words as given here. The conception of a primary module is a fundamental one in the theory of modular systems. Any irreducible spread determines a prime module, viz. the module whose members consist of all polynomials containing the spread. That this module is prime follows from the fact that no product of two M. 3 34 THE ALGKBRAIC THEORY OF MODULAR SYSTEMS [ill polynomials can contain the spread without one of them containing it (§ 22) and the module ; and the same is true if for polynomials we write modules. If M= {F-i, F^, ..., Fu) is the prime module of rank r determined by an irreducible spread of dimensions n-r, and if the origin be moved to a general point of the spread, the constant terms of Fi, F2, ■■■, Fk will vanish, and the linear terms will be equivalent to r independent linear polynomials, i.e. the sub-determinants of order > r of the matrix BF dF dF 3F dF IF will vanish, while those of order r will not vanish, at the origin. This will be equally true for any general point of the spread without moving the origin to it. Any point of the spread for which the sub-determinants of order r of this matrix vanish is called a singular point of the spread, and the aggregate of such points the singulai- spread contained in the given spread. The singular spread (if any exists) is therefore the spread determined by F^, F^, ■■■, F^ and the sub-determinants of order r of the above matrix. If M--{Fi, Fi, ..., Fk) is the l.c.m. of the prime modules deter- mined by any finite number of irreducible spreads of the same dimensions n ~ r, the same definition holds concerning singularities of the whole spread. In this case the singular spread consists of the intersections of all pairs of the irreducible spreads, together with all the singular spreads contained in the irreducible spreads considered individually. 30. The spread of any prime or primary module is irreducible. For if not the complete ««-resolvent has at least two factors corre- sponding to two different irreducible spreads of the module neither of which contains the other, and is the product of two polynomials neither of which contains the whole spread of the module, i.e. the module is neither prime nor primary. 31. There is only one prime module ivith a given {irreducible) sjrread, viz. the module ivhose members consist of all polynomials containing the spread. Ill] GENEBAL PROPERTIES OF MODULES 35 Let M={Fi,Fi, ..., i^i) be any prime module of rank r. It will be sufficient to prove tbat every polynomial which contains the spread of M contains the module M. The first complete partial M-resolvent of M other than 1 will be a power ii,™ of an irreducible polynomial i?„ in a;, iTr+i, •■-,«„. Also the complete «-resolvent is a member of (/i>/2; ■••j/*), §18, which is prime ; and every factor except ^,|"' is of too high rank to contain the spread of {fi,/^, ■■■,fk)- Hence Ru\ and therefore R^ itself, is a member of (fi,/^, ■■•,/*:)• Hence \Iiu)x=u,x,+...+itr,x,t is a member of 31, and also the whole coefficient of any power product of Mj, u-2, ..., Un in (^«)x=«,ai,+...+n„ic„- We have proved (§21) that (Ru)x=u,x,+...+UnXn= ■■■ + Ml-''"' (Mi Vl + ••• + «.-l fr-d + «/ 4>, where i/'i = a;i<^'— <^i, ...,il/r-i = a;r-iil(t>', Xq = -2/4>\ • ■ • I ^1-1 = 4>r-il4>' ') then F becomes a rational function of «,-, iTr+i, •., «n of which the denominator is '\ where / is the degree of F". This rational function vanishes for all points of the spread at which <^' does not vanish, and its numerator is therefore divisible by <^. "We have then U" " ■■■' ^' ' '■■' v ~ " ' where Xis a whole function oi x,-, av+i, •••, ^n! i.e. p(a; -"tl a:-"^ X -"^ x^ x\=^'^ or (^''i?'(a;i,a;2, ...,«„) = mod (i/'i, . . . , i/',_i , ^) = mod ili". Hence i^=Omodilf, which proves the theorem. It follows that a module which is the l.c. m. of a finite number of prime modules, whether of the same rank or not, is uniquely deter- mined by its spread, and any polynomial containing the spread contains the module. 32. If 31 is a primary inodule and 3Ii the prime module deter- mined by its spread some finite jyoiver of 31-^ contains 31. This theorem, in conjunction with Lasker's theorem (g 39), is equivalent to the Hilbert-Netto theorem (§ 46). The proofs of the theorem by Lasker and Konig are both wrong. Lasker first assumes the theorem (L, p. 51) and then proves it (L, p. 56) ; and Konig makes an absurdly false assumption concerning divisibility (K, p. 399). 3—2 36 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill By the same reasoning as in the last theorem it follows that Ru"^ (but not R-^ is a member of (A, A, ■■■,fk), and (Ru"')x=u, a;,+...+«na:„ = {•••+ «*/"' (««i l^i + . . . + M^-i '/'r-l) + m/ ^}"' = mod 31. Picking out the coefficients of u/"" and «/"'-*" Mi", we have i>'" = OmodM, and xp/" = Xcji mod M ; .: i/'i'"'=Omod Jf ; and similarly >/'2'""= •■• =^™'_-^= Omod J!f. Also if i^is any member of Ml, then, by the last theorem, ''F=0moi(4ri,...,>l/r-i,). Hence the product of any rm^ polynomials F and <^''"" is a member of (li,™', Vs , •■• ^w + ^iv ; and this can be done. 35. The L.c.M. of any number of primary modules ivith tlie same spread is a primary module with the same spread. Let Ml, M^, ■-., My: be primary modules with the same spread, and let M be their l.c.m. Then M has the same irreducible spread, since the product, which contains the l.c.m., has the same spread. Also if the product M'M" contains M, and M' does not contain the spread, then M" contains ili, and M«_... and il/j,., i.e. M" contains M. Hence il/is primary. The g.c.m. is not primary in general. 36. If M is primary and M' is any module not containing M then M/M' is primary and has the same spread as M. Let MjM' = M". Then since M'M" contains BI, and M' does not contain M, M" contains the spread of M. Also M contains M"; hence M" has the same spread as M. Also if M-^M^ contains M" then 38 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill M'MiM^ contains M'M" which contains M ; and if M^ does not contain the spread of M (that is of M") M'M^_ contains M, and M^ contains MjM' or M"; i.e. M" is primary. 37. Hubert's Theorem (H, p. 474). If F„ P.,, F^, ... is an infinite series of homogeneous polynomials there exists a finite number h such that Fi^ = mod (Fi, F^, . . ., Fj,) ivhm h> k. The following proof is substantially Konig's (K, p. 362). It must be clearly understood that Fi, F.,, F3, ... are given in a definite order. In the case of a single variable the series F-^, F^, F3, ... consists of powers of the variable, and if F), is the least power then F, = mod F,^ when h > k. Hence the theorem is true in this case. We shall assume it for n - 1 variables and prove it for ;; variables. The series F^,F^,F3, ... is called a modified form of the series F^, F,, F„ . . . if i^; =- F, and F,' = Ft mod {F, ,F„ ..., i'V,) for i>l. Thus the modules {Fi, F^, ■■■, F,) and (i'V, F^, ■■■, F-) are the same. The theorem will be proved if we show that the series Fi, Fi, ... can be so chosen that all its terms afcer a certain finite number become zero. We assume that Fi is regular in a?„, and we choose the modified series so that each of its terms FI after the first is of as low degree as possible in x^, and therefore of lower degree in Xn than i^/. The terms of the series F^, F^', ... of degree zero in Xu will be polynomials in .ri, x-^, .. , .t^.i-i and these can be modified so that all after a certain finite number become zero, since the theorem is assumed true for n — 1 variables. Let F'l^, F',,^, F'l^, ... be all the terms of i^/, F^', F3, ..., taken in order, which are of one and the same degree Z > in Xn ; and let f'l^tf'i.^, ■■■ be the whole coefficients of xj in them. Then f'liif'i.itf'hJ--- 3.re polynomials in w— 1 variables; and we cannot have f'l^ = Omod(f',^,f\,...,f',._^) for any value of i ; for if f\ = AJ-'i^ + AJ\,_+...+A,.J-\._^,thenF'i-A,F',^-...-Ai^,F\._^ is of less degree than I in x,,, which cannot be. Hence the number of the polynomials f'i^,f'i^, ■••, or the number of terms F'l., F'l^, ... in the series F^', F!, . . . , is finite. And the number of values of I is also finite, the greatest value of I being the value it has in F^'. Hence the theorem is proved. The theorem can be extended at once to an infinite series F^,F.i, ... of non-homogeneous polynomials since they can all be made homo- geneous by introducing a variable x^ of homogeneity. The following is an immediate consequence of the theorem : Ill] GENERAL PROPERTIES OF MODULES 39 Any module of polynomials Ims a basis consisting of a finite number of members. To prove this it is only necessary to show that a complete linearly independent set of members of any module can be arranged in a definite order in an infinite series. If / is the lowest degree of any member we can first take any complete linearly independent set of members of degree I, then any complete set of members of degree Z + 1 whose terms of degree l+\ are linearly independent, then a similar set of members of degree Z + 2, and so on. In this way a complete linearly independent set of members is obtained in a definite order. It does not matter in what order the members of a set are taken, nor is it necessary to know how to find the members of a set. It is sufficient to know that there is a definite finite number of members belonging to each set. 38. The //-module equivalent to a given module. Consider a complete linearly independent set of members of a given module M, not an £f-module, arranged in a series in the order described above ; and make all the members homogeneous by introducing a new variable x^. We then have a series of homogeneous polynomials belonging to an ^-module M^, whose basis consists of a finite number of members of the series. The module if„ is called the H-module equivalent to M, and a basis of M obtained from any basis of M„ by putting iTo = 1 is called an H-basis of M. The distinctive property of an .ff-basis (i^i, Fj, ..., F,) of M is that any member F of ilf can be put in the form A^F-,* A^F^^ ... + AJF^ where AiFi{i=l,2, ..., k) is not of greater degree than F. Every module has an H-basis, which may necessarily consist of more members than would suffice for a basis in general. The following relations exist between 31 and its equivalent H- module Mo ■ (i) to any member F oi M corresponds a member Fo of Af„ of the same degree as F, and an infinity of members x^^'Fo of higher degree; (ii) to any member Fo of J/o corresponds one and only one member oi M, viz. (i^o)a^o=i; ("i) tli^re is a one-one correspondence between the members of Mo of degree I and the members of M of degree $/. If XoFo = mod Mo, then (i^„):r.=i = mod M, and Fo = mod i)/„ by (i), i.e. there is no member XoFo oiMo such thati^o is not a member of Mo, and MoKxo) = Mo. Conversely an H-module M in n variables Xi, Xi, ■■■, Xn is equivalent to the module ilfr„=i if M/(xn) = M, and not othenoise. 40 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill In any basis (Fi, F^, ..., F^) of an ^'-module in which no member is irrelevant, i.e. no i^i = mod (Fi, ■.., Fi--^ , F^i, ..., Fi), the number of members of each degree is fixed ; as can be easily seen by arranging Fi, jPj, ... , i^s in order of degree. Hence in any H-basis of a module in which no member is irrelevant the number of members of each degree is fixed. On account of this and the other properties of an .ff-basis mentioned above an .ff-basis gives a simpler and clearer representation of a module than a basis which is not an .ff-basis. Example. Find an ^-basis of the module (a-/, x^ + x^x,). Take the iT-module (a;/, x^x^ + x^x^ and solve the equation XaXo = mod {x^, X2X0 + XiXs), or Xo 2Co = Xi" JCi + (Xi Xo + x-^ Xs) JT^ . Putting a-Q = we have (x^-Xi + XiXsX)x,=o = 0, i.e. JLi = XsJr, X^= — x-yX, whena;o = 0, i.e. Xx = XiX-^ x^Yy, X^= — XiX+XoY.i. Hence Xt,X„ = x{- {XiX + X(, Vi) + {x^Xii + x^x^ ( - XxX + Xf, Y^ = x„ {xl Fi - x-,x^_X-\- x-iX^ + x-x x-i Fo), i.e. Xo = mod («/, x-^x^, x^Xo + XiX,). Again, if we solve the equation Xo Fo = mod (xi^, XiX.,, x^Xo + x-^x-^, we find Fo= mod {x^, x-^^x^, x.?, x^Xo + x^x^ ; and if we solve XoZo = mod (xi^, X1X2, xi, XiXo + x^Xs), we find Z(, = mod {x^^, x-i^x.^,, xi, x^x^ + x-^x^. Hence {Xi, x-^x^i, x^-, x«x„ + x-^x^ is the ZT-module equivalent to {x^, X2 + X1X3), and {x-c, x-^x^, x^, Xi + x-xX:^) is an 5-basis of \Xi , VTg + X1X3). The extra members a^iA-j, X2' might of course have been found more quickly by multiplying X2 + x^x^ first by x^ and then by ^o. The method given is a general one. 39. Lasker's Theorem (L, p. 51). Any given module M is the L.c.M. of a finite number of primary modules. Let M be of rank r. Express its first complete partial z«-resolvent //„'''"^' in irreducible factors, viz. Ill] GENERAL PROPERTIES OF MODULES 41 and let Cj, C3, ..., Cj denote the irreducible spreads, of dimensions n-r, corresponding to Ri, R^, .., Rj respectively. Consider the whole aggregate Mi of polynomials F for each of which there exists a polynomial F', not containing Cj, such that FF' = mod M. We shall prove first that Mi is a primary module whose spread is (7i (« = 1, 2, . . . , j). Let Fi , Fi be any two members of Mi . Then since F^ Fi = mod M, andi?2-f2' = mod M, where neither Fi nor Fi contains C;, we have {A-^F-y + A2F^)F-^Fi=0modLM, where F^F^ does not contain 6',. Hence A-^F^ + AiF-i belongs to the aggregate Mi, i.e. Mi is a module. Again, since i^i^' = modi)/, i^ contains d, and Mi contains Cj. Now, if Fy, is the complete ?«-resolvent of M, (^''«)a:=M,.r, + ...+«,..T„ = mod M, while (^i"'0.r=M,.r.+..,+,i„a;„ IS the only factor of {Fu)x=iHx,+...+'u^^xn which contains Ci. Hence {Ri^"i)x=u,x,+ ...+wnin= ^ mod ilfj. But the polynomial {R^.i-=u,x,^-...+unXn does not vanish identically (i.e. irre- spective of M,, «2, •■■>««) for any point outside Ci (§21); hence Mi contains no point outside d, i.e. Ci is the spread of M^. Lastly Mi is primary ; for if F"F"' = mod Mi, then F'F"F"' = {)moAM, where i?" does not contain Ci ; hence, if F" does not contain d, F' F" does not, and F'" = mod Mi. Hence also if M" 3f"' contains 3Ii, and ibT" does not contain d, M'" contains Mf. Thus Mt is a primary module whose spread is Cj. Also M contains Mi, for every member of Mis a member of Mi. The module MjMi does not contain d ; for if Mi = {F^,F^,..., F^) and jP/, i^s', ..., Fk are polynomials not containing C; such that FFi' = Q mod M (I = l,2,...,k), then i'",i^i'i^;...i^k'=OmodiV/ (/= 1, 2,...,^-)- Hence FiFs'-.-F;! is a member of ilf/Jfj not containing Ci; and therefore ilf/itfj cannot contain d. Since il//ilfi does not contain d, {MjMi, MjM.2, ..., MjMj) does not contain any of the spreads Cj, C^, ■■•, Cj. We can now prove that if is any single member of {MjM^, MjM^, ..., MjMj) which does not contain any of the spreads Ci, C^, ..., Cj, then M=[M„3I,,...,Mj,(M,cl>)]. Since ilf contains [M^, M^, ..., it/,-, (i¥", <^)] it has only to be proved that the latter contains M, or that F= mod [ifi, M., ..., Mj, (M, <^)] requires F=0 mod M. 42 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [lH We have F=0 mod (AT, <^) =/^ mod M=/(t> mod ilf; ; but F= mod Mi; therefore /<^ = Omodil/j, and, since ^ does not contain C;, /= mod 31, = mod [3I„ 31., ..., 3Ij]. Rence/ = mod [3f„ 31,,..., Jij] (31/ M„ ..., 3I/3fj) = Omodi»/(§ 28), and F=fcl> mod 31 = mod 3J. Hence 31= [Jf„ 31,, ..., 31j, (31, <^)]. Now the spread of (3J, , ', -..) with no spread, which is the module (1). Hence -'''^=[^1, Q-2, •■■ , Qk] where Qi, Q2, •■■, Qk are all primary modules of ranks %r. 40. Comment on Lasker's Theorem. The above is in all essentials the remarkable proof given by Lasker of this fundamental theorem. He considers ZT-modules only and makes use of homo- geneous coordinates, in consequence of which his enunciation of the theorem is not quite as simple as the one above. Any module among Qi, Qo, •••, Q* which is contained in the l.c.m. of all the rest is irrelevant and maybe omitted. It will be understood in writing 3I={Qx, Qo., ■■■, Q^] that all irrelevant modules have been omitted. Those that remain will be called the relevant primary modules into which 31 resolves, and their spreads will be called the relevant spreads of 31. A relevant spread which is not contained in another of higher dimensions is called an isolated spread and the corresponding module an isolated primary module of 31. The other relevant spreads and modules are called imbedded spreads and modules of 31. All the relevant spreads of 31 whether isolated or imbedded are unique. Also the isolated primary modules are unique, but the imbedded primary modules are to some extent indeterminate. A process by which Q^, Qo, ..., Qt can be theoretically obtained, without bringing in any irrelevant modules, is described in (M). The isolated spreads are found from the irreducible factors of the complete !t-resolvent after rejecting all factors which give imbedded spreads. To these correspond unique primary modules of 31 which can be found. Let ili'°' be their l.c.m. The isolated spreads of 31/ 31^"'* are the relevant spreads of 31 imbedded to the first degree. To these correspond indeterminate imbedded primary modules of 31 which are chosen as simply as possible. Although not uniquely determinate the l.c.m. of each one and i(/'"* is unique, and the l.c.m. of them all and iV"' is Ill] GENERAL PROPERTIES OF MODULES 43 a unique module il/'''. The isolated spreads of i(//il/''' are the relevant spreads of M imbedded to the second degree ; and the l.c.m. of the corresponding (indeterminate) primary modules and J/'"' is a unique module M'-''\ The process is continued until a module il/''' is obtained such that 31/30^ = (1), when there will be no more relevant primary- modules to find. 41. An unmixed module is usually understood to be one whose isolated irreducible spreads are all of the same dimensions ; but it is clear from the above that this cannot be regarded as a satisfactory view. It should be defined as follows : Defiiiition. An unmixed module is one whose relevant spreads, both isolated and imbedded, are all of the same dimensions ; and a mixed module is one having at least two relevant spreads of different dimensions. An unmixed module cannot have any relevant imbedded spreads. A primary module is au unmixed module whose spread is irreducible. This cannot be taken as a definition because the meaning of unmixed depends on thfe meaning of primary. Condition that a module may be unmixed. In order that a module M of rank r may be unmixed it is necessary and sufficient that it should have no relevant spread of rank > r. This condition may be expressed by saying that 4>F=0 mod J/ requires F= QmoAM where ^ is any polynomial involving «,.+i, ..., x^ only. For if ilf contains a relevant primary module of rank >r a, can be chosen which contains it, and an F which does not contain it but contains all the other relevant primary modules of M, so that <^i'^=0m6dM does not require F-OraoAM; while if M contains no relevant primary module of rank > r there is no <^ containing a relevant spread of M and (f>F=0 mod M requires F=0 mod Ml(4>) = mod M (§ 42). A primary module Q has a certain multiplicity/ (§ 68). To a given primary module Q'**' of multiplicity /x corresponds a series of primary modules Q'"', Q^'\ ■■■, QW of mnltiplicities 1, 2, ..., /j. all having the same spread as Q'**' and such that Q'^' contains Q<^~'' and is contained in Q<^+''. Q'^' is the prime module determined by the spread of Q''^' and is unique; but the intermediate modules Q'^', Q''*', ..., Qi^~'' are to a great extent indeterminate (M, p. 89). Thus Q^'\ Q'^', ..., QM may be regarded as successive stages in constructing Q''^'. Two primary modules ivith the same spread and the sums multiplicity such that one contains the other must he the same module. 