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Cambridge Tracts in Mathematicsand Mathematical Physics

GENERAL EDITORSJ. G. LEATHEM, M.A.

G. H. HARDY, M.A., F.R.S.

No. 19The Algebraic Theory of

Modular Systems


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CAMBRIDGE UNIVERSITY PRESSC. F. CLAY, MANAGER

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THEALGEBRAIC THEORY OFMODULAR SYSTEMS

byF. S. MACAULAY, M.A., D.Sc.

Cambridge : i^//* j/^at the University Press 2>/ I*fp '


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PREFACEHPHE present state of our knowledge of the properties of Modular-^ Systems is chiefly due to the fundamental theorems and processesof L. Kronecker, M. Noether, D. Hilbert, and K Lasker, and aboveall to J. Konig's profound exposition and numerous extensions ofKronecker's theory (p. xiii). Konig's treatise might be regarded as insome measure complete if it were admitted that a problem is finishedwith when its solution has been reduced to a finite number of feasibleoperations. If however the operations are too numerous or too involvedto be carried out in practice the solution is only a theoretical one ;and its importance then lies not in itself, but in the theorems withwhich it is associated and to which it leads. Such a theoreticalsolution must be regarded as a preliminary and not the final stagein the consideration of the problem.

In the following presentment of the subject Section I is devoted tothe Resultant, the case of n equations being treated in a parallelmanner to that of two equations ; Section II contains an account ofKronecker's theory of the Resolvent, following mainly the lines ofKonig's exposition ; Section III, on general properties, is closely alliedto Lasker's memoir and Dedekind's theory of Ideals ; and Section IVis an extension of Lasker's results founded on the methods originatedby Noether. The additions to the theory consist of one or twoisolated theorems (especially 50 53 and 79 and its consequences;and the introduction of the Inverse System in Section IV.


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VI PREFACEThe subject is full of pitfalls. I have pointed out some mistakes

made by others, but have no doubt that I have made new ones. Itmay be expected that any errors will be discovered and eliminated indue course, since proofs or references are given for all major andmost minor statements.

I take this opportunity of thanking the Editors for their accept-ance of this tract and the Syndics of the University Press forpublishing it.

F. S. MACAULAY.

LONDON,June, 1916.


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CONTENTSART.

INTRODUCTIONI. THE RESULTANT

2. Resultant of two homogeneous polynomials .... 46. Resultant of n homogeneous polynomials ~8. Resultant isobaric and of weight L ..... 118. Coefficient of arLf ... anz in R is Rr'rlr+* -* ... 118. The extraneous factor A involves the coefficients of

(/^...../UXr^oonly ...... 119. Resultant is irreducible and invariant ..... 1210. The vanishing of the resultant is the necessary and sufficient

condition that F1 = ... =Fn=0 should have a propersolution .......... 13

11. The product theorem for the resultant . . . . .151 1. If (/*,, ..., Fn] contains (F^, ..., FH'), R is divisible byR .1512. Solution of equations by means of the resultant ... 1512. The it-resultant resolves into linear homogeneous factors in

X, j, It-), ..., U r ......... 16II. THE RESOLVENT

15. Complete resolvent is a member of the module ... 2015. Complete resolvent is 1 if there is no finite solution . . 2117. Examples on the resolvent ....... 2118. The complete si-resolvent Fu....... 2418. (/.)*= 1xl+...+i )^,=0 mod (Flt Fs,...,Ft) .... 241 9. All the solutions of Fl =Ft= . . . =Fk= are obtainable from

true linear factors of Fu ....... 2520. Any irreducible factor of Fu having a true linear factor is a

homogeneous whole function of x, Ml5 ..., un ... 2621. Irreducible spreads of a module ...... 2722. Geometrical property of an irreducible spread ... 28


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'Vlll CONTENTSIII. GENERAL PROPERTIES OF MODULES

iKT- PAGE23. M/M'= M/(M,M') 3023. If M'M" contains M, M' contains M\M" .... 3124. Associative, commutative, and distributive laws ... 3125. (J/, M')[M,M'] contains MM' 3226. M/M' and M/(M/M') mutually residual with respect to M . 3228. MI(M1 ,M,,...,Mk ) =[M/M1,MIM2 ,...,M/Mk] ... 3328. [Ml ,M,,...,Mk]/M=[M1/M,M2/M,...,Mk/M] . . . 3330. Spread of prime or primary module is irreducible . . 3431. Prime module is determined by its spread . . . .3432. If M is primary some finite power of the corresponding

prime module contains M ...... 3533. A simple module is primary 3634. There is no higher limit to the number of members that may

be required for the basis of a prime module ... 3634. Space cubic curve has a basis consisting of two members . 3735. The L. c. M. of primary modules with the same spread is a

primary module with the same spread .... 3736. If M is primary M/M' is primary 3737. Hilbert's theorem 3838. Relations between a module and its equivalent ^T-module . 39

38,42. Condition that an H-moduleMmay be equivalent toMXn=\ . 3938. Properties of an ZT-basis 4039. Lasker's theorem 4040. Method of resolving a module ...... 42

41, 44. Conditions that a 'module may be unmixed .... 4342. Deductions from Lasker's theorem 4442. When M/M' is M and when not .4442. No module has a relevant spread at infinity .... 4443. Properties of the modules J/M, J/W 4544. Section of prime module by a plane may be mixed . . 4746. The Hilbert-Netto theorem 48

UNMIXED MODULES 4948. Module of the principal class is unmixed .... 4949. Conditions that (Fi, F2 , ..., Fr) may be an #-basis . . 5050. Any power of module of principal class is unmixed . . 51

51,52. Module with y-point at every point ofM . . . .52


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CONTEXTS IXAKT. PAGE52. When a power of a prime module is unmixed ... 5353. Module whose basis is a principal matrix is unmixed . . 54

SOLUTION OF HOMOGENEOUS LINEAR EQUATIONS . . . . . 58XOETHER'S THEOREM 60

56. The Lasker-Noether theorem 61IV. THE INVERSE SYSTEM

58. Number of modular equations of an /7-rnodule of theprincipal class 65

59. Any inverse function for degree t can be continued . . 6759. Diagram of dialytic and inverse arrays 6759. The modular equation 1 =0. . . . . . .69

60, 82. The inverse system has a finite basis 6961. The system inverse to (F^ F2 , ..., Fk} is that whose Ff-

derivates vanish identically 7062. Modular equations of a residual module .... 7063. Conditions that a system of negative power series may be

the inverse system of a module 7164. Corresponding transformations of module and inverse system 7165. Xoetherian equations of a module 7365. Every Xoetherian equation has the derivate 1 = . . . 7365. The Xoetherian array . . 7566. Modular equations of simple modules 7">

PROPERTIES OF SIMPLE MODULES 7767. A theorem concerning multiplicity ..... 7769. Unique form of a Xoetherian equation ..... 7971. A simple module of the principal Xoetherian class is a

principal system ........ 8072. A module of the principal class of rank n is a principal

system 8173. n= p.' + p" 8274. ft'i'+ fj."i- =/ij-= /tr , where I'+l"= y \ -8375. ffm= l+fjLl + H.2 +---+fhn76. #Y-#Y =#V +r-#r = #r-#'Y + r, where /' +r=y-2. 84

MODULAR EQUATIONS OF UNMIXED MODULES 8577. Dialytic array of J/(r> 8678. Solution of the dialytic equations of J/(r> .... 88


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CONTENTSART. PAGE79. Unique system of r-dimensional modular equations of M . 8979. The %-dimensional equations 8980. Equations of the simple //"-module determined by the

highest terms of the members of an //-basis of M(r)81. IfR=\ and M is unmixed, M is perfect ....82. If J/(r) is a principal system so is M82. A module of the principal class is a principal system .83. J/(r> and M are principal systems if the module determined

by the terms of highest degree in the members of an /iT-basisof M(r) is a principal system ; not conversely ... 9184. Modular equations of an //-module of the principal class . 92

85. Whole basis of system inverse to M(r1 ..... 9386. Modules mutually-residual with respect to an //-module of

the principal class 9487. The theorem of residuation 9688. Any module of rank n is perfect 9888. An unmixed /^-module of rank n - 1 is perfect ... 9888. An //-module of the principal class is perfect ... 9888. A module of the principal class which is not an ZT-module is

not necessarily perfect 9888. A prime module is not necessarily perfect .... 9889. An .//-module M of rank r is perfect if the module

J/xr+2=...=j-n=o is unmixed 9990. A perfect module is unmixed 9990. The L. c. M. of a perfect module of rank r and any module in

#,. + 1 , ...,^,t only is the same as their product ... 9991. Value of-HI for a perfect module 9992. If J/, M' are perfect //-modules of rank r, and ifM contains

M', and MXr+l = ...=xn =o is a principal system, M/M 1 isperfect 100

NOTE OX THE THEORY OF IDEALS . . 101


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DEFINITIONSABT. PAGE

Modular system or module 1Member of a module 2Basis of a module ....... 2

1. Elementary member of (Fi, F-2 , ...,^V) ... 31. Resultant of (Flt /;,...,/*,) ... 46. Reduced polynomial 77. Extraneous factor 108. Leading term of resultant 108. Weight of a coefficient 118. Isobaric function 11

12. The tt-resultant FW 1(>12. Multiplicity of a solution 1713. Rank and dimensions 1813. Spread of points or solutions 1814. Reducible and irreducible polynomials ... 1914. Complete resolvent 2014. Partial resolvent .2016. Imbedded solutions 2118. Complete z-resolvent FH 2419. True linear factor of w-resolvent 2521. Irreducible spread 2721. Order of irreducible spread 2721. Equations of irreducible spread 2>23. Contained module 2923. Least and greatest modules 2923. Unit module 2923. G.c.M. of Jfl,M2t ...,Mk 2923. L.C.M. of Jfi, Jf2 > Jf* 3023. Product of Mlt Mz,...,Mk 3023. f", Oy, y-point 3023. Residual module 3029. Prime module . ... 33


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Xll DEFINITIONSART. PAGE29. Primary module 3329. Singular point and spread 3433. Characteristic number 3633. Simple module 3633. ^"-module 3638. Equivalent //-module 3938. .ff-basis of a module . . . . . . .3940. Relevant primary modules of M 4240. Relevant spreads of M " 4240. Isolated and imbedded spreads and modules . . 4241. Mixed and unmixed modules . . . . .4343. The modules M(r\ MM 4547. Module of principal class 4851. Basis consisting of the determinants of a matrix . 5254. Inverse arrays ........ 5856. Noetherian module ....... 6157. Dialytic array and equations 6457. Inverse array, inverse function 6457. Modular equations ....... 6459. Inverse system 6860. ^-derivate of E 6960. Principal system 7065. Noetherian equations 7365. Underdegree of a polynomial 7468. Multiplicity of a simple module 7868. Multiplicity of a primary module .... 7868. Primary module of principal Noetherian class . . 7868. Complete set of remainders 7968. Simple complete set of remainders .... 7977. r- dimensional modular equations 8677. Regular and extra rows of dialytic array of J/(r ) . 8777. Regular form of dialytic array of J/< r ) ... 87

77, 88. Perfect module 8785. Whole basis of system inverse to J/M ... 93


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LIST OF REFERENCESReferences to the following in the text are given by their

initial letters, e.g., (L, p. 51).(BX) A. Brill and M. Noether, " Ueber die algebraischen Functionen

und ihre Anwendung in der Geometi-ie" (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 Vorle&ungen iiber 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 angew. Math. 92 (1882), p. 181).

(E) E. B. Elliott, Algebra of Quanti&s (Clarendon Press, 2nd ed. 1913).(H) D. Hilbert, " Ueber die Theorie der algebraischen Formen" (Math.

Ann. 36(1890), p. 473).(H!) D. Hilbert, " Ein allgemeiues Theorem iiber algebraische Formen "

\M'


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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) E. H. Moore, " The decomposition of modular systems of rank n in

variables" (Bull. Am. Math. Soc. (2) 3 (1897), p. 372).(N) M. Noether, " Ueber einen Satz aus der Theorie der algebraischenFunctionen " (Math. Ann. 6 (1873), p. 351).(Ne) E. Netto, "Zur Theorie der Elimination" (Acta Math. 7 (1886),

p. 101).(S) C. A. Scott, " 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) Encykloptidie der Mathematischen Wissenschaften (Teubner, Leipzig,

Teil i, Bd. I, Heft 3 (1899), p. 283). G. Landsberg, " Algebraische Gebilde,etc."