44 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill 42. Deductions from Lasker's Theorem. A module of rank n resolves into simple {primary) modules of ivhieh it is the pro- duct (§ 25). If M' does not contain any relevant spread of M then MjM' = M. Let MjM' = 31". Then since 31' 31" contains M, and 31' does not contain any relevant spread of 3'I, 31" contains all the relevant primary modules into which Jf resolves, i.e. 3d" = 31. It follows that \i 31131' + M, i^/' must contain a relevant spread of 31. Thus if a polynomial F exists such that {xi—ai)F, (xo—a,^F, . . .,(«„-a„)i^ are all members of 31, while F is not, 31 contains a relevant simple module ivhose spread is the point P{ai, a^, ..., a„); for 3IJP + M. Example. The module 31 = {xi, xi, Xi + x^ Xi + XiX^x^) has a relevant simple module at the origin; for XiX^-x.f is a member of 31{i=\, 2, 3, 4), but x^-x? is not. The simplest corresponding im- bedded primary module, not contained in the l.c.m. of all the other relevant primary modules of M, is ixi, xi, x-^, x^ ; cf. Ex. iii, § 17. This example shows that it is piossihle for a mixed module 31 to contain a relevant primary module of higher rank than the number of members in a basis of 31. For the rank of {xi, xi, x^, Xt) is 4. If 31 is an fi'-module not having a relevant simple module at the origin the variables can be subjected to such a linear homogeneous substitution that Xn will not contain any relevant spread of 31, and we shall then have 31/{xny= 3f, and il/ will be equivalent to 3Ix„=i (^ 38). Thics the only condition {remaining permanent under a linear substitu- tion) that an H-module 31 may be equivalent to the module 31^,^=1 is that 31 should not contain a relevant simple module. A simple /T-module 31 is not equivalent to il/a;„=i ; in fact i(fj-„=i is in this case the module (1). Tf 31' contains any relevant spread of 31 then 31/31' =t= 31. Let 3/=[$i, Q2, ■■■,Qk\, and let 3f' contain the spread of Qi. Then some power 31''^ of 31' contains Qt (§ 32), and Qj/M"^ = (1). Hence the spread of Qi is not a relevant spread of 31/31^ = [Q,l3ry, Q,IM\ ... , q,l31"'], § 28 ; and consequently Mjld''^ =t= M. Hence also 31 jM' =t= 31; for if 31131' = 31 then 311 3r = 31. It follows that if 31/31' = 31 then 31' does not contain any relevant spread of 31. If 31,, is the ZT-module equivalent to 31 we know that 31ol{xo) = 3Io (§ 38) ; hence x^ does not contain any relevant spread of 31o, i-e. no module has a relevant spread at infinity. Ill] GENERAL PROPERTIES OF MODULES 45 If M, M' are any two modules such that M resolves into isolated primary modules only, viz. Q^, Q^, ■■., Qk, and (M, M') into primary modules Q/, Q,l, ..., Q{, of which Qi, Q^', ■■■, Qk Jiave the same spreads OS Qi, Qi, ■■■ , Qk respectively, then MlM' = [QJQ/,Q,IQ,\...,QklQk']. The spread of (M, M') is contained in the spread of M; and it is to be understood that ii{M, M') does not contain the spread of Qu then Qi={\). The spreads of Q'k+i, ■■■,Qi are contained in those of ft) ft, •■•, ft, but do not contain any of the latter. Now we have MIM' = MI^M, M') = [ft , ft , . . . , ft]/[ft', ft', .... ft']. Hence the theorem follows, by the second part of § 28, provided ft/[ft', ft2',...,ft'] = ft/ft'• This is true ; for Q-JQl contains ft/[ft', ft', • • ., ft'] , since [ft', Q.i, ..., Qi] contains ft', and, for a similar reason, ft/[ft', ft', ..., ft'] contains QilQ.'Q^'-Qi or QilQ-. 43. If a module M of rank r is regarded as a module M''^^ in s variables Xi, x^_, ...,Xs, while a^j+i, ..■,Xn are regarded as parameters ; and if i^'"' is a whole member of M^^\ that is, a wJwle function of the parameters as well as of the variables, then -F'"', regarded as a poly- nomial in Xi, i»2, ■■■, Xn, contains all the relevant primary modules of M of rank ■$ s ; and. conversely, any jwlynomial which contains all these primary modules is a member of M^^\ The most important case is that in which s = r. In other words, to treat a module ilf as a module in s variables has the sole effect of eliminating all the primary modules of M of rank > s ; and when s < r it reduces M to the module (1). Let M={F„F„ ...,Fk); then F^"^ = A,F, + A,F,+ ... + A^Fk, where ^i, A^, ..., Ak are whole functions of Xi, x^, ..., Xs and rational functions of x^+i, ..., x^, with a common denominator Z"'>. Hence jQWjJ'M = mod Al, and F^"^ contains all the primary modules of 31 of rank ■$ s, since i)'*' does not contain any of their spreads. Conversely, if i<^'*' contains all the primary modules oi Moi rank ^ s, and jD'"', a whole function oi Xs+i, ••■, x^ only, contains all the primary modules of ilf of rank > s, then Z>('')i'» = mod if, and F^'^ = mod M^'\ since 2)''' in respect to 3/'"' does not involve the variables. The module il/'''' resolves into simple modules, any primary module of M of rank r and order d contributing d simple modules to Jf' ''•. By finding these simple modules we are able to find the primary 46 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill modules of M of rank r ; and this completely resolves M if M is unmixed. 44. If M is a module of rank r < n and no-one of the modules M, {M, a'u - «,0, (^, «n-i - «,!-! , ^,<. - «)0, • • • (J^I, «,-+2 - «.+- , . . ., a;.„ - Un) con- tains a relevant simple module (a,-+2j ■■■, O'n having non-special values) then M is unmixed. In the contrary case M is mixed. This theorem will be used later for proving that certain modules are unmixed. We shall prove first that if M is mixed and does not contain a relevant simple module then {M, Xn — a,i) is mixed. Let M' be the prime module determined by a relevant spread of M of rank > ;• and < n, since M is mixed and has no relevant spread of rank n. To prove that {M, ar„ — a,i) is mixed it is sufficient to show that {M', x„ - a,i) contains a relevant spread of (31, .r,, — «„). Suppose this is not the case ; then (§ 42) {M, x„ - a„)l{M', x„ - an) = {M, x„ - a„), i.e. (M, x„ - a,)jM' = (M, x„ - an), and therefore MjM' contains {M, x,, - an). Let F be any member of 31! M' and (F^, I\, ■■■, I\) a basis o{3I; then F= AiFj + ... + Ai-Fk mod (x„ - «„), i.e. i'l„=»„ = {A,F,-i- ...-i- A^Fi)xn=an- Here we may regard a,i as a parameter replacing Xn. Hence i'^is a member of M regarded as a module in n — 1 variables, and therefore contains all the primary modules of 31 of rank $«— 1 (§43); i.e. F= mod 31. Hence 31/31' contains 31, which is not true. It follows that (31, Xn — Un) is mixed in general, i.e. if a,, has a non-special value. By the same reasoning, if (31, a",j — a„) does not contain a relevant simple module, (31, x^-i - a^-i , x,^ - a,,) is mixed, and so on. Finally if ( 31, ar,.+2 - a,.+2 , ■■■ , Xn — an) is mixed it must contain a relevant simple module since it is of rank n — 1. Hence if 31 is mixed one of the above modules contains a relevant simple module. It follows that if no-one of the modules contains a relevant simple module, then 31 is unmixed. Conversely if one of the above modules contains a relevant simple module (or more generally if one is mixed) then 31 is mixed. Suppose for instance that (31, Xn - an) is mixed. Then since (31, x^ — a,,) is of rank r + 1 it has a relevant spread of rank ^ r + 2. Hence there is a whole function <^ of .r,.+i, ..., Xn_i only containing this spread, and a polynomial i^in x-i, x^, ... , x,i such that 'l>F= mod (M, x„ - «„), while F+ mod (31, a-„ - a„). Ill] GENERAL PROPERTIES OF MODULES 47 Let (-FuFi,..., Fu) be a basis of M. Then F= A^F^ + . . . + A^Ft mod {x„, - a„), where we may assume that F, contains the relevant simple module of M sX (ai,a2, •■-, a„) ; and since J^ contains all the other relevant primary modules of 31 we have i^= mod M, which is not true. Hence {M, x„ — a„) does contain a relevant simple module at the point (a^ , a^, ■ ■ ■ , a^). 46. The Hilbert-Netto Theorem (Hi, Ne). If M' is any module containing the spread of a given module M some finite power of M' contains M. For M' contains all the relevant spreads of M and some finite power of M' contains all the relevant primary modules of M (^ 32) and therefore contains M. The theorem is proved in (Ne) for the case of two variables and in (Hi) for the general case. 47. Definition. A module of rank r having a basis consisting of r members only is called a module of the principal class (Kr, p. 80). Hence a module {F^, F^, ... , F,) of rank r is of the principal class. It is possible for the resultant of a module of the principal class to vanish identically. An example is given at the end of § 12. The .ff-module equivalent to a given module of the principal class is not necessarily of the principal class, e.g. the ^-module equivalent to (xi, iTo + x-iX-i) has four members in its basis (a;/, x-^Xi, x?, x^x^ + x^Xs), §38. A proper module is of rank ^ n and > 1. A proper module with a basis consisting of r members is of rank ^ r (cf. ex. § 42) ; for the module contains some point P in the finite region and a spread of dimensions w - r at least through any such point. Nevertheless a module with a basis of two or more members may be the non-proper module (1) ; e.g. (F, 1 + F) = (1). The unit module is sometimes said to be of rank n+ 1 ; but it is better to say that it is without rank, and that no module is of rank > n. In the absolute theory a module can be of rank « -•- 1. If {Fi, Fi, •••, F,) is of rank r it does not necessarily follow that {F., Fs, ... , F,.) is of rank r-1. Thus (ff +ff,f. +ff) is the same as (/,/i ,/2), and can be of rank 3, while (f +ff^,f.,_ +ff) contains (l +/) and is of rank 1. If however the series Fi, F2, ..., F,. is suitably modified beforehand (§ 37) then {Fg+i, ..., F,.) will be of rank r-s if (Fi, F2, ■■■, Fr) is of rank r. It will be sufficient to prove that (Fs + fhFi, F-i + a^Fi, ...,7^^,. + «,.i^i)isof rank r- 1 when a^, a^, ..., a^ IIlJ GENERAL PROPERTIES OF MODULES 49 are (at first) undetermined constants. If it is of ranlc si, <^2, •■-, s) is of rank s, where i. = K2 (F2 + a,F,) H- Xa (F, + asF,) +...+ A^,. (F,. + a,.F-,) = XnF + XinF2+ ... + Xi,.F,. (Xji = A.f2«2 + XisCKs + ... +X;,.a,., i=l, 2, ..., s), and the A,y are all arbitrary constants. We may regard the s relations Aji = AinOSj + . . . + Xi,.a,. as determining the s constants «2. «3, ■••, «»■+], leaving at least a,. (s + 1 $r- 1) quite arbitrary, whatever the values of the Aj; are. Now some spread of (^i, ^o, ... , 4>s} of rank s is a spread of (F-i + a.,Fi, ..-, Fr + ayFi) and is contained in Fr + a,F^ , and therefore in Fi (since Ur is independent of the Ay), and in each of F^, F3, ..., F,-. This would make (Ft , F«, ..., F,-) of rank s, which is not the case. Unmixed Modules 48. A useful test as to whether a given module is mixed or unmixed is proved in § 44. Theorem. A module of tJie principal class is unmixed. Lasker proves this for .^-modules (L, p. 58). The following is a general proof It is clear that any module of rank « is unmixed, since it resolves into primary modules which are all of rank n. Also a module of the principal class of rank 1 is unmixed. Hence the theorem is true for two variables, since in this case the module can only be of rank 1 or 2. We shall assume the theorem true for m — 1 variables and prove it for n variables. We also assume that the members of the basis have been modified if necessary so that, when (Fi, F^, •■■, F,) is of rank r, (F„, Fs, ..., F.) is of rank r- 1 (§ 47). We prove first that a module 31= (Fi, F.,, ..., F,) of rank r »■ + 2 the module {Fi, ..., F,-, Xs-cis, ..., .iv— a„) would contain a relevant simple module (§ 44) ; but it does not, because it is of the principal class. Hence {Fi, F^, ■■■, F,) is unmixed. 49. Deductions from the theorem. A basis (Fi, F^, ■■-, F,) of a module M of the principal class of rank r is an H-hasis of M or not, and an H-basis of i)/''' or not, accwding as the H-module determined by the terms of highest degree in Fi, F^, ■■■, F,. is of rank r or not. Let ilifo be the ^ff-module in «j, a'j, . . . , a-v, x^ corresponding to the basis (-fi, F^, ..., F,), so that (i)fo)a;,=o is the IT-module mentioned in the enunciation. Let {Mo)xa=o be of rank r. Then it follows by the same reasoning as in the theorem that XoF„ = mod Mo requires ^^(,=0 mod M„. Hence M^ is equivalent to i)/(§ 38), i.e. (F^, F^, ...,F,) is an ^-basis of M. It is also an jff-basis of il/'''. This follows in the same way by considering the jff-module il/o''' in Xi, x^, ..., x,-, Xf, corresponding to {Fi, F^,..., F,) regarded as a basis of il/<'''. The module (i)/o'''').i-,=o is a simple i]?-module not involving av+i, ..., Xn- If on the contrary {M^x„=o is not of rank r it is of rank < r, and x„ contains a relevant spread of Mo of rank ^ r, so that Mol(x^ 4= M^ and M'o is not equivalent to M (§ 38). Hence {F-^, F^,..., F,) is not an ^-basis ofilforofil/('->. If (i^i, Fi, ..., Fk) is an //-basis of a module of rank r the /T-module determined by the terms of highest degree in F^jF^, ..., Fy is of rank r. Ill] GENERAL PROPERTIES OF MODULES 51 But the converse is not true in general when k>r ; i.e. if the module determined by the terms of highest degree in F^, F^, ■■■, F^ is of the same rank r as the module (i^i, i^2, •■•, F^ the basis {F-^, F^,..., F,^ is not in general an ZT-basis when h > r. 50. Any power of a module of the principal class is unmixed. Let the module be M={Fi, F^, ..., Fr) of rank r. The spread of M'' is the same as the spread of M. Hence it will be sufficient to show that AF= mod M^ requires F= mod M"^ provided A does not contain any relevant spread of M. When y = 2 we have AF^OmoAlSP; hmc& F=QmoAM=A^F^+ ... + ArF,-, and A {A^F^ + ... + A,.F,) = QmoAM^ = F^F'''^mod.{F^, ...,F,)\ where i^'^' = mod M. Hence {AA^ - i^W) F^ = OmoA{F^, ..., F,.), AA,-F^'^ = Omod(F„...,F), AAi = mod 31, and ^i = Omodil/. Similarly ^, = mod 31, a.nd F=AiFi+ ...+ ArFr = mod 31-- Next suppose y = 3. Then since AF= mod 3I\ F= mod 3P = F^F^^^ + = 1Ap„ p, p, F,''''F.. ^'^ . . F/'', where p,+p,+ ... --pr = y-l. Take fi,4, •••! ^k for the variables instead of Xi, X2, •■•,>i'n. and move the origin to any point {x-^, x.^, ■■., «„) of M. Then F^, becomes ^'\(li + «i,---,f.t + a'J = ^i 0X1 OXn 0X-, and the terms of lowest degree in F^ are ^^g^ ex. ■ ■ + t. t oFry OX„J where Ap^,2^, ■■■. vr have their original values as functions of x^jX^,..., x^ ■ This last expression is of degree y- 1 in ^i, 4> •••, ^b and must vanish identically, since F has a y-point at every point of ]\J. Now the r quantities ti , 1- ^"S^^=^-^' r) are either capable of taking any r values (ii, ..., ^n being undetermined quantities and A'l,..., Xn fixed quantities) or they are not. If they are, every jd^j„p.,...,j,,. vanishes. If they are not, every determinant of the matrix dj\ dF dF uXi 0X2 o,r„ dF dF. dF, 1 uxi dx-2 '" dxii vanishes, i.e. (a-,, x.2, ..., «■„) is a .singular point of 31 (§ 29). Hence every Ap„p^_...,p,. vanishes for every non-singular point of il/ and is therefore a member of M (§ 22). Hence F'-^^ = mod J/^, which proves the theorem. 52. Definition. The module whose basis consists of all the determinants of the matrix th, U.2, ..., Ilk [ Vi, V2, ..., V,,. hi] general properties of modules 53 where the elements n, v, «•,... are polynomials, will be denoted by This is only an extension of the notation (Fi, F«,...,F\) for a module 31. If Ml is a prime module of rank r, and F\, i'l, ..., F,. any r members of 3Ii such that 31= (Fi, F\, ..., F,) resolves into 3/, and a second prime module i)// of rank r, then it may happen that i)// must have a certain fixed spread in common with 31^ irrespective of the choice of Fi, F^, ..., F,.. Such a spread (if any exists) must be a singular spread of 31^ ; but it does not necessarily follow from 31^ having a singular spread that 3Ii must contain the spread ; it depends on the nature of the singularity. If 31^ does not cut 31\ in a fixed sjiread then 3Ii is unmixed, and is the module whose memhers consist of all poly homials having a y-point at eoery point of 3Ii. In the contrary case some power 31^ of 3Ii will be mixed and will have the fixed spread in which 3Ii cuts it as a relevant imbedded spread, while polynomials i^'''' having a y-point at every point of 31, but not members of 31^, trill exist. Example i. The square of the prime module i)/i determined by an irreducible curve in space of three dimensions having a triple* point, the tangents at which do not lie in one plane, is mixed ; and there is consequently a surface having a 2-point at every point of the curve which is not a member of il/jl Thus if I = ( 2 j = GtiXs - av, x.,Xs - x^-, x-r - xyx-i), the surface {x^x-i-xff-{x}~XiX^{xi-Xx-x.^, after removal of the factor a\, will have a 2-point at every point of i)/,, but is not a member of 31^ ; for the surface has only a 3-point at the origin, whereas ever)"^ member of il/i" has a 4-point. /iH, U.2, u-i, u,\ Example i\. If 31 = { v,, v^, v,, v, \ = {Fi,F„Fl, F,), \u\, 2l\, Wi, %vj A triple point is not a 3-poiut. The general member of JlJi has only a 2 point at the triple point of the curve. It is evident that the module whose members consist of all polynomials having a 7-point at every point of a given irreducible spread is primary and unmixed. 54 THE ALGEBRA.IC THEORY OF MODULAR SYSTEMS [iH where each u, v, w is a homogeneous linear function of Xi, x^, Xs, X4, Ml being a prime module of rank 2, we have Mji^i + U2F2 + U3F3 + UiFi = 0, and two other similar identities. From these we can find the continued ratio Xi-.x^-.Xsi X4 as the ratio of four members of Mi^ by expressing each u, v, w in full. The common factor of these four members is a polynomial of degree 8 having a 3-point at every point of il/i, but not a member of M^. In this example M-^^ is mixed while if," is unmixed. 53. Theorem. The module with a basis of r roivs and k columns M=\ Ml, Ih, ■■ ., %. «1, V2, ■■ ., v„ Wj, W2, .. ■, Wk is of rank %k—r + 1 (0 < ^ - r+ 1 ^ n), and if of rank k-r+1 is unmixed. Also if Dp„p.j pr denotes the determinant formed hy the jOi"', ^2"', ..., />/' columns of the basis, the general solution of the equation '^Dp,,p^,...,p,.Xp^,p,,...,pr = (pi,Pii, ■■■,p,=i, 2, ..., k) p=k p=k 71=1 J5 = l ivhere Up^,...,p^_p,Vp^^...^2ivP' ••• <^''^ arbitrary polynomials subject with the unknotvns -i'})„j;j,...,p,. to the same law of signs as the determinants Dpi,Pi,...,pr, viz. eac/j Ap„pj,..,,p,., tp„p.;,...,p,.w" ■•• c/ianges in sign (but not in magnitude) for each intercliange of any pair of suffixes 2h, •■■ , p>i-, P- These two theorems will be proved together by a double process of induction. Assuming both theorems for ?■- 1 rows and k-\ columns, and also for r rows and k—l columns, we prove both theorems for r rows and k columns. Both theorems have been proved for r = 1 in § 48. It is understood that il/is a proper module, i.e. the determinants of its basis all vanish for some point whose coordinates are finite, but do not all vanish identically. After proving that AI is of rank $ ^ — r + 1 we assume that if M is of rank k — r+ 1 the module /u^ + a^Ui, ih + a-iUi, . . . , ut + auiii\ \V2 + a.2Vi, v-i + a^Vj, ..., Vk + a,;vA where a.^, a,, ..., aj, are suitably chosen constants or polynomials, is of rank k - r. This can be proved in a similar way to the corresponding property in § 47. We shall also suppose the matrix to have been so Ml Ul t«3 Z«4 ««5 Vi «2 »3 Vi «li Wi M.-2 W3 W4 1^5 and also for J// = ( ' \»2 III] GENERAL PROPERTIES OF MODULES 55 modified beforehand that if the first s^ k — r columns are removed the rank diminishes by s. It can be shown that the second part of the theorem is true before modification if it is true after. The same is true of the first part of the theorem, since the modification of the basis does not alter the module. The general proof will be sufficiently indicated if we suppose Mto have 3 rows and 5 columns. Then M=\ and we assume both parts of the theorem for the module Mi = { V2 V3 Vi «s ) U-i Ui MjX .»2 V-i Vi vj ' If A, B, C are the determinants of the matrix formed by the last two columns of the basis of M, we have Aui + Bvi + CiVi = Z>«5 («' = 1, 2, 3, 4, 5). Giving to i the values jWi, jC2> ^3 and solving for C (or Dt^ we have where Z>j.,», denotes the determinant *' ^- . This shows that every determinant Dp^p^j,^ when multiplied by Di^ is of the form XiDii5 + X,iD2a + X^Dsi^. Hence if there is a point of the module il/for which Dt;, does not vanish the module must have a spread of rank $ 3 (or ^ - r + 1) through that point. If however Dt^ contains the whole of the spread of il/we move the origin to a point of the spread and modify the last row of the basis by the other rows so as to make the constant terms in the elements of the last row all zero. After doing this we change 2/4, M5, Vi, v^ in the first two rows only to M4 + a, H5 + b, Vi + c, Vi + d, where a, b, c, d are constants. We thus get a new module containing the origin such that the new 2)45 does not contain the origin. This new module has a spread of rank $ 3 through the origin ; and since this is true for general values of a, b, c, d, it is still true when we put « = ^ = c = (i = ; for no diminution in the dimensions of the spread through the origin, i.e. no increase in the rank, could be produced by giving special values to a, b, c, d. Hence M is of rank ^ 3 ; and we have to prove that M is unmixed if its rank is 3. 56 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill Consider the equation '^Dp^p^p^Xp^p^.p^ = Q in which we suppose Pi45 we have In this the terms containing D^^ are obtained by putting ^a = 1 and giving p^, 2h the values (2, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5), viz. D,a (AsA'iss + AjA'im + As-X'lSS + -Z>34A'i34 + As Xi,,s + Z»4oX,45), and this is a member of (-D-245, D-ni) and therefore of M^. But Mi is unmixed and of rank 2, and Dan does not contain any of its relevant spreads; for if after modification of the last two columns of Mi by the first two, i)i45 contains a relevant spread of Mi then every Dip,p, contains the same spread, and consequently M contains the spread and is of rank 2, which is contrary to the data. Hence ^Dp^p, -^\p,p, = mod Ml = D.23i TF'234 + A35 W^'235 + A45Tr^'245 + D^r, ^^^'.iAr, = {D-mIv-i + Di.w~ + D^iWi) W'.u + ••• = Z>s3 {Wi TT'osj + M'5 ^-^'235) + • ■ • = 2 Dp,p.^ ( \V'p,p^«w.2, + ir'p.2P33*t'3 + f'T''j)2P>4"'4 + TI^'jj.ridWV,) ; or 2 Dj,,p^ Xp,-p^ = {2h < Ps = 2, 3, 4, 5), where ^p^v, = ^uhp.< - ^ W'p,p^p iVp (p = 2, 3, 4, 5). 1' The equation '^Dp.^p^Xp.p^ =0 stands in the same relation to J/,' as "^DptViP^^PiP^Pi = to M, and the general solution is ^p-.P:, = 2 U'p,-p^p Up + ^ V'p,p,p vp (p = 2, 3, 4, 5) which gives ^iPiP:, = 5 U'p^p,pUp + 2 V'p,p.^p Vp + % W'p,p^p Wp {p = 2, 3, 4, 5). p p p Substituting these values for X^p,p^ in the equation 2 iJp^p,p, ■^P\lHP-i ~ '^ it becomes, after simplifying, A34 (-i'234 + f' '234 Ml + T^'234 Vi + PK'034 Wi) + • . . = 0, an equation in reference to Mi of which the solution is X234 = - f^'234 Ml - y'i'n Vi - TF'234 M-i + S U.-up u„ +.. + .. (p = 2, 3, 4, 5) and similar expressions for X23-0, Xui, Xj4.,. If in these and the expressions found for Xip.p^ we put ~ U r>,PsP - '-lp.,PtP\i - ' PiPsP - ' P2Pa!Jl> " " PiPiV = ''' P>V,P\1 Ill] GENERAL PROPEETIES OF MODULES 57 we have, for all values ot pi, p-2,Ps = 1, 2, 3, 4, 5, ■^PlPlPl ~ ^ ^Pi PlPiP "p "*" ^ '\Pl PjP:iP "^P ''" ^ '' P\P:P.lP'^P \P ~ 1' 2, o, 4, 5), 7) p Ji which proves the second part of the theorem for BI. To prove the first part, that i)/is unmixed, it has to be shown that neither M nor {M, fCg-cis, ■■-, •'?'«- «k) can contain a relevant simple module, where s is any number ^ ^' - ?■ + 3 (§ 44). Let (a'l - Ci) i^= mod (J/, a's -a^, ■■■, x-,, - ««) = 22);,,,,,,,...,;,,, X;,,,,,, ,j,. mod (,r,, - «,, ...,«•„- a„). Ihen \^J-'PuP>.--,Pr -'^P\'p! 2),-).rj-<'i = .Ts-as=- = .Tii-aii = = 0. In putting a'l — Ci = 0:^-0,)= ■■■ = ^n ~ O'n = in 31 the number of variables is diminished but the rank remains equal to ^- - r + 1. Hence \-^Pi fik Sn.-, Ai • ■■ , flk Ill] GENERAL PROPERTIES OF MODULES 59 whose elements are polynomials, such that the general solution is Xi = A,fH + A^f2i+...+AJu (i=l,2,...,k) where Ai, A^, ..., A^, are arbitrary polynomials. The rows of this array are not independent. The general case of r equations can be reduced to that of solving a single equation. Consider first the single equation F,A\ + F,X,+ ...+ F^X,. = 0. The conditions imposed by this equation on X^ are merely that it must be a member of the module (i^'s , i^3 , ■ • • , Fk)I{F^). Let (/n , /21 , • • • , fvi) be a basis of this module. Then the general solution for X^ is -^1 = -4i/„ + ^2/21 + . . . +Ai'fr,. To each separate solution Xi =fji there corresponds a solution fj2,/j-3, ■■■,/jk for X2, X3, ..., Xk, giving a tow fj„/j^, ...,fjk of the array of solutions. The remaining solutions are those for which Xi = 0, when the equation reduces to X,F,+ ... + X,F, = 0. To each solution for X^^fj., {j' = I' + 1,1' + 2, ..., I") there corresponds a row 0,ff2,fj'3, •■■,fi'ii of the array of solutions in which the first element is zero. Similarly there are rows in which the first two elements are zero, and so on. The method may give more rows altogether than are necessary. Any row of the array which can be modified by the other rows so as to become a row of zeros should be omitted. In the case of r equations we eliminate X^-, X^, ..., Xy-i, obtaining Di^z vXr + Z)],2,...,i-n,r+i ^^",.+1 + ... + Z>,,2 ,._i,it^t = 0, and find the complete solution of this equation by the method just described. To each solution there is a unique set of values for X^, X^, •■• , Jf^-i which are in general polynomials. In an exceptional case the unknowns Xi, Xi, ..., Xu may be subjected to a linear substitution beforehand. The princiixd case. The principal case is that in which the module /«i ««2 • • • mA is of rank k-r + \. In this case it is seen from the equation in Xr, X+i, ■■•; Xt above that X^ is a member of the module A«l ?(2... i, where A', B' are ordinary power series with undetermined coefficients, were finite and determinate ; and that any polynomial F whose coefficients satisfied all these identical equations, when the origin was taken successively at each point of (/, <^), was a member of (/, <^). Thus the conditions which F has to satisfy in order to be a member of (/, <^) can be collected locally, so to speak, by going to each point of (/, <^) to find them. On going to a point not in (/, 4>) we get no conditions, for at such a point every polynomial is of the form A'f-^ B'. That the con- ditions are necessary is evident; for if i^=Omod(/, 4>) then F is of the form A'f+ B'4> wherever the origin is taken. Konig (K, p. 385) proved the theorem for the case of a module (/ij/a. •••>/>) of rank n in n variables; and Lasker generalized the theorem in the Lasker-Noether theorem given below. That the theorem is true for any module of rank n (not merely for a module of the principal class of rank n, the case proved b}'' Konig) follows from the Hilbert-Netto and Lasker theorems. For, Ill] GENERAL PROPERTIES OF MODULES 61 by Lasker's theorem, the module is the l.c.m. of a finite number of simple modules Qi, Q2, •••, Qr, and if 7 is the characteristic number oi Qi = (fi,A, ■■■,fh) and the origin is taken at the point of Qi, we have F= Pi/i + P.fi + ... + Pufh (where P^, P^, ..., P,, are power series) = A'J\ + A',f, + ...+ X,Ju mod 0' = mod Qt. Thus i^ contains [Qi, Q„ ..., Qi]. 56. The Lasker-Noether Theorem (L, p. 95). 1/ M = {F„F„...,F,) and F=P,F + P,F, + ... + P,F,, lohere Pi, Pr,, ■■-, Pk «>'« ordinary power series, there exists a poly- nomial 4> not containing the origin such that F<^ = mod M. Let Qi, Q,-2, ■■■, Qi be the relevant primary modules into which M resolves, and let Q^, Q^, ..., Qv be those which contain the origin, and Qi'+i, •-., Qi those which do not. Then, assuming the theorem to be true, it follows that F=Omod[Qi,Q„ ,..,Q,], since <^ cannot contain the spread of any of the modules Qi, Q^, ■■■, Qv. Conversely if i^= mod [4, ^2, ■■-, Qi^ and <^ = mod [Qm, •••, Qi\, where does not contain the origin, then F4> = mod M. Hence the aggregate of all polynomials F which are of the form P,F, + P,F, + ...+ PuFu constitutes the module [Qi, Q^_, ..., Qi^. Definition. A module which resolves into primary modules all of which contain the origin, such as the module [Qi, Q., ..., Q,] above, will be called a Noetherian module. Thus a Noetherian module, like an ^-module, ceases to be such in general when the origin is changed. Moreover an ^-module is a particular kind of Noetherian module ; for all the primary modules into which an ZT-module resolves are jST-modules and contain the origin. In order that a polynomial F may be a member of a Noetherian module {Fi, F., •••, F,i) it is sufficient that F should be of the form PiFi + P,F,+ ... + PuFu. Proof of the theorem. It is evident that the theorem is true for a module of rank n or dimensions (§ 55). We shall prove the theorem for a module of dimensions n-r assuming it true for a module of 62 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill dimensions n-r-l. It will be sufficient to prove the theorem for a primary module Q which contains the origin ; for it is clear that it will then be true in general. Let Q = {fuA, ■■;/,.), ^nd f=F,f, + PJ,+ ...+P,J„ where Pi, P2, •■•, A are power series. Let Qo = (fi,A', ■■-,/' id be the module whose members consist of all polynomials of the form of /, and Qp the like module obtained by moving the origin to P (and then back to 0). Choose a point P so near to as to come within the range of convergeucy of all the power series Pi, P^, ■■■, P% for each member// of the basis of Qo when expressed in the form of/. Then we have /' = mod <^p, i.e. Qo contains Qp. But it does not follow that Qp contains Qo however near P may be to ; for might be a special point of the spread of Q. We assume for the present that is not a special point of the spread ; and we choose P to be another point of the spread so near to that Qp contains Qo- Then Qo = Qp- Let u be a fixed arbitrarily chosen linear homogeneous polynomial, and/' any member of Q,o. Then /' = mod Qo = mod (Q, u)o. But (Q, m) is of n-r-\ dimensions ; hence, assuming the general theorem as regards (Q, «)> there exists a polynomial ^ not containing such that /'(/> = mod (Q, «) = 'pu mod Q, where ^ is a polynomial. Hence, since/',/', ■■-,/' y are members of qo, fi^i =PiU mod Q (t = 1, 2, ...,/*') ; hence PiU = mod Qo = mod Qp ; but u does not contain P, and - can be expanded as a power series when P is taken as origin ; hence 2}i = mod Qp = mod Qo = Ihifi + 1^12/2 +... + Piu-fu- Hence fl4>i = {Piifi +Pi-J2 + ■■■ +Pih'f'h) u mod Q (i =1,2, ..., h'). Solving these K equations for/',/', ...,/';,. we have i)/' = OmodQ, IIIJ where D = GENERAL PROPERTIES OF MODULES 63 Ph'iU PlfiU ■ Ph'h'U - u' = (- l)"'^i <^2 . . . <^;,' mod u. Now u contains the origin, but <^i, f^^, ..., (j>,,' and consequently D do not ; i.e. J) does not contain the spread of Q. Hence // = mod Q. Hence Qo contains Q, i.e., Qo=Q- This has been proved for a non-special point of Q. If is a special point, choose P a non-special point of Q so near to that Qo contains Qp. Then since Qp = Q we have again Qo = Q. The above proof only differs from the proof given by Lasker in the part relating to Qo = Qp- In this part Lasker's proof seems to be faulty. 64 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV IV. THE INVERSE SYSTEM AND MODULAR EQUATIONS 57. A considerable number of the properties proved in this section are to be found in (M) ; but tlie introduction of the inverse system is new. Definitions. The array of the coefficients of a complete linearly independent set of members of a module M of degree ^ t arranged under the power products mj, ("a, -.., w^ of degree %t is called the dialytic array of the module M for degree t. The linear homogeneous equations of which this array is the array of the coefficients are called the dialytic equations of M for degree t. Thus the dialytic equations of M for degree t are represented by equating all members of M of degree %t tn zero and regarding the power products of a:',, x^, ■■■, «„ as symbols for the unknowns. The array inverse (§ 54) to the dialytic array of M for degree t is called the inverse array of il/for degree t. The linear homogeneous equations of which this array is the array of the coefficients are called the modular equations of 31 for degree t. The modular equations for degree t are the equations which are identically satisfied by the coefficients of each and every member of M of degree ^ t. They may not be independent for members of degree < t and they do not apply to members of degree > t (see § 59). The sum of the products of the elements in any row of the inverse array for degree t with the inverse power products (i)i~\ w^-^, ..., w^-^ is called an inverse function of il/for degree t. Thus the modular equations of 31 for degree t are represented by equating all the inverse functions of 31 for degree t to zero, taking each negative power product {xf'^ x}^ ... .r/")-^ as a symbol for " the coefficient of a'/' xl'"^ ■.■ Xn " in the general member of 31 of degree t." We shall also say that a polynomial i^^= Sa^,,. .,,,,„*'/' ••■a;/" and a finite or infinite negative power series ^=2c,„ qni^i"' ■■■ x,,''")-^ are inverse to one another if the constant term of the product J^^JS vanishes, i.e. if 2«y,„p,,...,j,,,Cj,„p,, ...,,,„ = 0. Thus any member of 31 oi degree $ t and any inverse function of 31 for degree t are inverse to one another. IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 65 Any inverse function of M for degree t can be continued so as to become an inverse function of M for any higher degree (§ 59), and wlien continued indefinitely becomes an inverse function of M without limitation in respect to degree. If all coefficients after a certain stage become zero the inverse function terminates and is a finite negative power series. In the case of an ^-module the inverse functions are homogeneous (§ 59) and therefore finite. In order that a function may be an inverse function of M it is necessary and sufficient that it should be inverse to all members of M; hence if M contains M' any inverse function of M' is an inverse, function of M. The whole system of inverse functions of M can there- fore be resolved into primary systems corresponding to the primary modules of M. The inverse functions of a Noetherian primary module are all finite (§ 65) but not in general homogeneous. The inverse functions of a non- Noetherian primary module are all infinite power series (sj 65). We shall regard inverse function and modular equation as con- vertible terms, and use that term in each case which seems best suited to the context. A module is completely determined by its system of modular equations no less than by its system of members. The two systems are alternative representations of the module. Also the properties of the modular equations are very remarkable, and it is necessary to consider them in order to give a complete theory of modular systems. As there is a one-one correspondence between the members of a module M of degree $ t and the members of the equivalent ^-module of degree t, so there is a one-one correspondence between the modular equations of M for degree t and the modular equations of the members of the equivalent .ff'-module of degree t. These last are called the modular equations of the ^-module of (absolute) degree t. 58. Theorem. The number of independent modular equations of degree t of an H -module {F^, F^, ■■■, Fr) of rank r is the coefficient of x^ in (1 - x') ( 1 - ^'0 ■■•(!- «^'') (1 - •^)""' tvhere k, 4. •■■, 4 are the degrees of Fj, F^, ..-, Fr- Since the whole number of linearly independent polynomials of degree t is the number of power products of degree t, or the coefficient of x' in {l-x)-'% the theorem will be proved if it is shown that the M. 5 66 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV number iV(r, t) of linearly independent members of (i^i, F^, ■■■, F,) of degree t is the coefficient of x' in {1 - (1 - x^') {l-x'")...{l- x^'-)] (1 - «)-". This is easily seen to be true when r=l. Since any member of {Fi, Fr,, ••-, Fr) is a linear combination of elementary members, we have N{r,t) = A^(r-l,t) + p, where p is the number of polynomials iHiF,-, f^iFr, ■■■, <^pF,. of degree t of which no linear combination is a member of (i^i, F2, ■■•, -P,-i), orthe number of power products Wi, (1)2, ..., wp of degree t- I,, of which no linear combination is a member of (i^i, F2, ••■, Fr^i), § 48. Hence P + K(r - I, t — 1,~) = number of power products of degree t — l,. --= coefficient of a-* in x^ {\- «)~" ; and N{r, t)= N{r~\,t)-N (?•- 1, t - /,.) + coefficient of «' in a;''' (1 - a^)~"- Hence, assuming the theorem for N{r- 1, t), it follows tliat N{r, t) is the coefficient of a;' in { 1 - (1 - a;'0 . . . (1 - x^-^)] (1 - aO~" ( 1 - «''■) + cc'- (1 - xy\ or in {1 - {l-x') ... {l-x'"-') (1 -J')] (1 -«)-", which proves the theorem. This result is independent of the coefficients of Fi, F^, ■■■, Fr ; hence it follows that any member of {Fi, F2, ..., F,-) is expressible in one way only in the form X^'OF, + A'^'^F, + ...+ X^'--'^Fr, where X<'' (as in §§ 6, 7) is a polynomial in which Xi, x^, ..., xi occur only to powers as high as xi^-\ ..., x^i-^, the variables having been subjected to a substitution beforehand. The theorem can be applied to any module (i^i, F^, ■■-, Fr) of rank r if (Fi, F^, ■■■, Fr) is an iT-basis, i.e. if the i/-module deter- mined by the terms of highest degree in Fi, F^, ■■■, Fr is of rank r (§49). In this case the number of independent modular equations for degree t is the coefficient of x' in (1 -x^') ... (1 — .