(W2) Encyclopedic des Sciences Mathe'matiques (Gauthier-Villars, Paris,Tome I, Vol. 2, Fasc. 2, 3 (1910, 11), p. 233). J. Hadamard and J. Kiirschak," Proprietes generates des corps, etc."

This account of the theory is founded upon that of G. Landsberg butis much fuller both in subject matter and references.


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THE ALGEBRAIC THEORY OFMODULAR SYSTEMSIntroduction

Definition. A modular system is an infinite aggregate of poly-nomials, or whole functions* of n variables xlt a-*, ...,#, denned by theproperty that if F, JFl} F belong to the system Fr +F and ^.Falsobelong to the system, where A is any polynomial in #j, jc, ..., xn .Hence if Fl} F.2 , ..., Fk belong to a modular system so also doesA 1P1 + A 2F2 + ... + A kFk , where A l , A, ..., A k are arbitrary poly-nomials.

Besides the algebraic or relative theory of modular systems thereis a still more difficult and varied absolute theory. We shall onlyconsider 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 isa quantity not involving the variables, are not regarded as differentpolynomials, and any polynomial of degree zero is equivalent to 1. Norestriction is placed on the coefficients of Fl} F2 , ..., Ft except in sofar as they may involve arbitrary parameters u1} u.2 , ..., in which casethey are restricted to being rational functions of such parameters.The same restriction applies to the coefficients of the arbitrary poly-nomials A l} Ao, ..., A k above.

In the absolute theory the coefficients of F1} FZ, ..., A l} A 2 , ...are restricted to a domain of integrity, generally ordinary integers orwhole functions of parameters x , u, ... with integral coefficients;and a polynomial of degree zero other than 1 or a unit is not equivalentto 1.

* We use the term whole function throughout the text (but not in the Note atthe end) as equivalent to polynomial and as meaning a whole rational function.M. 1


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THE ALGEBRAIC THEORY OF MODULAR SYSTEMSDefinitions. 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 is a member of M in

the relative or absolute sense we shall write F = mod M, orF = mod M. The notation F~ modM only comes into use inthe sequel in connection with the Resultant.A basis of a module M is any set of members f 1} F2 , ..., Fk suchthat every member of M is of the form XlFl +XZF2 + ... +XkFk ,where X\, X^ ..., Xk are polynomials.Every module of polynomials has a basis consisting of afinite numberof members (Hilbert's theorem, 37).The proof of this theorem is from first principles, and its truth willbe assumed throughout.The theory of modular systems is very incomplete and offers awide field for research. The object of the algebraic theory is to dis-cover those general properties of a module which will afford a means ofanswering the question whether a given polynomial is a member of agiven module or not. Such a question in its simpler aspect is of im-portance in Geometry and in its general aspect is of importance inAlgebra. The theory resembles Geometry in including 'a great varietyof detached and disconnected theorems. As a branch of Algebra itmay be regarded as a generalized theory of the solution of equations inseveral unknowns, and assumes that any given algebraic equation inone unknown can be completely solved. In order that a polynomial Fmay be a member of a module M whose basis (F^ F2 , ..., Fk) is givenit is evident that F must vanish for all finite solutions (whether finiteor infinite in number) of the equations Fl = F2 = . . . =Fk = 0. Theseconditions are sufficient if M resolves into what are called primemodules*; otherwise they are not sufficient, and Fmnst satisfy furtherconditions, also connected with the solutions, which may be difficult toexpress concretely. The first step is to find all the solutions of theequations F^ - F2 = . . . = Fk = ; and this is completely accomplishedin the theories of the resultant and resolvent.

* Cayley and Salmon constantly assume this. Salmon also discusses particularcases of a number of important and suggestive problems connected with modularsystems (Sa).


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4 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [IThese remarks and definitions are equally applicable to any module

(Fi, Fz , -, Fk) of homogeneous or non-homogeneous polynomials;but the following definition applies only to the particular module(Fi, Ft, ..., Fn~).The resultant R of FI, F2 , ..., Fn is the H.C.F. of the determinantsof the above array for degree t=^l+ 1, where / = h + 12 + . . . + ln n. Itwill be shown ( 7) that R is homogeneous and of degree /!/2 ... /,/, inthe coefficients of Fi (i = 1, 2, ..., n).

2. Resultant of two homogeneous polynomials in twovariables.

Let F! = Oisf* + b-iXi l~^xz + ... + ki&f\JC 2 =


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l] THE RESULTANT 5where w (^ } is any power product of .TJ, x of degree / + 1. Thisexpresses the first important property of R.

3. Irreducibility of /?. The general expression for theresultant R is irreducible in the sense that it cannot be resolvedinto two factors each of which is a whole function of the coefficientsof F!, F. When this has been proved it follows that any wholefunction of the coefficients of Flt F.2 which vanishes as a consequenceof R vanishing must be divisible by R.R has a term o/J oj1 obtained from the diagonal of the deter-minant, and this is the only term of R containing a? l . Also, wheni = 0, R has a term ( l^^a^a/1 ' 1 , and this is the only term

of R containing aJ'~ l when ^ = 0. Hence, when R is expandedin powers of a2 to two terms, we have

where b = (- l)/s t, b^ mod al .Hence if R can be written as a product of two factors, we have*-(%**...)(*+...),where PI + qi = 4 and Pi + qs=li, and either pi or qv is zero ; for other-wise the coefficient b of a/1 " 1 would be zero or divisible by al , whichis not the case. Hence one of the factors of R is independent of thecoefficients of FI, since both factors must be homogeneous in thecoefficients of FI. Similarly one of the factors must be independentof the coefficients of F*, i.e.

*-(ft*+.-)(b*+,)-***\since the whole coefficient of a/* in R is a*t and of a/1 is a/1 . This isnot true ; hence R is irreducible.