■r'l-) (1 -^V't-i^ ^j^ important particular case is the following : The number of independent modular equations of a module {F-i, Fn, ..., Fn) of rank n such that the resultant of the terms of highest degree in F^, F^, ■■■, Fn does not vanish is kL...l„-i for I. 59. A711/ inverse function of 31 for any degree can he continued so as to give an inverse function of M for any higher degree. By carrying the continuation on indefinitely we obtain a power 6J, a)j gja - : -: -- -__=;: . _--=_\ isi :l+J:: 6+2 ~" ___ . t+z -- m -i- :i+r. r+2~ ---1 + 5 Oi ,iO_CO series (finite or infinite) which is an inverse function of M for all degrees without limit. Let {Fi, F,, ..., Fk) be an jfiT- basis of M. Then any member i^ of 31 is a linear combination of elementary members OiFj no one of which is of higher degree than F. Let I be the lowest degree of any member of 31. Write down the dialytic array of 31 for degree I, viz. the array of the coefficients of such members of the ZT-basis as are 68 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV of degree I. Their terms of degree I (corresponding to the compart- ment I of the diagram) are linearly independent, for if not there would be a member of M of degree < /, which is not the case. Next write down the rows of the array representing such members of the basis as are of degree 1+ I, and members obtained by multiplying members of degree I by Xi, x^, ..., «„, so as to obtain a complete set of members of degree l+\ linearly independent as regards their terms of degree ^+1, these terms corresponding to the compartment ^+1 of the diagram. Proceeding in the same way we can obtain the whole dialytic array for any degree. To obtain the inverse array for the same degree first WTite down square compartments 0, 1, 2, ..., ^-1 with arbitrary elements corre- sponding to degrees 0, 1, 2, ..., I— I, and then a compartment I inverse to the compartment I of the dialytic array. Each row of the com- partments 0, 1, 2, ..., ^- 1 can be continued so as to be inverse to the dialytic array for degree I, since the determinants of the compartment I do not all vanish. This completes the inverse array for degree /. All its rows can be continued so as to be inverse to the dialytic array for degree ^ -i- 1, and a compartment ^-i- 1 of new rows can be added inverse to the compartment Z -f 1 of the dialytic array. This completes the inverse array for degree l+l; and we can proceed in a similar way to obtain the inverse array for any degree. This diagram or scheme for the dialytic and inverse arrays of a given module M will be often referred to. The ease with which it can be conceived mentally is due to the fact that it is obtained by working with an i/-basis of M. Each pair of corresponding compartments l + i form inverse arrays, and in combination form a square array, showing that the combined complete arrays for any degree have the same number of rows as columns. In the case of a module of rank n the compartments of the dialytic array eventually become square and the total number of rows oj the inverse array is finite. To a square compartment in either array corresponds no compartment or rows of the other array. In the case of an ZT-module the compartments are the only parts of the arrays whose elements do not vanish, i.e. the inverse functions are homogeneous. Definition. The negative power series represented by the rows of the inverse array continued indefinitely will be called the members of the inverse system, and E^, E^, E3, ... will be used to denote them, just as Fi, E2, F3, ... denote members of the module. IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 69 The system inverse to (1) has no member. The system inverse to (iTi, .t72, ... , x,i) has only one member E=\; and the modular equation 1 = signifies that the module contains the origin. 60. Properties of the Inverse System. Before attempting to show in what ways the inverse system may be simplified we consider its general properties. 00 Definition. If ^^=2cj,j,j,„ j,^^{cci'^xt' ■■• Xn'"')~^ is a negative power series (no 2h negative), and A any polynomial, the part of the expanded product AE which consists of a negative power series will be denoted by ^.£'and called the A-derivate of E. Thus A negative power series E= '^Cp^,p^ ,,,^^ (a;/^r/^ ... .r/")"' is or is not an inverse function of a module M according as every member i^=2a,, ,p„...,p^ i^/^iTa^S.. a;/" of M, or not every member of M, is inverse to it, i.e. according as every 2ap^,j,,^ VnCp^^Pz Vn~^ °^' not. Suppose E an inverse function and F any member of M. Then Xi^x-r-.-x,!'^ F = '2aj,^^p„ p^Xi^^*'- ... A'/"'^" is a member inverse to -£■; hence every Sa,,^, p^ p,^ Cj,^+i^ p^^+i^^ = 0, and I'l+h Pii+'m ^ 1 ^^ •■•'''71 J > or Xi^ X2^ ■ ■ ■ xj" . E, is & member of the inverse system. Hence if E is a member of the inverse system oj M so also is x^'x^'^ ... ,t„". E, and if El , E,,, ..., Ej, are members so also is Ai. Ei + A^. E2+ ■■■ + A,,. E,, a member, where A-,, .4.2, ..., Au are arbitruQ-y polynomials. ■ In a slightly modified sense which will be explained later (§ 82) the inverse system of any module M has a finite basis [Ei, E^, ..., En] such tlmt any member of the inverse system is oftlieform X, . E\ + X, .E\+... + Xn . En, where Xi, X^, ..., Xn are polynomials. This theorem is evidently true in the important case in which the total number of linearly independent members of the inverse system is finite, viz. in the case of a module of rank n and in the case of a module of rank r when treated as a module in r variables only, or, in other words, in the case of n module which resolves into simple modules. 70 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV Regarding the inverse system as representing the modular equations of Mwe shall write 31= [Eu -EL, ••■, E,,] as well as 31= (Fi, F^, ...,F„). Here 31 is the l.c.m. of [^""1], [^2], ••■, [F,,] and the g.c.m. of (F), (F,), ..., (F,). Definition. A module 31 will be called a principal system if its inverse system has a basis consisting of a single member, i.e. if 3f= [E\ A module of the principal class is a principal system (§ 72), hut a principal system is not necessarily of the principal class. A principal system is however the residual of a module (i^ with respect to any module of the principal class which contains, and is of the same rank as, the principal system (cf § 62). 61. The system inverse to 31= (F^, F^, ..., F,^ is the system whose Fi-derivates (i = 1, 2, ..., k) vanish identically. In other words, in order that F may be a member of the inverse system of il/ it is necessary and sufficient that Fi.F {i=l, 2, ..., F) should vanish identically. For if E=1c^^^n^^_^q^{xl'^x^'^ ... Xn"-)~' is any member of the inverse system, and Fi = 2a„^^p^ p^x^^xi^ ...x'^'-, then Fi. E='S,ap^^j,^^,„^p_^^'S,Cp^j,.q^ p^^+q^ixi x^'.-.x^i j p ' Q -2,{Xi X2 ...Xn ) ^(tpi.ih Pn'^Pi+n+<2n~^' <1 P since every lap^ p,/pi+«, p„^q„ vanishes (a;/^ . . .r/"-Fi being inverse V to E). Conversely if Fi.E=0, then Sa^^ p„Cj,j+'-(i J',i+«„ = 0, p i.e. Xi'^ ... x,l'^Fi is inverse to E, and every member of 31 \s inverse to E, i.e. ^ is a member of the inverse system. Similarly if 31= [Ei, Eo, •■•, E„] the necessary and sufficient con- dition that F may be a member of 31 is that F. Ej (j = 1, 2, ..., h) vanishes identically. 62. The modular equations of 311 {F-^, F^, ..., F^) are the Fi-deri- vates of the modular equations of 31, i.e. [E, E,, ..., E,]I(F„ F„ ..., F0 = [..., F.Ej, ...]. For the necessary and sufficient condition that F may be a member of the residual module is FFi = 0mod3I (i = l, 2, ..., k) IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 71 or FF,.Ej=0 {i = \,2, ...,h; j = l,2, ...,k) or F.{Fi.Ej) = Q. Hence [..., Ft.Ej, ...] is the residual module (§ 61). 63. A system of negative power series with a finite basis \_E-i, E.2, ..., En] of such a nature that all derivates of Ei, E^, ..., E,,, belong to the system is an inverse system of a module if Ei (t=l, 2, ..., A) has an Fi-derivate which vanishes identically. For there are polynomials F such that the i^-derivate of each of El, E2, ..., -E71 vanishes identically, the product F-iF^-.-Fu being one such polynomial. Also the whole aggregate of such polynomials F constitutes a module M ; for if F belongs to the aggregate so does A F. Consider the dialytic and inverse arrays of M obtained as in § 59. Since every member of M is inverse to every member of [Ei,E2, ■••, E,,] all members of the latter are represented in the inverse array. If any other power series are represented, viz. if there is a row of the inverse array which does not represent a member of [Ei, E^, ■■■, En'], let it begin in the compartment I + i. Then if we omit this row we can add a row to the dialytic array representing a polynomial of degree l+i inverse to all members of \_Ei ,Ei, ..., Eii] but not a member of M. This is contrary to the fact that M is the whole aggregate of such polynomials. Hence the system inverse to il/is \_Ei, E^, ■■■, E^. CO Thus in order that E= ^Cp^^p^ p^^{xi^xt^...Xn^)~^ may represent a modular equation of a module it is necessary and sufficient that Cpj, pj, .... p„ should be a recurrent function of Pi, p-2, ■■■,Pn, that is, a function satisfying some recurrent relation 2aj,j,j,2,...,j,,^c^,j+!j j',i+i„=^ p for all positive integral values of l^, L, •••, 4, where the «pj, ,„ p are a set of fixed quantities finite in number. It may be that Cp ^.^ p satisfies several such recurrent relations not deducible from one another ; but it is sufficient if it satisfies one. 64. Transformation of the inverse system corresponding to a linear transformation of the modular system. If the variables in the modular system M are subjected to a linear non-homogeneous substitution with non-vanishing determinant by which M is transformed to M' it is required to find how the inverse system [Ei, E^, ..., En] is to be transformed so as to be inverse to M'. 72 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV In other words, if the negative power series E is inverse to the polynomial F it is required to find a power series E' inverse to the transformed polynomial F'. It will be shown that an E' exists which can be derived from E in a way depending only on the substitution and not on the polynomial F. Let F= 2a,^ „,^ccfK..x/\ F' = %a\^ ,,..^V'^ ■ ■ a^«"", and let the coefficients Cj,^, j,^ j,,^ of E be represented symbolically by Ci^'c/^-.c/". Then we have ^=2c/'^..c/"(^a^^..a-/")-i; and "*")'j. Pa. -. !',i^i ''2 ■■■I'll — "> since E, F are inverse to one another. Let the inverse substitution be .r/ = a'iiX^ + . . . + «',■„«„ + a/ (z = 1, 2, . . • , n). Then Sa',, ,^{anX,+ ...)"K.. («'„,.r,+ .../" = 2a,,, ,,_^,r/'^...T/", and we have {coeff. ofa•^''^..a:/"in2a',^...,,,^(a'nirl+...)''••■(«'.aa■l+•■•f'} = 0> i.e. 2a'5j...,^^(a'iiCi+ ...)''K.. {a\aC-,+ ...)''" = 0, i.e. the power series E'=1{a\^c^+ ...)*^.. (a'„iCi + ...)"" (a'/" ... x/")"' is inverse to the polynomial F'= Sa'^^, ...,q^^Xi^ ... «,*'. Hence the coefficient of {x^'^x^- ... a:'/")"' in the transformed power series E' is c'4j,,2 5„ = («'nCi+ ...)''i(a'2iCi+...)42...(a'„iCi+...)S where, after expanding the right-hand side, cf^cf-^ ... Ctl'n is to be pat equal to Cj,^, p^ y,^, the coefficient of (;r/i «o^2 . . . a;/")"^ in £". For such a transformation of F and E, when not inverse to one another, 2«/)i,j2, ...,3-,iC2,i.P2 2'u i'^ ^'^ absolute invariant. The most important transformation is that corresponding to a change of origin only. In this case, if F= ^a,^...,,y^'...x,f- and E= %c^\..cl\xl'\..x:^r\ and the new origin is the point (— «!, -a™, ... , -a,i), then i^ ' = 2 «pj ... ,,^^ (a^i - aj/' • ■ ■ (^n - fliO''" and ii" = 2 (c, + aO"' • • • (c« + «,./" («/'' • • • «;>/'")"■ It is to be noticed that if ^ is a finite power series it nevertheless transforms into an infiuite power series E' . In particular if E= 1 IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 73 then E' ='Xa^^ ■■■an"'{a;i^xi-... Xn'^)~\ the inverse function of \Xi fti , . . . , a*,, — a,jj. For homogeneous substitutions another way of considering corre- sponding transformations of F and E can be given, which however excludes a cJmnge of origin. Represent E by CO ■2, ... Ih ■«■> - ■■■ Pn'- instead of Scj,^, IJo, •• ■.j'„(*'i ' ...a-/")-S and let the : new E be defined as inverse (or conjugate) to F= Sa., v-2 j',ri ....'/" when the same relation 2 ^P,,- -.Vifi'i,- ■'I'n = holds as before. Then for contra- gredient substitutions of Xi, x.^, ..., x^ and Ui, u.,, ••■, «« the poly- nomial i^and power series E will always remain inverse (or conjugate) to one another if they are so originally. Also the members E of the inverse (or conjugate) system of a module il/, when expressed in the new form above, are the power series with respect to which the members (of the basis) of the module M are apolar (§ 61). 65. The Noetherian Equations of a Module. The modular equations ^Cp^,p^ ly^^yx''^ X2^'^ • ■ ■ x,''")'^ = of a module M for degree t are finite because they are only applicable to members of degree ^t, and the coefficients (^i''\r/^ . . a:/")~^ in the general member of degree t vanish when j}^ + ... + 2}u> i- A modular equation may however be finite in itself, i.e. every Cp^,2>.,. -.Pn ^o^' ■which P1+P2+ ■■■ +p,i exceeds a certain fixed number I may vanish. If such an equation is applied to a polynomial of degree > / it only affects the coefficients of terms of degree ^ /. Definition. The Noetherian equations of a module are the modular equations which are finite in themselves. There are no Noetherian equations if the module does not contain the origin. For if ^= is a Noetherian equation of absolute degree I, and 0)-' a power product of absolute degree I which is present in E, the derivate equation w .E = is 1=0, showing that the module contains the origin. Eveiy Noetherian equation has the equation 1=0 as a derivate. On the other hand Noetherian equations always exist if the module contains the origin, for the equation 1 = exists, and so does the equation (u~' = 0, where w is any power product of less degree than any term which occurs in any member of the module. 74 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [IV The whole system of Noetherian equations of a non-Noetherian module il/ forms only a part of the whole system of modular equations, and is exhibited by a scheme similar to but different from that of § 59, with which it should be compared. In this new scheme the rows of the dialytic array represent the members of the module arranged in order according to their underdegree (or degree of their lowest terms) instead of their degree (or degree of their highest terms). The first set of rows represents a complete set of members of underdegree k which are linearly independent as regards their terms of degree k, "lZ~ *r J i -1 I I'A' ' ii^/t?":: - — 7 4-^ - t,+ •->■ - - ^ Zr' V{: :ltV i,±2-.-: l,±5\\i where li is the lowest underdegree of any member of M. These are obtained from any basis of 31, which need not be an ^-basis. The next set of rows represents a complete set of members of under- degree ^1 + 1 which are linearly independent as regards their terms of degree A + 1, obtained partly from the basis of i)/ and partly from the set of members of underdegree li by multiplying them by Xi, Ws, ..., Xn, and similarly for succeeding sets. The compartments Z], /i+l, ... correspond to the terms of lowest degree in the suc- cessive sets. IVj THE INVERSE SYSTEM AND MODULAR EQUATIONS 75 To obtain the corresponding inverse (or Noetherian) array first insert square compartments 0, 1, 2, ...,li—l with arbitrary elements (or with elements 1 in the diagonal and the remaining elements zero) corresponding to degrees 0, 1, 2, ..., li—l; and then a compartment li inverse to the compartment ^ of the dialytic array. This completes the array for degree li ; all its rows are inverse to all members of 3/ and represent Noetherian equations. Next insert a compartment ^i + 1 inverse to the compartment /j + 1 of the dialytic array, and continue its rows backwards so as to be inverse to the first set of rows of the dialytic array. This completes the array for degree h + l ; and we can proceed similarly to find in theory the whole of the Noetherian array. The object of the diagram is merely to exhibit the whole system of Noetherian equations, which it evidently does. If i^ is a polynomial for which all the Noetherian equations for degree t are satisfied, then, up to and inclusive of its terms of degree ^, i^is a linear combination of members of the module of underdegree ^ t, i.e. F is expressible as far as degree t in the form X-^Fi + X^iFi + ■ ■ ■ + J^icFk, where Xj, Xa, ..., Xk are polynomials, and i^= mod (ilf, O'+O- Conse- quently if F satisfies the whole system of Noetherian equations it is of the form P,Fj + P,F^ + ... + P^F^, where P„ P.,, ..., P„ are power series. Hence FFg = mod 31, where Fo has a non-vanishing constant term (§ 56) ; and, if 31 is a Noetherian module, F=0 mod 3f. Hence the wJiole system of modular equations of a Noetherian module can be expressed as a system of Noetherian equations. 66. Modular Equations of Simple Modules. If in the last article the rows of the compartment k + « of the dialytic array should be equal in number to the power products of degree k + i there will be no Noetherian equations of absolute degree '^li + i. In this case the Noetherian equations are finite in number and can be actually determined (at any rate in numerical examples). This can only happen when the module contains the origin as an isolated point, and the Noetherian equations are then the modular equations of the simple Noetherian module contained in the given module. The simple module itself is {31, 0''"^') and /j + i is its characteristic number. Thus the simple modules at isolated points of a given module 31 can all be found by moving the origin to each point in succession and finding its Noetherian equations and characteristic number. 76 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV Let M have a simple module at the pomt (ai, a.,, ..., a„). Move the origin to the point and find the Noetherian equations. They will be represented by finite negative power series and all derivates of the same. Also any such system represents a simple module at the origin ; the fact that the coefficients of El, Ei, ..., El, are recurrent functions (§ 63) placing no restriction on them when finite in number. Let Et = 'S, Cp^^ 3,^,, ..,, p^^ (x^'^xt^ ■ ■ ■ «„ ")"' be of absolute degree yj-1. Moving the origin back to its original position, that is, to the point (— aj, — as, ..., — «,;), the equation Ei = becomes (§ 64) 2 (ci + aO"' {c, + «,)"' ■ ■ • (c» + «.)"" {a;l''xi' . . . a-/")"^ = 0, where, after expanding (ci + a^p . . . (d + a,if''\ each c^^ . . . c„ " is to be put equal to the known constant c^^.q.^ g„ which it represents. Also Cg^, ,2 ,,^ = if gi + g-j + . . . + q,, > y,-. Thus {ci + a,/' (c. + a,r ■ ■ ■ (c„ + ««)"" = (1 + 7-T- ■ (l + ^)"" «/''«/^ ■ • • a,,"" \ ill / \ (*,i / — "-Jjj, ))2, ■■■, P,j"l "2 •..U',1 , where kp^.p.