4. The necessary and sufficient condition that the equationsFl =F2 = may have a proper solution (i.e. a solution other than#i = x.3 = Qi) is the vanishing of R.This is the fundamental property of the resultant. If theequations Fl =F = have a solution other than ^ =#.2 = it followsfrom

Xxl* 1 = mod (F, , FJ, Rx2l+l = mod (F1} F2),that R =0, by giving to xly x the values (not both zero) whichsatisfy the equations Fl = F^ = 0.

Conversely if J? = we can choose Xj, A.2 , ...,X,+2 so that thesum of their products with the elements in each column of the


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6 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ldeterminant R vanishes. Multiplying each sura by the power productcorresponding to its column, and adding by rows, we have

where Xl5 A2 , ... Xl+2 do not all vanish. Hence, since A.^/2 ' 1 + ... isof less degree than F2 , F must have a factor in common with FZtand the equations Fl =F2 = have a proper solution.

In the following article another proof is given which can beextended more easily to any number of variables.

5. When R=$ there are 1 + 2 linearly independent members of(Fi, F2) of degree l+l, and / of degree /. When .# = there areonly I + 1 linearly independent members of degree 1+1 and stillI of degree I, i.e. in each case 1 less than the number of terms ina polynomial of degree 1+ 1 and / respectively. Hence there willbe one and only one identical linear relation between the coefficientsof the general member of (F1} F2) whether of degree l+l or /.Let this identical relation for degree / + 1 be

Cl+i, o Zl+i, o + fy i Zt> i + + C , i+i Z0< i+1 0,where Z LJ denotes the coefficient of x*x in the general memberof (Fi,F2} of degree i+j, and the c fj are constants. Then, ifF is the general memberof (Fi,F^) of degree /, x^F is a member of degree l+l whosecoefficients must satisfy the relation above. Hence

Ci+i.0 Zi


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I] THE RESULTANT 76. Resultant of n homogeneous polynomials in ~

variables.The general theory of the resultant to be now given is exactly

parallel to that already given for two variables, although it involvespoints of much greater difficulty as might be expected. Anothermethod of exposition depending on a different definition of theresultant is given in (K, p. 260 ff.).

Let FI, F2 , ..., Fn be n homogeneous polynomials of degreesl\, 1-2, , la, of which all the coefficients are different letters. Inparticular, let al} a.2 , ..., be the coefficients of x^, x**, . . . , a-,*" inFI, Fz , -, Fn respectively, and c1} c.2 , ..., cn the constant terms ofF1 ,F*,,...,Fn when xn is put equal to 1, so that c = aH . LetI=li + l2 ~. .. + ln -n, Z = /i/2 ---4, Ll =L/h, L2 = L/l.2) ... LH = L/ln .

The resultant R of F1} F,, ..., Fn has already been defined ( 1)as the H.C.F. of the determinants of the array of the coefficientsof all elementary members of (Flt F.2 , ..., Fn} of degree /+ 1.

We shall first consider a particular determinant D of the array,viz. that representing (1) the polynomialX^Fl + A^F2 + ... + A^*-VFn of degree /+ 1,where ^(4) denotes a polynomial in which all terms divisible byxl l or x-2 ... or #, ' are absent, which may be expressed by sayingthat X is reduced in dr1} #2 , ... , -r,-. The polynomialis represented by the bordered array


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8 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [lthat the same is true of the columns, we notice firstly that thereis no power product to ('+1) of degree / + 1 reduced in all the variables,for the highest power product of this kind is #1

?1 ~ 1 #2'2~ 1 ... XH

U ~ Iwhich is of degree /


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l] THE RESULTANT 9coefficients of Fn , and homogeneous and of degree > Z, in thecoefficients of Ft (1=1,2, ...,-!). It follows that R. which is afactor of D, is at most of degree Ln in the coefficients of Fn . Weshall prove that R is of this degree, and consequently of degree Z, inthe coefficients of Ft.

Let D' be any other non-vanishing determinant of the array, viz.

a,


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10 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [lfor the unknowns r, F, ..., Y^n~2\ X^n~ l\ This equation has aunique solution, since the more particular equation

F x} 1 + F'1 ' x^ + . . . + F afc\ + X^- 1 ) = A nhas a unique solution (for any given polynomial A n can be expressedin one and only one way in the form on the left) and shows that thenumber of the coefficients of F(0), F (1 >, ..., F have been found. Next solve the equation

Z F, + ZWFa +...+ Z = A _! + F. We canproceed in this way till X, X^, ... , X^n~ l\ i.e. Xl5 A2 , ... , X^, have allbeen found.

In this method of solving the unknowns on the left are associatedwith F1} F2 , ... , Fn-i only and not with Fn . Hence ( J is a rationalfunction of the coefficients of Flt F^ ..., Fn whose denominator isindependent of the coefficients of FH , and the same is therefore trueof jr = ( } . Hence every determinant D' of the array has a factorin common with D which is of degree Ln in the coefficients of Fn >The resultant R, which is the H.C.F. of all the determinants D', istherefore of degree L t in the coefficients of Ft (i- 1, 2, ... , n).

If we put D = AR, A is called the extraneous factor of D, Wehave proved that A is independent of the coefficients ofFn ; and itis proved at the end of 8 that A depends only on the coefficients of

8. Properties of the Resultant. Since D has a termr /" ( 6) R has a term a^ of* anLn . This is called theleading term of R.

Since D vanishes when Ci, c2 , ..., cn all vanish ( 6) the same is trueof R ; for D =AR and A is independent of clt c2 , ... , cn .

The extraneous factor A of D is a minor of D, viz. the minorobtained by omitting all the columns of D corresponding to power


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I] THE RESULTANT 11products reduced in n - 1 of the variables and the rows which containthe element* a^,a^ ..-, in the omitted columns (M^p. 14). ThusD/A, where A is this minor of D, is an explicit expression for R.