^ p,, is a whole function of pi, p^, ..., ^5,1 of degree yj— 1. Hence the modular equations of any simple module at the point («i, «2, ..., a,i) are represented by power series ^^jjj, Jjg. ••■■ Pn ^'^^ ^2 '•'0/n \J^\ X2'---Xn ) =0 and their derivates, where kp^,p^ j,,, isa whole function of jUi,jt?2,...,^,j. Conversely any system of equations (finite in number) of this type with all their derivates is a system of modular equations of a simple module at the point (csj, 0-2, ■■■, «„,). The following is a consequence of the above. The general solution for the recurrent function Cj,j, j,^, ..., p,, (§ 63) satisfying a set of re- current equations Sap^.p^ Vn%+h--'Pn*hi^^ ^'°'' ^^^ positive p integral values of h, 4i •••, 4, when the corresponding polynomials 2aj,j,p2 p,i «^/'^^2^'^ • • • ^Z" bave only a finite number of points («!, oSs, ..., a„) in common, is ^Aai'^af^... aj'"; where 4 is a whole function of pi, p,, •••, Pn dependent on the point (fflj, a.2, ..., a,,) and involving linear parameters. When the polynomials have an infinite number of points in common there can scarcelj' be said to be a general solution for fj,j, P2 p^. IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 77 Properties of Simple Modules 67. Theorem. If the resultant of {Fi,F.i, ..., F.^) does not vanish identically the number of Noetherian equations of any simple module of (Fi, F^, ■■■,F,i) is equal to the multiplicity of the cmre- sponding solution of F^- F^^ ■■■ = F„ = given by the resultant. This theorem is proved for the case m = 2 in (Mi, p. 388) and for the general case in (L, p. 98). Both proofs are very complicated ; and a simpler proof is given here. By the resnltant of {Fi , i^a , • • ■ , F^) we shall understand the resultant with respect to x^, x^, ..., x^-i, viz. a polynomial in x^, the variables having been subjected to a homogeneous linear substitution beforehand. Move the origin to any point of {Fi, F2, ■■■, F,,). Then, if x^ is the highest power of x^ which divides the resultant, C is the multiplicity of the solution of Fi = Fo = ■■■=-- Fn=0 corresponding to the origin. Let Q be the whole simple module of (Fi, Fi, .2, ■ . ■ , 4>n and <^i', (^2', • • • ) 't>n', none of which vanish at the origin, such that cj>tFi = Omod(F,',F,',...,F:) and <)>; F/ = mod {F„ F„ . . . , F„). Hence the module {4>iFi,4>iFi, •■■,'i'nF.,,) contains {F;, F^, ..., F^), and the resultant of the former is divisible by that of the latter (§11). 78 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV But the resultant of (<^ii^i, <^2-P'2. ■•■ . "/"ai^n) is the product of 2" resultants of which one only, the resultant of {Fi, F^, ..., Fn), has x,i as a factor. Hence the resultant of (Fi, F^, ... , Fn) is divisible by as high a power of x^ as the resultant of (-f /, i'V, • ■ • , Fn), and vice versa ; i.e. the two resultants are divisible by the same power of Xn. Now the resultant of the terms of highest degree 4, 4, ■■•, 4 in Fi, F«', ..., Fn' does not vanish, for the coefficients of these terms are absolutely arbitrary if li, h, ■■■, L are all chosen as high as the characteristic number of Q. Hence the equations Fi=F2= ■■■=Fn=Q have no solutions at infinity, and the number of their finite solutions is lik-.-In, taking multiplicity into account. Also the sum of the values of N for all the points of (i'V, F,', ■■■, FJ) is A 4 ■ ■ • 4 (end of § 58), i.e. is equal to the sum of the values of C. Also each point of (Fi, Fs, .-•, F„') except the origin comes under the simple case considered above ; for even if the curve (F2', F3, ..., Fn) has any singular points other than the origin, F-l does not pass through them, since the origin is the only fixed point of F^. Hence the values of C and N are equal at each point of {F-l, F^, ..., Fn) other than the • origin, and are therefore also equal at the origin. This proves the theorem. 68. Definitions. The multiplicity of a simple module is the number of its independent Noetherian equations. This number has a geometrical interpretation when the theory of the resultant is applicable; but in general it has only an algebraical interpretation. The multiplicity of a primary module of rank r is the multiplicitj' of each of the simple modules into which it resolves when regarded as a module in r variables only. Thus there are four important numbers in connection with any primary module, viz. the rank r, the order d, the characteristic number 7, and the multiplicity /n. A primary module of rank r will be said to be of the irrincipal Noetherian class if there is a module (Fi, F^, ■■■, F,.) of rank r which contains it and does not contain any primary module of greater multiplicity with the same spread. On moving the origin to any general point of the spread any member of the primary module will be of the form Pji^i + P, J^a + • • • + PrF,., where Pi, P2, . . • , P,. are power series. In other words, the primary modules into which a module of the principal class resolves are said to be of the principal Noetherian class. IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 79 Any prime module is of the principal Noetherian class ; but in general a primary module is snch that any module of the principal class which contains it determines a primary module of greater multiplicity. For example, 0^ is of multiplicity n + 1, but any module of the principal class of rank n containing 0^ contains a simple module at the origin of multiplicity 2" at least. If itf is a module of rank n the number of its modular equations is finite and equal to the sum S^tt of the multiplicities of its simple modules. In order that we may have F =Q mod M the coefficients of i'''must satisfy the S/x equations (which will not be independent unless i'^is of sufficiently high degree). Any set of Sju. linearly independent polynomials such that no linear combination of them is a member of M is called a complete set of remainders for M; and has the property that any polynomial F which is not a member of M is congruent mod M to a unique linear combination of the set of remainders. The simplest way of choosing a complete set of remainders is to take the polynomial 1 of degree 0, then as many power products of degree 1 as possible, then as many power products of degree 2 as possible, and so on, till a set of Sju. power products has been obtained of which no linear combination is a member of M. We shall call any such set a sim2?le complete set of remainders for M. li M=\_Ei, E2, ..., E^ is a simple Noetherian module no member E of the system [£',, iiz, .•-, A\] can have the same coefficients (assumed real) as a member F oi 31; for if E and F had the same coefficients the sum of their squares would be zero. Hence if the members of the system [Ei, E^, ..., Ei^ have their power products changed from negative to positive they will form a complete set of remainders for M. 69. A Noetherian principal system \E^ is uniquely expressible as a system [^] such that the polynomial F with the same coefficients as E is a member of the module [-E']/0. Let E2, E3, ..., Eij. be a complete set of hnearly independent derivates of Ei all of less absolute degree than Ei, and let Fi,Fi, ..., i^^ be the polynomials having the same coefficients &5 Ei, E.,, ..., E^. Then E2, E3, ■■■, E^ are the members of the system [E,'\lO={x^.E„ x,.E,,...,ii\.E,]; and F^_, Fs,---, Fy. is a complete set of remainders for the module [EiyO. Hence there is a unique F such that F=F^ + X.J\+... + KF^ = 0modi[E^'\IO. 80 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV The member E of {Ei~\ with the same coefficients as F is unique, and the system [£'] is the same as the system [£"1]. A homogeneous Noetherian equation is ah-eady in its unique form. 70. If E is homogeneous and of absolute degree I the numbers of linearly independent derivates of E of degrees V and I -I' are equal. Let Ei, E-i, ... , En be the members of the system \_E^ of degree l' , and Fi, F,_, ■■■ , Fj, the members of the module [£"] of degree I', and Gi, (to, ..., Gn the polynomials which have the same coefficients as El, E^, ..., En', so that Fi, ..., Fl, Gi, ..., Gjf form a complete set of linearly independent homogeneous polynomials of degree I'. Then the -f 1-, F2-, ■■■, i'i-derivates of E vanish identically, and the 6*1-, G^-, ..., 6rj\r-derivates are the derivates of degree l-l', and are linearly inde- pendent ; otherwise some linear combination of G^, G-'s,..., Gn would be a member of the module [.E*]. Hence the numbers of derivates of E of degrees l' and I -I' are equal. 71. The modular equations of a simple module Q of the princip)al Noetherian class consist of a single equation E=Q and its derivates; that is, a simple modide of the principal Noetherian class is a principial system (M, p. 109). Take the origin at the point of Q. Then the modular equations of Q are Noetherian, and the characteristic number 7 of Q is 1 more than the absolute degree of the highest modular equation. Also since Q is of the principal Noetherian class it is the whole Noetherian module contained in a certain module M={Fi, F'2, ■■■, F^ of rank n. By choosing the degrees /], 4, •••, 4 of i^i, i^a, ..., i^,i to be > y we may assume {F^, F^, ..., F,,) to be an iT-basis of 31 (§ 49). Now if F is any polynomial of degree 4 + 4+ ■•• +l,i — n-l such that XiF, X2F, ...,£c,iP are all members of 31 then F itself is a member. "We prove this for 2 variables referring for the general proof to (M, p. 1 10). When n = 2, we have XiF=A,F, + A,F„ x,F=B,F, + B,F„ where Ai, Bi are of degrees $4-2 and A^, B._ of degrees ^1^-2. Hence x, iA,F, + A,Fn) = x, (B, F + B, F,), or {x,_Ai-XiBi) Fj = {x^B.,- XnA^ F^, or x.2Ai-Xi^Bi = Q = XiBr. — X2A2, since x^Ai-x-^Bi is of degree <4 and cannot be divisible by F.2. Hence A^, A., are both divisible by x^, and F= mod {F^, F^). IV] THE INVEKSE SYSTEM AND MODULAR EQUATIONS 81 Suppose Q={Ei, Ei, ..., En], where each Ei is relevant, that is, not a member of the system [E,, ..., Ei^^, Ei+u ■■■, E,,]. Then the conditions that XiF, x^F, ..., x^F are to be members of M require only that the coefficients of F should satisfy all the derivates of Ei = E.2= ... = Eh = (but not these equations themselves) and all the modular equations of the other simple modules of 31; i.e. lik...ln-h equations in all. But these conditions require P=Omodil/, or that the coefficients of -F should satisfy all tlie /1/2...4 modular equations of M, which are equivalent to ^1 4- • ■ 4 - 1 independent equations as applied to i^ (§ 58). Hence the IJ^.-.l^-h equations as applied to i^ are equivalent to no less than IJ^.-.l^—l independent equations. Hence k = l, and Q = {E,'\. The converse of this theorem, viz. that a simple principal system is of the principal Noetherian class, is true in the case of 2 variables (M:,), but not true in the case of more than 2 variables. Thus \_Xi " + X.2 + X-i J = {^Xi X-2', Xi' tTa", A'2^'3, X-^Xi, X-iX-i) is a iiriucipal system which is not of the principal Noetherian class. 72. A module of the principal class of rank n is a principal system. Let [-£"1], \_E.^, ... , [£"„] be the simple modules into which the given module resolves, and -yi, 72, ■••,7a the characteristic numbers, and «i, a,) •••) «« the aycoordinates of the points of [-£i], \_E^,..., [_E„]. The given module \^Ei, E^, ■■., E^.] will be proved to be identical with [E^ + E,+ ...+E.]. Since x^ — «{ contains the spread of [Ei], (xi - a^''* is a member of the module [^j], §32, and {x-^- ai)''^ . Ei vanishes identically (§ 61). Hence from the equation i;\ + iil, + . . . + Ea = we have (x, - a,y'- (x, - a,y\. . (x, - a.)^« .E^ = 0. The operator on the left hand is a polynomial in .r, - «i in which the constant term does not vanish ; hence if we apply the inverse operator (xi-a.2)~''-...(xi-a^y'^'^ expanded in poAvers of (x^-a,) as far as («i-ai)^'"^ we shall obtain E, = ; since {x^-a,J.E, vanishes identically when / > 71 . Hence Ey, and similarly E-., E^, ..., E^, are all derivates of ^i+A'^+.-.+iJa and the given module [£'„£'„...,-£'J = [^, + £'o---.-^^o]. If M is a module of the principal class of rank r then j^Vl''' and all its simple modules are principal systems. Hence any module of the principal class, and its primary modules, are 2}rincipal systems (§ 82). 6 82 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV 73. If a simjjle module M^ of multiplicity /u, is a principal system [^], and M'^' is a simjjle module of multiplicity ft! contained in M^j., and M^jM\. = M"^:, then My.lM\.. = M'^., and /x + /jl" = fi (M, p. 111). The modular equations of 3IJ3I'^' are the i^-derivates of E=0, where F is any member of M'^. (Ji 62). Let F^, F«, ..., Fi^., be a complete set of remainders for i)/'^.. To these can be added i^fi'+i, • • ■, i^^ so that Fi, F'2, ■■■, F^ is a complete set of remainders for 31^. Also each of i^^+i, •.., F/j. can be modified by a linear combination of Fi, Fi, ..., Ffj.' so as to become a member of il/V' ; and we will suppose this to have been done. Then the Fij.'+i-, ■■■, i^w-derivates of F= are modular equations of il/'V"! and are linearly independent, since no linear combination of i^^'+i , . . . , i^^ is a member of 31^. Also any other i''-derivate of E^O, where F is a member of il/V' , is dependent on the ju. - (u-' equations already found, since F=KF, + \,F,+ ... + A^J; mod 31^, which requires, since F= mod 31'^; X^F, + KF.2 + ... + A^.i^^. = mod 31'^., or Ai = A„ = . . . = A^. - 0. Hence the ii^-derivate of 2^=0 is the (A^.+ii^^.+i+ ... + A^i-'''^)-derivate, and the number of modular equations of 7)/"^" is |U, - /j.', i.e. ju. = /a' + /*". Also since 31'^'3I"^'' contains il/^, 31' i^' contains 3I^j3I"^«\^'\i\c\\ is of multiplicity /x. — ij." = fj.' . Hence i)/ V' = 31^,131" ^.^ . It is true in general for unmixed modules of the same rank that if ilf is a principal system containing 31', and 31/31' = 31", then 31', 31" are mutually residual with respect to il/(cf. § 24, Ex. ii). In (M, p. 112) the opinion is expressed that if 31^ is any simple module of multiplicity /x, and il/V any module contained in 31^, then the multiplicity of 31^131' !,.■ cannot exceed /^-/x'. This is not correct, as the following example shows. Example. Let 31^ = {E„ E,] = [(a-i.rO-' + (x.a:,)-', (w.x,)-' + ix.A'e)-'], and i)/V' = («, , «2, 0") = [x;-\ x^-^, x,-\ .rr'], so that /x = 2 + 6 + 1 = 9, /x' = 4 + l = 5. Then 3IJ3r^.= [E„ E,]l{x„ x,) = [x,.Eu x«.E„ x,.E„ x,.E.;\ = [ii's , Xi , x^ , Xe J = 3J ij_'. Hence (since 31^1 31' ^. = 31' i^) 3f'^, and 31'^. are mutually residual with respect to il/^ ; and the multiplicity of 3I^/3I'i^- is /x' > /x - ^u,'. IV] THE INVERSK SYSTEM AND MODULAR EQUATIONS 83 It can be proved that if il/^ is simple and contains i)/'^' the multiplicity of 31^31'^' cannot exceed 1 + |(/a — /ji-')" or f + i (/^ - /«■')" according a,s /j.- i^.' is even or odd. 74. If a simple H-module M^ of multiplicity fx is a princijml system [i?] v)ith cha/racteristic number y, and if il/V', M"^« are mutually residual luitk respect to M^, and /xj, /x';, ix'i are the numbers of linearly independent modular equations of M^, M' ^^ il/'V- of degree I, then iJ.' I, + fji."i„ = ixi, = fj.i«, where r + 1" = y-l (M, p. 112). Here S is homogeneous and of absolute degree 7 - 1 ; and we have already shown that fx-i.^fxi,, (§70). The /a'V- modular equations of M"ij.- of degree /" are i^'-derivates of E=Q, where F' is a member of il/V' of degree l'. Hence /u,",- is the number of members F' of M'^' of degree l' of which no linear combination is a member F of i)/^ ; for F.E vanishes identically. There are fi-i- polynomials in all of degree I' of which no linear combination is a member of M^, and /u-V of these are such that no linear combination of them is a member of il/V', while the remainder /«,,.-/;. can be modified by the /u-V so as to be members of i)/'^'. Hence m'V' = ^i' - /^V, or ^V + /^"!" = ft' = f*r-. Thus the values of ft!'i are known for all values of I in terms of the values of i^i and /^'j for all values of I. lb. If M is any module of rank n in x^, x^, ••■ , x^, and Mo the equivalent H-module in x^, ■■■, a:,,, x„, and ij.„, the number of modular equations of{M^xo=f> of degree m, then the number of modular equations of Mfm- degree m is This is immediately seen by considering the scheme of § 59 carried as far as degree m. The number of rows in the compartments 0, 1, 2, ..., Z- 1 of the inverse array is the number of power products of degree ^ Z-1, and each such power product inverted represents a modular equation of (il/o):,„_o- This number is therefore 1 +^j + /x2+ ... +/i;_i. The numbers of rows in the succeeding com- partments are /u,, i^i^,, ...,/-,„ ; and H,„ is the total number of rows, viz. 1 + yUi + /^2+ ... +Mm- Also the total number of modular equations of M, or the sum of the multiplicities of its siinple modules, is equal to the multiplicity of 6—2 84 THE ALGEBRAIC THEORY OF MODULAR SYSTKMS [iV 76. If M' is any module of rank n and /i' the sum of the multi- plicities of its simple modules, we can choose n members F^, F^, ■■■, Fn of M' such that the resultant of their terms of highest degree does not vanish. If then the sum of the multiplicities of the simple modules of M= (F^, F^, ..., F,^ is n- the sum of the multiplicities of the simple modules of il//il/' is /i-ya' (§§ 71, 73), and if MjM' =M" theu MjM" =M'. The important point is that M' is unrestricted except that it is composed of simple modules. The simple modules of M are principal systems, but not those of M' . These remarks are intended to point out the generality of the following theorem. If (Fi, Fs, ■••, Fn) is an H-basis of a module M of rank n, and M' any module contained in M, and M" the residual module MjM' , then M', M" are mutually residual with respect to M, and H'i~ H"v = H'l'+i" — Hi'i = Hi' — H"v + ;», wh&re V + 1" +n+l is the sum of the degrees of F^, F-2, ■•■ , F,i, and Hi, H'l, H"i are the numbers of modular equations of M, 31' , M" for degree I. This gives the values of H"i for all values of I in terms of the values of H'l for all values of I ; for Hi is known by § 58. The theorem is a generalization of the Brill-Noether reciprocity theorem (BN, p. 280, §5, "Der Hiemann-Roch'sche Satz"). It ex- presses the reciprocal relations between the numbers of the conditions which must be satisfied by members of iW and 31" in order that the product 3T3I" may contain 31. A somewhat more general theorem is the following : If {Fi, Fs, ... , F,c) is an H-basis of a module 31 of rank n such that the H-module determined by the terms of highest degree in Fi, F2, ■■■, Fi- is a principal system with characteristic number y, and if 31' is any module contained in 31, and 31" the residual module 311 31', then 31' , 31" are mutually residual with resjoect to 31, and H'l' - H"i« = H'l'+v -Hr = Hv - H"i'+i", where l' + 1" = y- 2. We shall prove this more general theorem which includes the other. We must prove first that the simple modules of 31 are all principal .systems*. Let 3I„, 31^, 3Io" be the ^-modules in x^, x^, ..., x„, x^ equivalent to 31, 31', 31". Then (i)io)j;^=o is a principal system ; and * The converse that if J/ is u module of rank n whose simple modules are all principal systems (-1J'o):c„ = o is a principal system is not true. For example, if M is the module in 2 variables determined by 3 points in a plane, then (jI/qJj. =0 has the modular equations a;j~'=.ro~' = l = 0, and is not a principal system. IVj THE INVERSE SYSTEM AND MODULAR EQUATIONS 85 the multiplicities ^, ^', ,." of (7l/o).x,=«, W)x„=-o, (il/o").„-=o are the sums of the multiplicities of the simple modules of M, M', M" (§ 75). Let Q' be the module determined by the ^ points forming the spread of BI, and Q' the residual module Miq. Also let Qo, Q„" be the ^-modules equivalent to Q', Q". Then, since Q'Q" contains 71/, Q„'Q„" contains Mo, and (Qo'Q„"\^o contains (3fo\=o, which is a principal system. Hence also (Q„").