Each coefficient a of Fl} F2 , -.-, Fn is said to have a certainnumerical weight, equal to the index of the power of one particularvariable (say .*) in the term of which a is the coefficient. In the caseof non-homogeneous polynomials the variable chosen is generally thevariable # of homogeneity. Also the weight of ap is defined as ptimes the weight of a, and the weight of aptV ... as the sum of theweights of ap, bq,(f, .... A whole function of the coefficients is said tobe isoburiL- when all its terms are of the same weight.The resultant is isobaric and of weight L. Assign to jcl , #,, ..., xnthe weights 0,0, ...,0,1. Then the coefficients of F+, Fz,..., F*have the same weights as the power products of which they are thecoefficients. The z'th row of the determinant D represents the poly-nomial wi^ = a,u>^+1) + & lWm) +---+*Wm) - Tnu s the weightsof ai,bi, ...,, are less than the weights of hff+1>,*(l+1),...,V+1)respectively by the same amount, viz. the weight of u>;. Hence, onexpanding D, the weight of any term is less than the sum of theweights of oi!('+1) , OVM) , -,


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12 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [l9. The resultant of F^ F2 , ..., Fn is irreducible and invariant.

It has been proved that the resultant is irreducible when n = 2 ( 3) ;and the proof can be extended to the general case by induction.Let jftn = the resultant of (F1} F2 , ..., Fn-i) Xn =o ;

.F = the resultant of the homogeneous polynomials F-^,F2(0\ ..., F^\ in sc-i, #2 , ...,#n_2 ,#o obtained from FltFa,-.., Fn -i by changing xn- l} xn to x^x*, xnx

-&n \-^n)x l = ...xn-^= Q Kn Xn _ j 4- ... + Cln Xn ',R' = the resultant of Fl , F2 , ..., Fn^ , Fn ' ;M = the resultant of F , Fn', two polynomials in xn-i, xn ;

J^l '] = J^2 h = ~ Is n-l


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I] THE RESULTANT 13factor involving the coefficients of Flt F, ..., FH-i- This musttherefore be a factor of I?'.

Again R is what R becomes when all the coefficients of FH otherthan those of F* are put equal to zero. Hence R has an irreduciblefactor of the form Rnl*anq + . . . , where q > p. The remaining factor ofR is independent of the coefficients of F1} F.2 , ..., Fn- lt and thereforealso of the coefficients ofFn when n > 2. Hence R is irreducible.

It easily follows that R is invariant for a homogeneous linearsubstitution whose determinant ( , \ does not vanish. Suppose that

R = and that this is the only relation existing between thecoefficients of Fl , F%, ..., Fn . Then not more than one relationcan exist between the coefficients of FI, F2', ..., F*', the poly-nomials into which Fi,F,...,Fn transform. Since 7? = thereare less than p. linearly independent members of (F^, F2 , ..., Fn)of degree / + 1 and therefore less than p. linearly independentmembers of (Fi, F2', ..., Ff) of degree /+!, and the only singlerelation between the coefficients of FI,F2', . . . , F* which will admitthis is R = 0. Hence R = Q requires R = 0, and R is divisible byR. The remaining factor of R' is independent of the coefficients of

fx\LFJ, F2 , ...,Fn , and can be shown to be f ,J A proof that R isinvariant without assuming it irreducible is given in (E, p. 17).

10. The necessary and sufficient condition that the equationsFI =F2 = ... = FX = Q may have a proper solution is the 'vanishingof R.

In the general case, when the coefficients are letters,AMxnl+l= mod (F,,F,,...,F^Put ,r,t = 1 and change* Ci to c, -F< (i = 1, 2, . . . , n) ; then A does notchange, being independent of cl ,c.2 , ...,cn (8); but R changes toR -A^ - A 2F.2 - ... A nFn , and this must vanish ; henceR=Omod(Fl,F2,...,FH)Xn=l .Hence R vanishes if the equations Fl = F2 = = Fn - have asolution in which xn = 1, i.e. if they have a proper solution.To prove that 72=0 is a sufficient condition, we shall assume that .S=0is the only relation existing between the coefficients of Fl} F2 , -..,Fn .There are then less than p. linearly independent members of(F1} F.2 , ...,Fn} of degree 1+ 1. Hence the coefficients 5|>1 ,pi ,... J pm of

* Called the Kronecker substitution.


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14 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [lthe general member of degree l+l must satisfy an identical linearrelation 2Cp t .p-2,...,PnZp l ,P2 ..... Pn = Q> Pl+P2 + +Pn=l + 1-

The coefficients of the general member of degree I also satisfy oneand only one identical linear relation, whether R vanishes or not. Toprove this it has to be shown that the number Nof linearly independentmembers of (Fl ,F2 , ..., Fn~) of degree / is 1 less than the number pof power products of degree /. If no relation exists between thecoefficients of fl , F2 , ... ,Fn the equationcan always be solved by the method of 7, where A l} A 2 , ..., A n arearbitrary given polynomials. Hence Nis not greater than the numberof coefficients in X, X*\ ... , X^~ l\ or inX^X^ + A^W*+ ." + A^- 1^,viz. p-l, since, when this expression is of degree /, every powerproduct except Xi '~ lxz "~ l ...xnln~ l occurs once and only once in it.Hence N^p- 1.

Any particularity in Fl , F2 , . . . , Fn can only affect the value ofN bydiminishing it. Hence for the remainder of the proof it will be sufficientto show that N- p I in a particular example in which M = Q. LetFl =(xl -x2}xlll~\ F2 =(x*-x3}xt*-\..., Fn ^(xn -xl}xnln ~ l .Then R = since the equations F^ = F2 = . . . =Fn = have the propersolution Xi= xz- ... =xn -\. Let x^x^ . . . xnpn be any power product ofdegree /. If pi ^ A changex^x* to xl l ~ lx^ where pl +p2 = l - 1 + q% ;this is equivalent to changing x^x^ . . . xnpn to x^x/"1 ... x*n + A^.Again if qz ^ L change x^xf* to x^~ lx*3 ; and if q2 < 4 proceed to thefirst pr ^ lr and change xrVr x^r^ to eef*~ laf . If we continue thisprocess, going round the cycle xlt x%, ...,# as many times as isnecessary, the power product xf'xj'* ... x* will eventually becomechanged to Xi l

~ lx^~ l ... i . Hence these two power products arecongruent mod (F1} F.,, ..., Fn), while neither of them is a memberof (Flt F2 , ...,Fn), since they do not vanish when xl = ...=xn -l.Hence N--=p-l.