„.o contains (il/„),x.„ _„/(§„'):.„. o whose multiplicity is /A -a; i.e. the sum of the multiplicities of the simple modules of Q"^fx- a. This is only possible when the simple modules of 31 are all principal systems ; for if [£",,... , E„] is the simple module of 31 at the point P (say), the corresponding simple modules of Q', Q" are P and [E,,K,..., Bh]/P, and the multiplicity of the latter is h less than that of [E,,Bi, ...,!;„]; so that yn - S/* > /v. - a, ^h%a = a, and h=l. It follows that 31' and 31" are mutually residual with respect to 31. It also follows that /x = /t' + ju", and that (il/„')^„=o and (3Io"\^o are mutually residual with respect to (3f„%^^„. Hence At'r+i+/'i- = ft.+i =/ir' (§74). Also ff'r=l+fi,' + ix^'+...+ /, . (§ 75). Hence (/7V.r--irV) + 5-'V- = (mV+i+j".V+2+ ••• +/r + j") + (l + Ml" + /"="+ ••• + Ai'V') = 1 + (/r+r + /J-i") + (/r+!"-i + A^-j") + ■■• + (/^V+i + m'VO = 1 +/ii+/i2+ ... +ni.. = Hv; i.e. /TV - ^"," = H'r^r - Hr = ^r - fl"'r+,". Modular Equations of Unmixed Modules 77. We have hitherto specially considered modules oj" rank n, that is, modules which resolve into simple modules. The J?-module of rank n is of a special type, since it is itself a simple module, and its equations are homogeneous. The general case of a module of rank n is therefore that of a module which is not an iiT-module. When however we consider a module of rank < n it is of some advantage to replace it by its equivalent /f-module, which is of the same rank but of greater dimensions by 1. We shall not avoid by this means the consideration of modules which are not jET-modules, but the results obtained will be expressed more conveniently. We shall therefore assume that the given module 31 whose modular equations and pro- perties are to be discussed is an .ff-module in n variables .Tj, ci\, ..., x,,. 86 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV By treating any ^-module ilf of rank r (whether mixed or unmixed) as a module il!/'''' iu r variables x-^, x^, ..., Xr'\'^ will resolve into simple modules and have only a finite number of modular equations, viz. a number /a equal to the sum of the multiplicities of its simple modules. The unknowns in the modular equations will be represented by negative power products oi Xi, x.^, ..., a^,. while the coefficients will be whole functions of the parameters a;,.+i, ■■■, x^- The module determined by these modular equations will be unmixed, viz. the l.c.m. of all the primary modules of M of rank r (§ 43) ; and will be the module M itself if M is unmixed. We proceed to discuss these equations and shall call them the r-dimensional modular equations of ilf (or the modular equations of ilf'' ') since they are obtained by regarding the module M as a module it/'''' in space of r dimensions. AT''"' is not an iZ-module. The dialytic array of ilf' '"'. We choose any basis {F„F^, ..., t\) of M as the basis of M ''''. This is not in general an //"-basis of 3P'\ The module Mx,. j=..,=a;,t=o determined by the highest terms of the mem- bers of the basis of ilf''' is of rank r (assuming that Xi, x^, ..., Xn have been subjected to a linear homogeneous substitution beforeliand) and is therefore a simple ^-module whose characteristic number will be denoted by y. Construct a dialytic array for J/'''' whose elements are whole functions of «,.+i, ■■■,x,i in which each row represents an elementary member wiFj of ilf'', where Wi is a power product of x-^, x^_, ..., «,. (cf § 59). The first set of rows will represent the members of the basis which are of lowest degree I, the next set a complete set of elementary members of degree Z + 1 which are linearly independent of one another and of the complete rows in the first set, the next set a complete set of elementary members of degree Z + 2 linearly independent of one another and of the complete rows in the first two sets, and so on. In comparing this with the scheme of § 59 there is the obvious difference that the elements of the array are whole functions of Xr^\i •••, ^11 instead of pure constants ; and there is the more important difference that tlie compartments I, I +1, ... do not necessarily consist of independent rows, because the array is not constructed from an H- basis of J/'''. It is only the complete rows of the array that are inde- pendent. The elements in the compartments are all pure constants independent of Xr+i, ■.., Xn- The diagram of § 59 serves perfectly well to illustrate the dialytic array although its properties are now different. IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 87 111 each compartment we choose a set of iiadependent rows such that all the remaining rows of the compartment are dependent on them, and we name them regular rows and extra rows respectively, and apply the same terms to the complete rows of which they form part. In the compartment y the regular rows will form a square array, and the same will he true of the compartments y + 1 , y + 2, .... Eventually a com- partment S > y will he reached such that the number of rows in the whole array for degree 8 is exactly /a less than the whole number of columns, where /a is the number of modular equations of il/'""' as men- tioned above. After this all succeeding compartments 8+1, 8 + 2, ... will consist of square arrays only without any extra rows. We can now modify any extra row of the array by regular rows so as to make all its elements which project beyond the columns of degree y— 1 vanish, and this leaves its elements in the columns up to degree y— 1 whole functions of «,. + i, ...,x,iOl the same degrees as they were before. If this is done with all the extra rows projecting beyond the columns of degree y — 1 the array may be said to be brought to its regular form in which the whole number of rows of the array for degree y- 1 IS, fA. less than the whole number of columns, and all the compart- ments y, y + 1, ... are made square. The extra rows, modified so as to end at the columns of degree y- 1, represent members of ilf'' of degree y - 1 which are not elementary members m, .?}. We may further modify the regular form of the complete array for degree y- 1 so as to reduce the number of rows in each compartment y-1, y-2, ... successively to independent rows. The elements of some of the rows of the array for degree y-1 may thus become frac- tional in ^,. + 1, ..., Xn, and the whole number of compartments will in general be increased, so that the last (or first) compartment will be numbered r consisting of all power products of ,ri, ^2, ■■■,Xr of degree l' as the number of columns of the compartment I" exceeds the number of rows of the same. We denote these power products in ascending degree by m-,, o>., ..., m^ (so that u)i = l) and all remaining power products to infinity in ascending degree by w^ + i, w^+2, ■■•■ The two series coi.w^, ■■■,<^^ and w^ + i, u)^ + o, ... overlap in respect to the degrees of their terms. The basis of M used for constructing the dialytic array of vlf') must be one in which each member is of the same degree in Xj, x.^, ..., x,. as in x-i, X., ... , x,^. We shall say that M is a perfect module if the array 88 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV of il/l'' as originally constructed has no extra rows, i.e. if the basis (F„ F,, ..., F,) is an iT-basis of 3P\ 78. Solution of the dialytic equations of J/'''. We return to what has been called above the regular form of the dialytic array of i)/'*''. Each row represents a member of il/'''' and supplies a congruence equation mod il/''''. Solving these equations, regarding fO/i+u <"n+i>, •••as the unknowns, we have D<^j, + -Dpi 0)1 + Z>ja<"-2+ ■•• +2>j,wWn = 0mod AT'''' (p = t^ + i, h-+'2, •••)• There are two slightly different cases according as the degree of 0)^, is then the determinant of this array formed from the columns corresponding to wm-i, w^+i, •••, and Dj,i the determinant formed from the columns corre.sponding to (u^+i, •.., (Dj,_i, (D,-, a)j,+], — If Wp is of degree > y we must use the array up to the degree of Wj, in order to solve for Wj,. D is the same as in the former case except for a factor independent of av-n, •■■! *'« (since the compartments y, y + 1, ••• are square and all their elements are pure constants) by which the equation can be divided. Also Djn is a sum of products of determinants of the regular form of the array for degree y — 1 with determinants from the remaining rows of the larger array, so that the H.c. f. of the determinants of the array for degree y - 1 can be divided out, and we obtain in both cases (A) Ru>j, + Epi<>)^ + ...+Ej,^(Oij, = OmoA3I'-'"' {p = fJt. + \, ix+2, ...). This equation is homogeneous in a^i, a'^, ..., *-,„ and each Rpt is homogeneous in a,v.fi, ■••, .r„. Also, owing to the fact that the re- mainders (Di, (1)2, ..., 0)^ are a simple set, each oip is congruent modil/'''' to a linear combination of those power products wj, - Rpn<^ij.) mod J/'*''. p=IJ-+l IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 89 Here coefficients of Wi, (u., ..., to^ on both sides are equal, i.e. (B) R'^-i= 2 i?,,.cu_, (v = l, 2, ..., ^). 7%2S M ^^e complete system of modular equations of M^^'\ or r-dhnen- sional modular equations of M, and the system includes all its oivn derivates. R and all the Bpi are definite ivhole functions of a',.+i , . . . , x„ . If any other complete system were given and solved for co_i , lo.j, . . . , (o,.,^ in terms of w_|^_i, (o_^_2, ... the result would be the unique system (B). Since in (A) Rwp and i^pjiOj are of the same degree in cCi, x^, .. ., x^, so in (B), Rny_^ and Iipiw_p are of the same degree, i.e. all terms in one equation (B) are of the same degree in a\, x«, ..., a-,,. Also since (§ 78) Epi vanishes if the degree of _j ; but every ui-p of the same degree as w-; and not among w-i, (o_2, ..., iti_^ will appear on the right-hand side of (B). (B) is the complete system of r-dimensional equations of the L. cm. of all the primary modules of 3T of rank r ; and will decompose into separate distinct systems corresponding to the separate primary modules of rank r if Mha,s more than one irreducible spread of rank r. The n-dimensional equations. We can obtain the whole system of !?-dimensional equations of M corresponding to the system (B) as folloAvs : <•)_,, or (xi'^xi'^...x,?'')''^ represents the whole coefficient ot x^'x-f^.-.x,!''' in the general member of i1/'''', i.e. it stands for 5 C^ Pi ^ Vn\-l „Vr-\l Pn ^y'i'i ■■-It'll ) r+l •'•*^'« > the summation extending to all values of ^;,.+i, ■.., j),, only. If this be substituted for each {x^^ ...Xy'')" in each of the equations (B) the whole coefficients of the power products of x,.+-i, ..., x,, will represent the ^-dimensional equations. This loill be the whole system of n-dimensional equations of M if M is unmixed, as tve shall assume liereafter is the case. The whole system of modular equations of a mixed module may be regarded as consisting of the separate systems corresponding to the primary modules into which it resolves. 80. The system of homogeneous equations (C) R">~i = ^Rpi'^-v {i=h 2, ..., /x) obtained from the system (B) by retaining only those terms on the right hand in which Bpi and oi-p are of the same degrees as R and «_,• 90 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV respectively is the complete system of equations of the simple H-module determined by the highest terms in x^, x^, ..., x,. of the members of an H-basis of M^'^K This can be seen by considering the diagram of § 59 assuming that it had been constructed from an ^-basis of il/''''. The compartments I, 1+ 1, I + 2, ... in the two arrays in §59 are the dialytic and inverse arrays of the simple ZT-module determined by the highest terms of the members of the ^-basis ; and the modular equations of this simple /^-module are represented by the compartments 0, 1, ■■.,/, /+1, ... of the inverse array. The system (C) is that which is represented by the compartments of the inverse array. 81. If R = l the module M {assumed unmixed) is perfect. Since J/ is unmixed every whole member of i)/'"'' is a member of il/(§43). Also, since ^=1, there is an inverse array of J/'''' each of whose compartments consists of independent rows in which all the elements are pure constants. Hence there is a corresponding dialytic array having the same property. From this it follows that M is perfect (§77). 82. The r-dimensional and /7-dimensional equations of M. If the system (B) is a principal system, i.e. if all its equations are derivates of a single one of them, each simple module of i)/''"' is a principal system ; for if i?" is a polynomial containing all the simple modules of J/'''' except one, then J/ ''''/( Z^) is the last one, and is a principal system (§ 62). The converse is also true (see § 72). Also the unmixed module M in n variables is a principal system, as we proceed to prove. Let the ^--dimensional equation of whicli all the equations of the system (B) are derivates be where i^j,^, p^ ,j,, is a homogeneous polynomial in ir,.+i, ..., a'a of degree ^i +JO2+ -•• +^"1+8. The integer 8 may be negative, but the more unfavourable case for the proof is that in which it is positive. Let Cj,j_j,, ^^ be the coefficient of ar^'^^ ... .r/" in -K^^.^j^, ...,jj,., so that jOr+i + ... +jo„=^i +...+/>,. + 8. To convert the equation into an n- dimensional equation we put 1 IV] THE INVERSE SYSTEM AND MODULAR EQUATIONS 91 as in § 79, and we have or, equating the whole coefficient of .r'l'^J . . . xj" to zero, p which is homogeneous and of absolute degree /,.^.i+ ... +4-8. Similarly the general «-dimensional equation obtained from the coefficient of .t™;+' ■ • • «, J"" in the «/'... ,r,''-derivate of (1) is 2c,,.j,,,....p,,(^/''-^..^/''-''-<';«-^'-^^...r;"»-''")----0, (3) „ '■+! where ^,, ..., t,., in,.+i, ..., m^ are any n fixed positive integers (in- cluding zeros) such that rincipal system. For the basis equation of the system (C) must be the homogeneous equation and all the other equations of (C) must be of less absolute degree. Now the system (B) is unique and any equation obtained from it Bl U)_i + i?2W_2 + . . . + i?pa)_p + . . . = must be theresult of multiplying the equations of (B)by^^,^,,-i, ■■■,Bi and adding and dividing out R. Hence the equation 00 Rui-i= 'S,Rj,t-p is exactly the same derivate of Z2ii)_^ = 2^j,^.co_p as the corresponding homogeneous equation JSa)_i = S/i!^i Zi + ... +l,. — r, and consequently Pi > li for one value at least of i. The corresponding ?j-dimensional equation is (§ 82) -y* = ^Kp^^p^ p^ {w,^'x:k..x:^y\ ...(5) p where p-t^+po_+ ... +p,i = Zj + 4 + • • • + 4 - «, and at least one p-, > U (/=!, 2, •..,?•) and every pj 4 being unspecified numerically. More ex- plicitly it is the unique modular equation of the simple module (Fi, Fo, ..., F,., «'r+i, ■•■, 'T,/"); for it is a relation satisfied by the coefficients of the general member of (Fj, F^, •■-, Fr) of degree /i + 4 + ■ • • + 4 - '» in which Pj< Ij (j -r +1, ..., n), i.e. it is the unique relation (§ 58) satisfied by the coefficients of the general member of of degree 1^ + ... +l,i-n. The coefficients Kp^^p^ p^^ are whole functions of the coefficients of Fi, F^, ..., Fr of a similar kind to the resultant of (Fi, ..., F,., x^^'l\' ■■■' *""'") ^"^^ °^ degree 1 less than this resultant in the coefficients of each of Fi, F^, ■■■, Fr, viz. of degree Li- 1 in the coefficients of Fi where LJi = lj.2...l,i = L. The vanishing of Kp ,,,.^ ;,^j is the condition that ^,'''^/^..a.-/" = Omod(i^„ F„ ..., Fr, x';i\, ..., xj'^) (§ 61, since the «i^'a:.2^-...a;/"-derivate of (5) then vanishes), whereas the non-vanishing of the resultant is the condition that every power product of degree 1^+ ... +ln-n+l is a member of the module. It is probable that some of the quantities Kp^,p^ p,^ factorise but that they have not all a common factor. The resultant of {F,F.,, ...,Fr,x;:^\,-,x,h is 4':v'" (§8). r+l The 85. Whole basis of the system inverse to il/'''. simplest wM6< basis [£"1, E., ..., F,,] of the r-dimensional system inverse to an unmixed i^-module iJf of rank r, or the simplest expression for the system of equations (B), satisfies the following conditions : (i) each Ei (i=l, 2, ..., h) is a tvhole member of the inverse system, i.e. its coefficients are whole functions of the parameters .Zr+ + 1) •••) ^7i 3 94 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV (ii) all the members E^, E^, ..., En are relevant; (iii) any whole member of the system [E^, E.2 ii",,] is of the form X,.E, + X,.E,+ ...+Xn.Eu, where X^, X^, ..., X^ are whole functions of «,.+!, ■••, Xn as well as of Xi, X2, ..., Xr] (iv) El, E2, ■■■, En have as high absolute under- degrees in x^, x^, ..., x,. as possible. A whole basis, as distinguished from a simplest whole basis, is defined by (i) and (iii). A basis (Pi, F^, ..., Fi) of il/ furnishes a whole basis of If"', and any whole basis of J/''' satisfying the condition corresponding to (iii) above is a basis of M. A simplest whole basis* of il/'''' is one in Avhich the degrees of Fi, F^, •■•, F^'mxi, x^, ..., x,. are as low as possible. If (Fi, F2, ■■■, Fr) is any module of rank r containing 31 such that (Fi, -F'a, ..., i^,.)x,.+i=...=c«„=o is of rank r, and 3f= {Fi, F^, ..., F^), and the degrees of i^,-+i, •••, i*^;- in Xi, x.2, ■■■, x,- are as low as possible, the basis {Fi, F^, ■■■, i*!') will be called a whole basis of il/'''' in reference to (Fi, F^, ..., F,). All of jP,.+i, •■■, -^V are to be relevant, but some or all oi Fi, F^, ..., F,- may be irrelevant for a basis of 31. 86. Properties of //-modules mutually residual with respect to an //-module of the principal class. Let Fi, Fi,.-., Fr, of degrees li, 4, •••, l,-, be any r members of the unmixed ZT-module 31 of rank r such that is of rank r ; and let 31' be the residual module (Fi, F^, ■■■, F,.)I3I. Also let (Fi, ..., F; F',.+i, ..., F'r^n) be a whole basis of J/'C' in reference to (F^, F^, .-■, i^,-) = [-£]• Since -F',.+i is of as low degree in Xi, x.2, ..., x,. as possible the terms of i^V+i of highest degree in Xi, X2, ..., X,. do not form a member of the module and are therefore of degree /',.+» ^li + I2+ ■■■ + Ir — r in Xi, x^, ..., x,.. Also, since E begins with terms which represent the modular equation " A simplest whole basis of il/C' is a whole basis which approaches most nearly to an ff-basis ; but is not necessarily an H-basis. For example, (.Ti'', Xi^X2, XiX^}, X2*, X'^X-^ + X^X^Xi, X^ X^ + X-^x{-X^ is the basis of a module jl/ of rank 2, and a simplest whole basis of Jl/(2), but not an H-basis of il 121 ; since x^x-^ - xi , 4>-l be general mem- bers of M, M', Ml, M{ with coefficients involving linear parameters. Then i^'i^/=OmodiVi»/' = Omod(i?'i, F., ..., F,) = FFimoi(F„ ...,Fr), (1) where i^ is a polynomial of the same degree as Fi, Fi, F'. Also Fii=Qmod.MMi = XiF;mQA{F^, ...,Fr), (2) and ^^i' = mod MMi = XF{ mod {F.^, .■.,Fr); hence by cross multiplying and dividing out t^/i^/, Xi = XFimoA{F„ ..., F) = Omod(Fi, F,, ..., F,), .-. X; = mod (i?'„ F,, ..., F)!iV=OmodM'. Similarly F't>i =0mod3r3Ii = X'F' moA{F.^, ..., F,.), (3) where X' = mod 31 ; .-. X'Xi = mod 3I3I' = XFimoi{F„ ..., F,) (4) Multiplying (1), (2), (3), (4), and dividing out F'F^Fi^XXi, 4>iF be a member of M, where <^ is a whole function of the parameters only. Then i^ is a member of M '"■' and if we insert a row in the array representing i^it will be dependent on the rest, i.e. where Fi, F«, ..., Ff, are the members of i)/'''' represented by the rows of the array, and Xj, Xj, ..., Ap are rational functions of x,.+i, ..., Xn- Equating coefficients on the two sides of power products of Xi, Xo, ■■■, x,. corresponding to the columns of the determinant D mentioned above, and solving for Aj, X,, ..., Xp, we see that XjZ) and consequently X; is a whole function of «,.+i, ■■■, Xn- Hence i^ is a member of M, since F-i, F2, ■■; Fp are all members of M; and ^i^=Omodilf requires F= mod 31. Hence M is unmixed. If Mis a perfect module oj rank r and M' a module in Xr+i, ■■., x,, (indepiendent of x-i, x^, ..., x,) the l.c.m. of M, M' is the same as their p-oduct MM'. For if the F above is a member of the l. c. m. of M, M' the elements in the row representing F are all members of M', and the Xj are linear functions of them and therefore also members of M' HfincG i?'=2Xii^i = OmodilOf', i.e. [M,M']=MM', since X^ =0 mod ilf' and i'"'i = mod ilf. 91. The number Hi of modular equations of degree I of a perfect H-module M of rank r is the coefficient of a^ in (1 + yUi« + /x,^'^ + ... + iXy-ix'^~'^) (1 -ir )'■"", where y is the characteristic number, and /i^, the number of modular equations of degree on, of the simple module Mx,.+^=...=xn=o- For the general member of M of degree I is (§ 90) KFi + KF.+ ... +\Fp, where K, K, •••, \ are whole functions of x,.+i, •■•, x.,„ and cannot vanish identically unless Xj, Xj, ..., Xp all vanish identically. Hence 100 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [iV the number of linearly independent members of M of degree I is the total number of terms in \, Aj, •••, \- Now the number of the polynomials F^, F2, ..., F^ which are of degree m is /i,„ less than the number of power products of Xi, x^, ..., «,. of degree m, and the number of terms in each corresponding X (of degree l — ni) is the coefficient of «' in a;'" (1 — a;)"""" Hence the number of linearly independent members of M of degree I is less than the number of power products Qi Xx, x^, ..., Xn of degree I by the coefficient of a^' in (1 + /a,* + ft.^x' + . . . + fn-ix'') (1 - a^)''~" ; and this coefficient is the value of Hi. § 75 is a particular case. 92. If M is a perfect H -module of rank r such that the simple module Mx^. j=...=a^i=o is a principal system, and M' a perfect H- module of rank r contained in M, the module MjM ' is perfect. The ju. and /u' r-dimensional modular equations of M and M' begin with the ix and jj! modular equations of ili!r,.+i=...=a;„=o and M'x,. i=.-.=a:„=o- Also the /i— m' r-dimensional modular equations of MjM' are the i^'-derivates of the modular equations of if, where F' is any member oi M', and begin with the i^'a:,,^j=...=,v,.=o-derivates of the modular equations of ilfa;^+j=...=x„=.o, that is, with the modular equations of il/x,.^i=...=i„=o/if 3:^+1=... =a,-,i=o- These are /a-/^' in number, since il/x,,^j=...=a:„=o is a principal system containing i)/'.rr^,-...=rr„=o- Hence MjM' is perfect (§81). 93. We may sum up some of the relations between different kinds of modules. A module of the principal class is unmixed and a principal system, and in the case of an iiT-module is perfect. Any power of a module of the principal class is unmixed, and in the case of an ZT-module is perfect (§ 89, end), but is not a principal system ; e.g. {x-i, x^y is not a principal system. A module of rank k — r + \ whose basis is a matrix with r rows and k columns is unmixed, and in the case of an i^^-module is perfect ' Xi X, (§ 89, end), but is not a principal S3fstem ; e.g. the module . is not a principal system. A primary module of the principal Noetherian class is a principal system, but not perfect. NOTE ON THE THEORY OF IDEALS The following is a brief explanation of the theory of ideals of algebraic numbers* and functions and the relation in which the theory given in the preceding pages stands with respect to it. Gauss {Disquisitiones Arithmeticae (1801)) was the first to consider the laws of factorisation in a domain of whole numbers other than that of rational whole numbers 0, ±1, ±2 He proved that two given complex whole numbers a + b >J —I, c + d J — I (a, 6, c, c/ rational integers) have alwaj's an H. c.F. and that any such number is a unique product of prime factors. Kummer {J. reine angeiv. Math. 35 (1847), 40 (1850), 53 (1857)) in extending the research to a larger class of whole numbers found that these properties were no longer absolutely true. Nevertheless he succeeded in making such numbers amenable to all the simpler laws of rational integers by introducing certain ideal members not existing in the domain considered ; and thus laid the foundation of the theory of factorisation of whole algebraic numbers. Finally Dedekind (D), by using ideals instead of ideal numbers, extended the theory to the whole numbers of any algebraic corpus and to whole algebraic functions of one variable (DW) ; while Kronecker (Kr) extended the same theory of factorisation to algebraic functions in general. Kronecker went still further ; he gave the first steps of a general theory of ideals of algebraic functions (Kr, p. 77) under the name of modular systems. In this general theory factorisation plays only a subsidiary part, since an ideal which is not prime is not in general a product of prime ideals. Modules of whole rational functions (as defined pp. 1, 2 above) are ideals and modules in the sense of Dedekind ; and the theory of such modules is the necessary starting point of the general theory of ideals. An algebraic number is any root a of an algebraic equation " The following are notable general accounts of the theory of algebraic numbers : D. Hilbert. "Berioht iiber die Theorie der algebraischen Zahlkorper" [Jahrcsh. d. deutschen Math.-Verein., Berlin (1897), Bd. iv). H. Weber. Lehrbuch der Algebra (Brunswick, 2nd ed. (1899), Bd. ii, p. 553). G. B. Mathews. " Number" {Enctj. Brit., Cambridge, 11th ed. (1911), Vol. 19, p. 847). For other references to the arithmetic theory of algebraic numbers and functions see (D), (DW), (K), and (Kr), p. xiii. 102 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS in which the coeflSoieiits ffloi o'l, •••, «m are rational integers. We may suppose that oSq is positive, and that ao, ai, ...,a^ have no common factor other than 1, and that the equation is irreducible in the rational domain. There is only one set of values of a,,, oii, ...,«,„ satisfying these conditions for an assigned a. a is called a whole (algebraic) number if ao = ^, and is called a, fractional number if ao=t= 1. Thus an algebraic (as well as a rational) number is integral or fractional, but cannot be both. In any case aoa is a whole number j3, and a=/3/ao, i-e. the denominator of any fractional algebraic number a can be rationalized, while the numerator remains a whole algebraic number /3. All roots of any equation x'^ + CiXf^~^ + ... + Cn = (whether reducible or not) in which Ci, C2, ... , c,i are rational integers are whole algebraic numbers. For all irreducible factors of the left-hand side are of the type a;"' + ai«™-i + ...-)-«„. We omit the proof of this as of most other properties to be stated. Hence any number is whole if it satisfies any equation of this tj'pe. If a, j3, y, ... are whole numbers a + /3 and a(3 are also whole numbers (D, p. 145); and so also is any whole rational function of u, /3, y, ... with rational integral coefiBcients. A whole number a is said to have another /3 as a factor, or to be divisible by /3, if a=)3y, where y is a whole number. A whole number e is called a unit if it is a factor of 1 ; or f is a unit if e and 1/f are both whole numbers. Thus if in the above equation ao= +a,„ = l all its roots are units. Two whole numbers a, 13 are said to be eqidvalent (as regards divisibility) if a = f/3 where € is a unit ; for then any whole number which divides either a or /3 divides the other. Such equivalence of a, fi is denoted by a ~ /3. A corpus of algebraic numbers is the aggregate of all rational functions (with rational coefficients) of any finite set of given algebraic numbers ai, 02, ..., nj;. All numbers of the corpus are rational functions of a single element a = Cjai-|-Coa2+...+Ctai., where c-^,ci, ..., Cj.. are rational integers so chosen as not to be connected by special relations. The corpus generated by a is denoted by n (a) and the aggregate of alge- braic integers included in the corpus by a (a). The order of the corpus and of a is the degree of the irreducible equation of which a is a root. Thus G(l) is the corpus of rational numbers and a (1) the aggregate of rational integers. Any rational function of any finite number of elements of ii (a) is an element of O (a), and any whole rational function with rational integral co- efficients of any finite number of elements of a> (a) is an element of a> (a). Any corpus fi(a) includes O (1), for aja—l. If aQa"'-|-ai.r'"~'-l-...-t-a,„ = is the irreducible equation of which the element u of the corpus Q. (a) is a root, the other roots d, a", ... , a("'~') are NOTE ON THE THEORY OF IDEALS 103 called the conjugates of a, and Q (a), ... , Q (al™"')) the conjugates of Q(a). If a is an element of Q. (a) then Q (a) is the same as Q, (a), and if not, not. The corpus generated by a, a, ..., a('"~') is called the Galoisian domain coiTesponding to il (a). The conjugates of any number ^=f{a) of Q (a) are ^'=/(a'),...,j3('"-')=/(a("'-')). The product 0/3' ... /3(™~^) is a rational number (being a symmetric function of (I, a', ..., a("'~')) and is called the norm of and written norm 0. Since and norm are both numbers in fi (a), norm /3/0 is a number in JJ (a). Moreover if is a number in a> (a), then |3', ..., j3("'~') are whole algebraic numbers, and norm /3//3 is a number iu {a). m — \ Norm (aM + 0w+...) is defined as n (a(')M + /3('>v +...), «, ?),... being inde- i=o terminates. O (a) is a domain of rationality. &> (a) is called a proper holoid domain (Kiinig), that is, a domain in which every sum, difference and product, but not every quotient, of two elements is an element of the domain. A proper holoid domain in which every pair of elements a, have an h. c. f. in the domain (defined as a factor S of a and of j3 such that every common factor of a, /3 is a factor of 8) is called a complete holoid domain, a (a) is not neces- sarily complete. The simple.st example of this last statement is the domain a>{J — b) which is fully discussed by Dedekind (D, p. 73). li x=a + h J -b {a, b rational) then {x-aY + 5b^ = 0, and in order that a; may be whole 2a and cfi + 5b^ must be rational integers, i.e. a and 5 must be integers. Consider the two whole numbers 9, 3 (1 + J'^). If these have an h.c. F. in co (V - 5) it must be 3S, where 8 is a whole number in d^ is 9 and the other 1, since neither can be 3. Also if a2 + 562 = 9 the only solutions are a= ±3, 6 = and a= ±2, 6= ±1 of which the latter must be rejected since ±'2.±J-b does not divide 3. Similarly for 1 + J^^. The numbers 3, 1 + sT-b have however a common factor (1 + J^)lJ2or J -2 + J'^ not in to (V - 5). Another point requiring notice is the distinction between a uon- factoriSable number and a prime number. A non-factorisable number in D) (a) is one which has no other factors in a («) than such as are equivalent^to 104 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS itself or 1. A prime number is one which cannot be a factor of a product ^y without being a factor of /3 or of y. Thus 3 and 1 + V -5 are both non- factorisable in a)(V — 5), but neither of them is prime; 3 is a factor of (l + «y-5)(l — \/ — 5) but not a factor of i+^^~5 or 1-^-5; and 1 + V — 5 is a factor of 6 but not a factor of 2 or 3. In a complete holoid domain every non-factorisahle element is prime and every prime element is non-factorisahle (K, p. 15). Let n be any element of the domain which is non-factorisable, and let o3 be divisible by tt and /3 not divisible by n. Then the h.c.f. of j3, 7r~l ; E.G.]?. of a/3, OTT ~a ; H.C.F. of a/3, n ~ H.C.P. of a;3, ott, 7r~ H.C.F. of u, tt ; i.e. H.C.F. of a, TT ~ TT, or a is divisible by -n ; hence -w is prime. Again if TV is prime and equal to 7ri7r2, one of ttj, tr^ is divisible by tt and the other is a unit ; hence -n is non-factorisable. It is to be noticed that the proof depends only on the notions of product, quotient, and h.c.f., and is therefore applicable to any domain in which each pair of elements a, /3 has a product a0, and an h. c. f. S (defined as above), and may or may not have a quotient y, defined by a=/3y. In a complete holoid domain any element which is not an infinite product of factors {not counting unit factors) is a unique product of prime factors if equivalent factors are regarded as the same factor. For any element which is not prime is a product of two factors neither of which is a unit ; each of these again if not prime is a product of two factors, and so on. Hence any element which is not an infinite product is a product of prime factors pi'pj" ...p,!''- This resolution into factors is unique in the sense of equivalence; for if pi ... p,.'''~?i"'' ••• qj"'^ where qi, q«, ..., q^ are primes and none of them units, p^ must be a factor of q-^ or q^ ... or q^, and if a factor of q^, then Pi~qi\ from which the rest follows. The domain of all algebraic integers is a complete holoid domain (D, p. 247) but contains no prime numbers, since any number a has an infinite number of factors, e.g. ^a. Tliis property of completeness is peculiar to numbers ; it does not hold for functions, not even for relatively whole algebraic functions of a single variable. No number in a, (a) can be an infinite product, for otherwise its norm, which is a rational integer, would be an infinite product of rational integers. Hence if a (a) is complete each number in it is a unique product of prime factors. All the above remarks concerning algebraic numbers (with the exception noted) apply mutatis mutandis to algebraic functions. The only difi'erence is that there are two kinds of whole algebraic functions, relative and absolute. An algebraic function is anj' quantity u which satisfies an algebraic equation AoZ«' + Aiz"'-'^ + ... + A^ = in which the coefficients Ao, Ai, ... , A,n are whole rational functions of n variables a'l, .■C2, •■■, •'^«- NOTE ON THE THEORY OF IDEALS 105 a is called a relatively whole (algebraic) function if A^ does not involve the variables. In this case the numerical coefficients of Ax, A^^, ..., A,,^ may be any real or complex numbers, whether algebraic or not. Moreover z is whole relatively io .Vi,X2, ..., x,. \i A^, involves Xr^.\, ... , !o„ only. a is called an absolutely whole (algebraic) function if Ao = \ and the numerical coefficients of A^, A.^, ... , A,n are rational integers. In the case of functions fl(l) is the corpus of rational functions, and (B (1) the aggregate of whole rational functions. In still continuing to speak of algebraic numbers it will be understood that what is said applies equally to algebraic functions. In considering the properties of algebraic numbers we naturally regard the numbers of a corpus Q (a) and the domain (1). Q (a) and more especially a> (a) are further subdivided. Dedekind defines a module in O (a) to be the aggregate of all numbers (or functions) where ni, a^, ... are fixed elements of Q (a) and a,, 012, ... any elements of a> (1), that is, rational integers in the case of number modules and whole rational functions (relative or absolute) in the case of function modules. If ai, 02, ... are whole numbers, that is, elements of a [a) instead of J2 (a), the module is a module of whole numbers. Any module of whole numbers (or functions) has a finite basis (/x;, /ij, ..., /m^) ; and any module effractions with a finite basis («!, 02, ... , oj.) is practically the same thing as a module of whole numbers, since aj, a2, ... , ai can be multiplied by a rational integer a so as to become whole numbers /ii, ;i2j.-.) Mti^"'^ then any element of the module (aj, a2,...,ai.) is equal to the corresponding element of the module (ft], /iji ... , fJ-k) divided by a. There are modules of fractions with infinite bases ; but they seem to be unimportant, and it would be simpler to restrict the meaning of the term module to a module of whole numbers or functions. A module would then be defined as any aggregate of elements of m (n) such that if a^ , a^ are any two elements of the module, oj + oj and aai are also elements of the module, where a is any element of a> (1). An involution of whole functions is any aggregate of elements of o> (a) such that if aj, 02 are any two elements of the involution, 01+02 and cai are also elements of the involution, where c is any constant. In the absolute theory the elements of w (a) are absolutely whole functions and c a rational integer. Dedekind's definition of an ideal is similar but still more fundamental. An ideal is any aggregate of elements of a (a) such that if oi, 02 are any two elements of the ideal, 01 + 02 and /xoi are also elements of the ideal, where n is any element of a (a). Every ideal has a finite basis (oi, oj, ... , oj.) and is a finite module (oj, 02, ..., a,); but not every module of whole numbers or functions is an ideal. In the domain of whole rational functions an ideal and a module are identical. 106 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS Kumnaer had found that the integers of a corpus did not necessarily satisfy all the simple laws of rational integers, or in other words they need not form a complete holoid domain. It occurred to Dedekind (and apparently independently to Kronecker) to consider in this case not the individual integers of a corpus only but sets of integers. For this purpose Dedekind made use of the ideals already defined. We shall in the first place consider ideals from a rather abstract point of view. The remarks apply also to some extent to modules and involutions. Theaggregate of elements of an ideal (ui, a^, ..., a^) constitutes an image of the properties possessed in common by all the elements of the aggregate, and especially of properties of divisibility (if any) common to ai, a^, ...,01. The term ideal should strictly be applied to these properties common to all the elements, whatever they may be ; but it is more convenient and concise to define the ideal as the aggregate of elements itself. This point of view, viz. that the ideal is a set of properties rather than a set of numbers or functions, is the justification for saying that each element of the ideal con- tains or is divisible by the ideal, since it possesses all the properties in question. Kronecker makes use of another image, in some respects simpler, viz., aiMi + 02^2 + ... + ai-Mj.