Let F= 2zql ,qt,...,q*Xi9l &2 lt XT?" be the general member of(Fi,F2 , ...,Fn} of degree /; then XiF is a member of degree /+!in which the coefficient of xf lx** ... xnpn is %,,p2 .....pi-i,...,pn - Hence

^cp t ,ih...-,pn^ ..... w-i..... pn =0 (=1,2, ..., w),Or 2C,,. ,...,+!..... qn Zqlt9t ..... qn = (f = 1, 2, ...,).


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l] THE RESULTANT 15These equations in ~9lf9s ,..., 9 are therefore equivalent to one only;and the continued ratio C9l+il9s

This result can easily be extended to the case in which any or allofFl} F2 , .- , FH resolve into two or more factors.

If F^Fz,...,Fn are att members of the module (F^ F,, ...,F*')the resultant R of Fi,Fs,...,F* is divisible by the resultant R qfFi, F2', ..., Fa'. For if R = then R = 0.

12. Solution of Equations by means of the Resultant.The method of the resultant for solving equations can only be appliedin what is called the principal case, that is, the case in which thenumber r of the equations is not greater than the number n of theunknowns, and the resultant F of the equations with respect to ar,, a^,..., xr-! (after a linear substitution of the unknowns) does not vanishidentically. When F vanishes identically the method of the resultantfails, but the equations can be solved by the method of the resolvent,due to Kronecker, as explained later. The method of the resolvent isalso applicable to any number of equations whether greater or less thanthe number of unknowns.

Homogeneous Equations. Let the equations beFl =Ft= . . . =Fr-of degrees d, 4, , 4, where r $ n. We assume that their resultantF. with respect to xl} ar2 , ---, #r_i does not vanish. We regard a^, xt).-., XT as the unknowns, the solutions being functions of xr+l , ..., j-,.But instead of solving for one of the unknowns xl , x^ ..., xr we solvefor a linear combination of them, viz. for x= UiX^ + u^xs + ... + UTXT *where MI} u*, ..., ur are undetermined quantities. Let Fu stand for

... -urxr . Then we regard F1 =F2 = ... = Fr = Fn =* Called the Liotrville substitution.


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16 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [las the given system of equations with #1} . .., xr , x as unknowns, andtheir resultant F (u) with respect to xl , xz , ..., xr gives the equation

Definition. F (u) is called the u-resultant of (Flt F2 , ..., Fr~).Applying the reasoning of 9 it is seen that F is the resultant

(with respect to x^, ..., xr^, x^} ofFlt Fz ,*..,Fr when xr,xr^, ...,#vare changed to xr% , #r+1# , , #n#o> and is a homogeneous polynomialin xr , xr+l , ..., xn of degree L = IJ2 ... lr) viz.F =Br+lZrL + -,where Rr+l is the resultant of (Flt F2 , ..., Fr)Xr+i= ... =Xn=o, and doesnot vanish; for a homogeneous substitution beforehand betweenxr , xr+1 , ..., xn only would be carried through to F .

Similarly F^w) is the resultant (with respect to xl , ..., xr , ZQ) ofFU Ft, , Fr , Fu when x, xr+l , ..., xn are changed to xx^, xr+l x ,...,xn x , and is a homogeneous polynomial R'r^xL + ... in x, %r+i, , %nwhere R'r^ is the resultant of (F1} F2 , ..., Fr , Fu}Xr+l =...^Xn=o. It iseasily seen* that fi'r+1 = fir+ i. HenceF^ =Rr+i XL + . . ., where r+1 4= 0.

To each solution xr -xri of F = Q corresponds a solution (x^, x^iy..., xri) of the equations Fj, =Fz = . . . = Fr = for ^ , x^, . . ., XT ( 10).There are therefore L solutions altogether, and they are all finite,since Rr+\ * 0.

Similarly to each of the L solutions x = x t ofF (u) - there con-e-sponds a solution (#li} x^, ..., xri , x^) of Fl - . . . =Fr -Fu = ; and asregards (a^, x^, ..., xr^) the L solutions must be the same as thoseobtained by solving F = 0. Hence it follows that

Xi ~ Ui^u + U-iXx + ... + UrXri ,where xlit a?2i , ..., xri are independent of ult u.2 , ..., ur . HenceF (u] = Rr+ l "[i(x-U lXli - ...-UrXrl) ( = 1, 2, .-., Zr).

Thus F (n) is a product of L factors which are linear in x, ul} u.2t..., ur , and the coefficients of uly u->, ..., ur in each factor supply asolution of the equations FI F2 - . - . = Fr = 0.

Also the number of solutions is either L = ll li ...lr or infinite, thelatter being the case when F vanishes identically.

* By introducing a as coefficient of x in Fu it is seen that JJ'r+i is divisible byaL by considering weight with respect to x. Also the whole coefficient of a^ in jR'r+l is Er+i ( 8). Hence R'r+ i= aLRr+i = J?r+i .


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l] THE RESULTANT 17If /) is the determinant for (F,, F3 , ..., Fr , FH ), regarding jr,,*i, ...,

:is the variables, like the D of 6, we have Du = AF {U}. The ex-traneous factor .4 depends onlyon the coefficients of(Flf FZt -, jPr)x.=o,that is, of (Flt F: , ..., Fr),p+1 = ...= Jr,,=o. Hence A is a pure constant,ndmt fxr^, ..,*! andofu lt u.2 , ..., it,, and ire may take DH ='(.< ff>>- i"i>int'in for x.

Definition. The number of times a linear factorx - u^ x^ ... - urxriis repeated in F (u} or /) is called the multiplicity of the solution(*u, **, > *) This term has a definite geometrical interpretation;it is the number of solutions or points, in the general case distinct,which corne into coincidence with a particular solution or point in theparticular example considered.

In the case of n homogeneous equations in n unknowns such thatRQ, the complete solution consists of the non-proper solution(0, 0, .-., 0) with multiplicity L = li L ... ln .