- or ai + a2M+... + aiM'~\ where u, Mi, ii^, ..., % are indeterminates. This is not called an ideal because the term had already been appropriated by Dedekind with a different meaning, but it takes the place of Dedekind's ideal. Thus at the outset we can form a natural conception of what should be meant by saying that an ideal (aj, 02, ..., at) contains or is divisible by another Oi,/32, ...,/3i). The conditions should be that each of ai, 02, ..., aj,. is an element of (/3i, 02, ••■1 ft) ; for all elements of (ai,02, ..., a^) will then possess all the properties possessed in common by all the elements of (j3j, 02, ..., ft), and this apart from the fact that we may be unable to state explicitly what these properties are. Again we can give a natural meaning to the g.c.m. and the l.c.m. of two ideals. The g.c.m. or h.c.f. of (nj, 02, ...,aj,.) and (ft, 02, ...,ft) should be an ideal (-yj, y^, ...) contained in both such that every ideal contained in both is contained in (yi, y2> •••)■ There is one and only one such ideal, viz. the ideal (ai, a2, ..., ai.,ft,/ii2, .-.ift), cf § 23. The L.c.M. should be an ideal (■)'i,y2i •••) which contains both and such that every ideal which contains both contains (Tii 72) •••)■ Again there is one and only one such ideal, viz. the ideal whose elements consist of all elements of m (a) containing both (ai,02, ..., aj.) and (/3i,/32, ...,0;). These elements constitute an ideal by definition. But the crux lies in the difficulty of attaching a natural meaning to the term ■product. The product of two ideals should be an ideal whose properties consist of the product of the properties of the two ideals, and to this product of properties we cannot attach a meaning a priori from the definition of an ideal. Moreover the aggregate of the products of any element of (oi, 02, ..., ai) and any element of (ft, 02, •••jft) does not constitute an ideal. The best that can be done is therefore to define the product of these two ideals to be the NOTE ON THE THEORY OF IDEALS 107 ideal ( . . . , oj^j, . . . ), * = 1 , 2, . . . , ^, ^ = 1 , 2, ...,l. This ideal includes all products a/3 of elements of the two ideals, and in addition aU sums of such products. It could not be told beforehand to what a theory based on so tentative a definition might lead. We may say that the fact of an ideal containing another is a case of true divisibility if it always follows as a necessary consequence that the first is the product of the second and a third ideal (the conver.se being true by definition). This is exactly what Dedekiud proved to be the case for all ideals of algebraic numbers and relatively whole algebraic functions of one variable, but only by means of a long series of subsidiarj' theorems. It followed that any such ideal could be uniquely expressed as a product of prime ideals. We know however that this is not true for ideals of functions of more than one variable, since it is not true for modules of rational functions. Also it is not true for ideals of absolutely whole algebraic functions of one variable ; e.g. {x) contains {x, 2)* but is not the product of {x, 2) and a third ideal ; for the residual {x)j{x, 2) is {x), and {p;) is not the product of {x, 2) and {x). Kronecker's theory (Kr) concerns whole algebraic functions in general, and one of its remarkable features is that it applies to absolutely whole as well as to relatively whole functions. The absolute theory is based on the following fundamental theorem, which is proved by Konig (K, p. 78) : If fxtfi: •■•,fk o-i's o.'n-y ^' polynomials in Ui,zu,..., n any product of coeficients of f-^,f^, ■■■ifk taken one from each, and IIi, 112, •■■ '^6 coefficients of the polynomial f I f^ ...fii', then n satisfies identically an equation of the type np+n<''n''~^+n'^'n''"^+ ... +nW=o, where II(') is a homogeneous polynomial of degree i (i= 1, 2, ..., p) in Hi, Ho, ... with rational integral coefficients. Kronecker gives the theorem in the second of the two memoirs referred to in (Kr), having discovered it after the first memoir was written. He states it for two polynomials /i,/2 in a single letter it or x. Konig gives the theorem in the more general form above. It is not generally necessary to introduce more than one letter or indeterminate u. If we suppose /i,/2, ...,/j; to be polynomials of degrees l\,li, ...,li: in a single letter u the number of the quantities n is (?, + l) (Z2 + I) ... {l^+'i), while the number of the quantities Hi, Ha, ... (which are .sums of the quantities 11) is only I1 + I2+ ... + lji + l; andt _{h + h + ...+h)\ P=- h\U\...l^\ ' In the relative theory (x, 2) = (1), but not in the absolute theory. In the absolute theory a module in n variables can be of rank n + 1 (of. § 47) ; such in fact is any module which has some rational integer (but not unity) as a member, or any module which has no spread and is not (1). In both the absolute aud relative theories the non-proper module (1) is without rank. t The value found for p by Konig for the case fc = 2 is {li + l.;^+l)\lli\(l^ + l) !, which is not symmetrical in li,h- It can be proved that p need not be greater 108 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS Let the coefficients of fi,/^, ■■■,fk be absolutely or relatively whole algebraic functions of n variables .Tj, a;2, ..., ^„. These all belong to and determine a corpus of functions. Let Mi, M^,, ...,M^ be the ideals deter- mined by the coefficients of fi, fi, .-..fk respectively, and M the ideal (ni,Il2,...) determined by the coefficients of /1/2 ... /it. Then n is any element of the basis of the ideal MiM^... Mk,a.nd n(*) is an element of the ideal M'- (and of the involution M'). Kronecker says that a quantity n which identically satisfies an equation of the above type (where nl') is any element of the i"' power of a given ideal M) contains M in the wider sense of the word. 11 contains M in the strict sense if it is an element of M, i.e. if it satisfies a linear identity of the above type. One ideal contains another (in the wider sense) if each element of (the basis of) the first contains the second (in the wider sense) ; and if each of two ideals M, M' contains the other (in the wider sense) M, M' are .said to be equivalent in the wider sense. We denote such equivalence by M~ M', having already denoted .strict equivalence by M=M' Kronecker also remarks that (in the wider sense) if M contains M' and M' contains J/" then M contains M" Consequently if M~ M' and M~ M" then M' ~ M" If 21, Ml, J/2, ..., Mf. have the meanings given to them above we have M~MiM^...Mk. This conception of wider eqiiivalenoe is of considerable importance, and is specially applicable to Kronecker's theory. To any ideal M oi & given corpus of functions there corresponds a unique closed equivalent ideal M(, within the corpus. The elements of Mf, consist of all whole functions n of the corpus which satisfy identically an equation of the type n''+n(')n''"i+nP)n''-^+... + n''''=o, where n(') i.s an element of i/'. Any n which satisfies identically an equation n''+no(i)n'-^+no(2)n''"V...-l-no''' = o, where IIoW is an element of J/o', satisfies a linear identity of the same type and is an element of i/,,. All ideals in the corpus equivalent to M are equivalent to J/q. A closed ideal may have relevant imbedded spreads; the closed module {xi^, XiX^) is an example. If {speaking in the wider sense) M' contains M" then MM' contains MM", and conversely if MM' contains MM" then M' contains M" ; conseque^itly if MM' -MM" then M'~M". This theorem is not true for strict (or linear) equivalence, i.e. if MM' = MM" it does not follow that M' = M" (see § 24, Ex. i) unless M is an unmixed ideal of rank 1 (defined below). than the smaller value given above, while for some of the products H the value of p is less. In the cases of the first and last product H it is evident that p = l. If ?j = Z„ = 2, p is 3 for the middle product H, and 6 for the others, except the first and last. NOTE ON THE THEORY OF IDEALS 109 Let M= (ai , 02, . . ., oi), J/' = (oi', 02', . . ., a' 7c')) ^l" — ("l") 02"! • • •> «"*")• Then if M' contains M" each element 1' of the basis of M' satisfies an identity a''' + aWa'P-l + Q(2)a'P-2+... + n(P)=0, where a(') is an element of jV"', i.e. a homogeneous polynomial in ai", 02", . . ., d'v of degree i with whole functions of the corpus for coefficients. Putting aaj = a we have where a(') 0/ is an element of {MM'y ; hence a contains MM", i.e. i/ji/' contains MM". Conversely, given that MM' contains MM", a'o,- contains MM" where a is any element of M', i.e. we have an identity where ^j(') is an element of (i/i/")'=(ai, 02, ..., a^Y ("i"; "2", •.., "'V')'- Hence this identity is homogeneous and of degree pj in ai , 02,..., a^, and arranging it in power products of these, each coefficient is homogeneous and of degree pj in a', ai", 02", ..., a'V'- There are k such equations _for the same a', viz. when _;■ = !, 2, ...,k. The resultant of the k equations with respect to 01,02, ...,04 is a homogeneous equation in a', a-l', 02", . • . , a'k" of degree kp^p^-.-pt^, since it is homogeneous and of degree pip^-.-pi-lpj in the coefficients of they* equation. Also since the resultant is homogeneous in a', ai", a{', ..., a'V', and is found by a purely algebraical process, we can find the coefficient of a ''''''"'''' in it by supposing all of a/', 02", ..., a'l-- to be zeros. The resultant then becomes the resultant of (aa^)"; (a'ag)"-, ..., (o'ai.)"*, viz. a'*"'"^- ■"'■•. The coefficient of this term is therefore 1. Hence a contains M", i.e. M' contains M". The above properties are true not only for ideals but also for modules and for involutions whether of absolutely whole or relatively whole algebraic functions. Kronecker's way of considering a set of whole algebi-aic quantities an,ai,..., 1; (numbers or functions) is more direct than Dedekind's. He sets them in a frame or form* u„ = ao + ai M + a2 M^+ ... +a;tt^ where u is an indeterminate. Instead of power products of one indeterminate u we could use I indetermi nates u^, U2, ..., ui or power products of any less * We use the term form here and later as meaning a representation which is not a function but is subject to algebraic laws and operations. The form becomes a {unction of « if m is regarded as a variable or parameter. The notation a„ for a form is copied from Konig ; and the notation for an ideal M=(ao,ai, ..., a;) is the same as that used in the text for a module. Kronecker's notation is quite different ; e.g. he uses M for an element which is denoted above by a, and the term modular system as equivalent to divisor system or basis. 110 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS number. The indeterminates serve merely to separate the quantities ao, ai, ..., a;. Kronecker then expands norma,, in powers of u, viz. norma„ = (ao + niM + ... + a;2j')(ao' + ai'tt + ... + a;V)... ...(ao'"'"" + nj^"'-»M + ... + a,('"-V) = F„ + Fiu+ F2ii'+ ...+Ftu>' {k = lm), where Fo,Fi, ..., Fj^ are whole rational functions of Xi,X2, ...,a',j. If i^O) -^ii •••) Pk have an h.c.p. B, which may be a rational integer only, or a whole rational function of x-^, x^, ...,sc,^, then, having regard to the fact that norm a„ is the product of the above factors, we may say that ao, aj, ..., a; have something in common of the nature of a factor, which may be called tlieir ideal common factor, and may be represented by the form a„. So long as this factor D, which is the complete partial resolvent of rank 1 of the module (i^Oj -^i! •••) -^J-)) is °i^ly taken into account, while the partial resolvents of higher rank are neglected, Kronecker's theory is a theory of factorisation only. Dedckind had established a theor}' of factorisation of whole algebraic numbers, which he subsequentl}' extended to relatively whole algebraic functions of one variable. Considering that the factorisation of whole rational functions is exactly parallel to that of whole rational numbers the question naturally arises whether the factorisation of whole algebraic functions is parallel to that of whole algebraic numbers. Kronecker proved that it was absolutely parallel. Kronecker says that u„ and norma,, are primitive or iinit forms if Z) = l. This is legitimate in a theory of factorisation. Later be says that they are properly primitive only if the module* {F^, Fi, ... , F,i) = (l). If B+l then norm a„/Z> is a unit form. Kronecker names a„/(norm a,,//)) an "algebraic modulus or divisor," which may be interpreted, an "equivalent of a," in respect to divisibility and factorisation. Konig names a„/e„, where t„ is any unit form in m (a), an ideal whole quantity of a> (a) ; and accepts the rather absurd paradox that the sum of two such quantities is their h.c.f. It would be preferable to name a„/€„ an ideal whole form. He proves that such ideal forms can be uniquely resolved (in the sense of equivalence) into products of prime ideal forms, and shows how for a given form the prime factors can be actually found. To compare two forms a„ and a,,' Kronecker considers the fraction a^'/'a,, and rationalizes the denominator by multiplying numerator and denominator by norm aja^, which is a (strictly) whole form in a (a). If the new numerator a,,' norm ajou is divisible by the Z* of the new denominator norm a„ then the form a„' is said to be divisible by the form u„. If further the quotient of * The module (Fq, Fi, ... , F^.) is the aggregate of all whole rational functions AqFo + AiFi+... + J^F^, and the ideal {Fq, Fi, ..., F^] in the domain w(a) is the aggregate of all funotious /3oFo + /3i.F'i + ... +/3iJ?j., where ft, ft, .,.,ft are elements of w(a). NOTE ON THE THEORY OF IDEALS 111 u,,' norm a„/a„ by Z) is a unit form, then a,j7a„ = e„7^ii> where fn, f,/ are both unit forms, and the ideal forms a,„ u,,' are equivalent as regards divisibility. The divisibihty of a,/ by u,j is the same thing as the divisibility of the ideal (qq', aj', ... , nV) by the ideal (qq, ai, ... , a;) in the case of algebraic numbers and relatively whole algebraic functions of one variable ; but not in other oases. This, as we show below, is a consequence of the fact that in these two cases (i^oi -^n ••■ j Fj,) = {D). Let Mht the ideal (oq, oi, ..., a;), and, as before, let norm n„ = F(, + FjU + ... + F^u'' {h = Im). Then, in the Galoi.sian domain Q(a, a', ..., al'"-')), the ideal (Fo, Fi, ..■■, F^)is eqviivalent to the product of if and its conjugates M', 31", ..., iVI'"*?), by the fundamental theorem; and another ideal (Fa, F-{, ..., F\,) obtained in a similar way from any other basis of M is equivalent to [F^, Fi, ..., F^), i.e. a homogeneous equation of degree p exists between Fi, Fq, F^, ..., F^., in which the coefficient of Fi'' is 1, and the other coefficients are whole elements of the Galoisian domain. By rationalizing the equation it follows that the modules (Fa, Fi, ..., Fi:), {Fo',Fi, ..., F'^i) are equivalent. Hence we may define the rank of the ideal M and of the form u,j to be the rank of the module {Fq, Fi, ..., Ff^. We may also say that the ideal M is unmixed in the wider sense if the closed module equivalent to {F^, F^, ..., Fic) is unmixed. A principal ideal is an ideal (/3) having a basis consisting of a single element /3. It can be proved without difficulty that the only ideal in a given corpus a (a) equivalent to a principal ideal (/3) is the ideal (/3) itself. The ideal J/" above is called an unmixed ideal of rank 1 if (Fq, F^, ..., F^) is a principal ideal, i.e. if {Fq, Fi, ..., F^) = {B) + {1). Suppose now that J/=(ao, aj, ..., ai) is an Unmixed ideal of rank 1, and that the form a,/ is divisible by a,i in the sense defined above. Then, putting we are given that (ao' + ai'M+... + a'j.M'') (/3o+/3itt+...+/3i_i?t*~') is divisible by B. Hence atPj is divisible by D, i.e. ai^j=D^ij. Hence Multiplying by a,„ and putting Fi = D4>i, we have a/(<^o+^i'« + --- + <^A.-M') = (/3io + fti« + -"+ft, il■-^«*"')(ao + al«+•••+a^M')• Hence (fto, ^il! •••, Pi,k-i){a(y, "1, ..., a;)~(a/)((^o, 4>u ■■-, 'i>k) = {.ai), since {(jjo, 4>i, ■■■, ic) = (V> ^.nd (qj') is a principal ideal. Hence (ao, a/, ..., a'v) = (..., fiij, ...) (oq, oj, ..., a;) (i=0, 1, ..., V, j=0, 1, ..., i:-l). Conversely, if an ideal (ao', oi', ..., a';.) contains an ideal (ao, ai, ..., aj) the form a,,' is divisible by the form a„, since nj norm a„/a„, and therefore also a/ norm aJcL,,, is divisible by B. 112 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS Hence an ideal (qq', a-l, ..., ay) which contains an unmixed ideal (ao, ai, ..., m) of rank 1 is the product of {a^, uj, ..., a,) and a third ideal (..., ^ij, ...), i-e. it has (oq, aj, ..., oj) as a trite fa/itor. This includes Dedekind's principal result, since all ideals considered by him are unmixed ideals of rank 1. If the ideal (ao', a{, ..., dy) is also unmixed the quotient (..., /3y, ...) is also unmixed. Hence there exists a theory of factorisation for the -whole aggregate of unmixed ideals of rank 1 in a corpus Q (a). It follows that an unmixed ideal (ao, aj, ..., ai) of rank 1 can he nndtiplied hy a second unmixed ideal of rank 1 so as to become the principal ideal (fi), where fi is any element of (of,, aj, ..., a;). If it he true that any ideal of rank 1 must contain an unmixed ideal of rank 1, which is obvious in the two cases considered by Dedekind, but has not been proved in general so far as I know, then any unmixed ideal of rank 1 is a unique product of unmixed prime ideals of rank 1. For, assuming the truth of the hypothesis, it can be shown that any two given unmixed ideals M, M' of rank 1 which have a common factor must have an h. c. f., viz. the unmixed ideal M" of rank 1 such that (i/, M') = M"M"', where M"' is either (1) or of rank > 1. It can be easily proved that the ideals M", M'" thus defined are unique, and that any unmixed ideal of rank 1 which is a factor of M and of M' is a factor of M"\ hence M" is the h. c. f. of M and M'. In the cases considered by Dedekind {M, J/') is itself an unmixed ideal of rank 1, and M" = {M, 31'). I cannot say whether this resolution into prime factors is exactly what is meant by Krouecker in his statement XIII, p. 89 ; and I cannot attach any true meaning to the parallel statement XIII°, p. 92, regarded as an extension of XIII. Kroneoker also considers another kind of divisibility of a form «„' by a form a„, which is more adaptable to the general theory of ideals. A form u,,' might be defined as divisible by a„ if the ideal J/' = (ao', oi', ..., aV) contains the ideal M={aQ, «j, ..., ai) in the strict sense. This definition is open to the objeotion that a,i'/3„ could be divisible by a,i/3„ without a,/ being divisible by a„. The objection disappears when a wider definition is taken, viz. that a,,' is divisible by a„ if M' contains M in the wider sense. The necessary and sufficient condition that any given ideal M' = {ao', ai, ..., a'y) may contain any other given ideal J/=(ao, aj, ..., ai) in the wider sense is that the ideal corresponding to a„' norm a„/a,i contains the ideal (Fo, Fj, ..., F^.) corresponding to norm q„ in the wider sense, which is the same thing as containing the module (i'oi .^i, ■■•) -^t) iu the wider sense. In other words, it is necessary and sufficient that each of the k-l + l' + l coefficients a" of the form aJ norm a,Ja,i should satisfy identically an equation of some deo^ree p which is homogeneous iu a", Fg, F^, ... , F,,, the coefficient of a"" being 1, and the other coefficients whole rational functions. CAMBKIDGE : PF.IKTED BY J. B. PEACE, 31. A. AT THE DNIVEESITY PRESS