X'-n-humogeneous Equations. In the case of non-homogeneousequations a linear substitution beforehand affects only #1, xz , ..., xnand not the variable x of homogeneity. Hence it is possible for Rr +ito vanish identically, while F and jF (M) do not, no matter what theoriginal substitution may be. In this case there is a diminution in thenumber of finite solutions for x, but not in the number of linear factorsof FQ" } . To a factor i^x^ + u.^Xn * ... + urxri of .Fo'"' not involving xcorresponds what is called an infinite solution of Fl = F.2 - . . . =Fr =in the ratio x^:x^: ... :xri . Infinite solutions are however non-existent in the theory of modular systems ( 42). An extreme case isthat in which F ' u> does not vanish identically, but is independent of xtwhen all the L solutions are at infinity.

It may happen that a system of non-homogeneous equations hasonly a finite number of finite solutions while the resultant ^ vanishesidentically. In such a case the method of the resultant fails to givethe solutions.

Example. The equations xf = x.2 + .^ x3 = x3 -f j-j x* - have thefinite solution xl - x.2 = x3 - ; but the resultant vanishes identicallybecause the corresponding homogeneous equations

.rf = x x., + x1xs = X x3 + xt x., =are satisfied by xt> -x1 = 0,a, system of two independent equations only.


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18 THE ALGEBRAIC THEORY OF MODULAR SYSTRMS [ll

II. THE RESOLVENT13. We shall follow, with some material deviations, Kiinig's

exposition of Kronecker's method of solving equations by meansof the resolvent. The equations are in general supposed to be non-homogeneous; and homogeneous equations are regarded as a particularcase. Thus a homogeneous equation in n variables represents a coneof n 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 ofequations FI =FZ - = Fk ~ in n unknowns #1} x^ ..., xn . Theunknowns are supposed if necessary to have been subjected to a homo-geneous linear substitution beforehand, the object being to make theequations and their solutions of a general character, and to preventany inconvenient result happening (such as an equation or polynomialbeing irregular* in any of the variables) which could have been avoidedby a linear substitution at the beginning. In theoretical reasoningthis 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 x^when ^2 ) #3 5 , %n have arbitrary values; (ii) those which exist forofi, #2 > not included in (i), when a?3 , ..., xn have arbitrary values;(iii) those which exist for x, x2 , x-,., not included in (i) or (ii), when#4 , ...,xn have arbitrary values; and so on. A set of solutions for#j, x-i. ..., xr when ar+1} ...,xn have arbitrary values is said to be ofrank r, and the spread of the points whose coordinates are the solutionsis of rank r and dimensions n - r. If there are solutions of rank r andno solutions of rank < r the system of equations F^- F2 = . . . = Fk = andthe module (Flt F2 , , Fk) are both said to be of rank r.

14. The polynomials FI, F.2 , ..., Fk , and also all their factors areregular in x^. Hence their common factor D can be found by theordinary process of finding the H.C.F. of Ft , F2 , ..., Fk treated aspolynomials in a single variable xl . If D does not involve the variableswe take it to be 1. If it does involve the variables the solutions ofD = treated as an equation for #x give the first set of solutions of theequations F! =F2 = ... =Fk = mentioned above.

* A polynomial of degree I is said to be regular or irregular in .r ( according asthe term xj is present in it or not.


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II] THE RESOLVENT 19In the algebraic theory of modules we regard any algebraic equation

in one unknown, whether the coefficients involve parameters or not, ascompletely soluble, i.e. we regard any given non-linear polynomial inone variable as reducible. A polynomial in two or more variables iscalled reducible if it is the product of two polynomials both of whichinvolve the variables. A polynomial which is not reducible is called(absolutely) irreducible. Any given polynomial is either irreducible oruniquely expressible as a product of irreducible factors, leaving factorsof degree zero out of account. It is assumed that the irreduciblefactors of any given polynomial are known. Thus the polynomial Dabove may be supposed to be expressed in its irreducible factors in#1} jc, ..., xn , and to each irreducible factor corresponds an irreducibleor non-degenerate spread.

Put FI - Di (i = l, 2, . . ., ). Then fa , .2 , ..., < t have no commonfactor involving the variables, and the same is true of the twopolynomials

Aj^j + A,,,


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20 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [llNow any solution of f\ = F2 = . . . - Fk = is a solution of D = or of

(1 > = or of faW = < 2< = . . . = ^^ = 0.Hence any solution of Fl =F2 = ... = Fk - is a solution of D = or of/Jl1 ) = o or of !(1) = < 2(1) = ... = < fc , (1) - 0. Proceeding in a similar waywe find that any solution of Fj- ... =Fk = is a solution ofDD^ ... D(n-^ = Q, since < 1(n~ 1) , ^ ("" ]) , - , ( ~ 1} are polynomials ina single variable xn at most and have no common factor.

Conversely if 3 , #4 , ... , #n is any solution of D(2) = the resultantof 2A.j;(1) and 2/ij(1) ... Z>("" 1) = 0,say a solution >, a?i+1 , ...,# of D^^^O, there corresponds asolution^, 4, -, ^, #+i, > of the equationsF1 =F2 = ...=Fk = 0.Hence from the solutions of the single equation DD(l) . . . />< (l~ 1 > = i wocan get all the solutions of the system F =FZ = ... =Fk = 0, sinceall the solutions of the latter satisfy the former.

Definitions. DD(l} ... ZX71" 1 ' is called the complete (total) resolventof the equations F-^ = F*= . . . =Fk = and of the module (Fl ,F.,,..., Fk).j)(i-i) [& called the complete partial resolvent of rank i, and any wholefactor of D(i ~ l} is called a partial resolvent of rank i.

15 . The complete resolvent is a member ofthe module(F} , F, , . . . , F/j.For 2 PiFM = mod (3 A


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Il] THE RESOLVENT 21no common factor, we can choose polynomials , in the single variableso that Stf,^"' 1 * = 1. Hence

/)/>'> ... Z*"- 1^ mod (Flt F2 , ... , Fk}.If the equations FI- t\= ... = Fk = have no finite solution the

(.'j/ttj'/rf? resolcent is equal to 1 ; consequently 1 is a member of(Flt F.>,...,Fk), and every polynomial is a member.

16. We have seen that to every solution x t = & of Z)* 1'" 1 * = therecorresponds a solution fl5 >,..., j, zi+1 , ... , j.'n of the equationsF1 =F = ... =Fk - 0. It may happen that there is an earlier completepartial resolvent Z) " 1 ' which vanishes when #, = ,-,...,#, = ,-. Insuch a case the solution , ... , ,-, .zvn, ..., # of Fl =...=Ft=0corresponding to a solution of D^'^ - is included in the solutionscorresponding to D(J~ l) - 0, and may be neglected if we are seekingmerely the complete solution of F1 =F = . . . =Fk = 0. Such a solu-tion is called an imbedded solution. All solutions corresponding to anirreducible factor of Z)^ 1 ' 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 givenset of algebraic equations into definite irreducible spreads ( 21). Ithas been supposed that the complete resolvent also supplies a definiteanswer to certain other questions. The following examples disprovethis to some extent.

Example i. Find the resolvent of n homogeneous equationsFl = F, = . . . = Fn = of the same degree / and having no propersolution.

Since there are no solutions of rank < n the complete resolvent isZ*"- 1 '. The first derived set of polynomials FJ l\ F.V, . . . , Fkl(l) are homo-geneous and of degree l~, the 2nd set FJ-\ F.2 (~} , ...are homogeneousand of degree l\ and the in - 1 ,th set F^"^, F.,(n ~ l\ ... are homogeneousand of degree /" . This last set involve only one variable #, and4iiitherefore have the common factor ar , which is therefore therequired complete resolvent.We should arrive at a similar result if we changed x^ toXi + ai(i= 1, 2, ..., n} beforehand, thus making the polynomials non-homogeneous. The complete resolvent would then be (# + )'The resultant would be (j-n + anf. The difference in the two resultsis explained by the fact that the resultant is obtained by a process


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22 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [llapplying uniformly to all the variables, and the resolvent by a processapplied to the variables in succession.

Example ii. Konig (K, p. 219) defines a module or system ofequations as being simple or mixed according as only one or morethan one of the complete partial resolvents I), Z>(1), ... , D(n~^ differsfrom unity. Kronecker (Kr, p. 31) says that the system of equationsFI=FZ =-^fc=0 is irreducible in this case ; and the Ency. des Sc. Math.(W2 , p. 352) repeats Konig's definition. We give two examples toshow that this definition is a valueless one.

If u, v, w are three linear functions of three or more variables, anypolynomial which contains the spread of u = v - is of the formAu + Bv; if it also contains the spread of u = w = Q, B must vanishwhen u = w = (), hence B must be of the form Cu + .Dw, and Au + Bvof the form A'u + B'viv; if it also contains the spread of v = w = V,A' must be of the form C'v + D'w, and A'u + B'vw of the formC'uv + D'uw + B'vw. Hence a polynomial which contains all threespreads is a member of the module (vw, wu, nv], and also any memberof the module contains the three spreads. This module, althoughcomposite, is not mixed in any proper sense of the word.

Besides having partial resolvents of rank 2 corresponding to thethree spreads the module has a partial resolvent of rank 3 corre-sponding to its singular spread u = v = w - 0. This last partialresolvent does not correspond to any property of the module which isnot included in the properties corresponding to its partial resolventsof rank 2 ; in other words the partial resolvent of rank 3 is purelyredundant.

The resolvent Z>(1) />(2) can be found as follows : Supposeu = a + a lxl + aXi + ...,v- l) + iX l + .,ic.2 + ..., w =Then the resultant of \vw -t- \^wu + A.,Mt> andwith respect to #1} apart from a constant factor, is

Ciu) (b-iU- a v)( GI v - bl w a, it: - c, } u b^u a l v ]

its four irreducible factors corresponding to the spreadsv = w=0, w = u = Q, u =v-0,

(AaMs " X*^ u = (^8/*i- ^i/*s) v * (Ai^ " ^sMi)Hence D(l] = (^ v ~ b } w) (a l w - ct u) (^ u - a l v) ;and fa


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II] THE RESOLVENT 23from which we -obtain

J)M = (blC - k>fi) u + ( - Cortj) v + (arj)z - o-A) ic.Example iii. Compare and find the resolvents of the two modulesM = (x-i3, x-23 , .r/2 + x-2 + x1 x.2X3),

M'=(x*, xfx*, .zvr,-, .r.23, x? + x* + x^x^x^.The resolvent ofM will be found by obtaining the resultant with

respect to xl of the two equationsA^j3 + \.,xfx.2 + Ajo-j-r-2 + \4x.? + A5 (x? + x? + x^x^x^ = 0,

and frx? + n-2xi2x.2 + p-zX-^x.? - p.4 x


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24 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ll#3 1 to indicate these, but it has no such factors. The completeresolvent may indicate imbedded modules which do not exist as inEx. ii, or it may give no indication of them when they do exist as inEx. iii.

Example iv. It is stated in the Encyk. der Math. Wiss.(Wx , p. 305) and repeated in ("W2 , p. 354) that if only one completepartial resolvent D(r) differs from 1, and Z>(r) has no repeated factor,the module is the product of the prime modules corresponding tothe irreducible factors of D(r) . The absurdity of this statement isshown by applying it to the module (u, vw}, where u, v, w are the sameas in Ex. ii. The complete resolvent is Z>(1) - (b^u a^y) (c^u a-^w),and the product of the prime modules (u, v), (u, w} corresponding toits two factors is (w2, uv, mv, vw} 4= (u, vw}.

18. The (/-resolvent. The solutions of Fj. = F2 - . . . = Fk =are obtained in the most useful way by introducing a general unknowna; standing for u-^x^ + u2xz + ... + unxn , where MI, u, ..., un are undeter-mined coefficients. This is done by putting

X ~

in the system of equations Fl =F2 - . . . - Fk = 0. We thus get a newsystem/!=/2 = ... =/fc = in x, x, xa , ..., xn , where

// 77T I& UnX.^ UnXn \ , . j ^

(= u^Fi(- ,x,--->xn \ (i=l,2,...,A),\ n /

the multiplier uji being introduced to make // integral in i^. Thereis evidently a one-one correspondence between the solutions of the twosystems, viz. to the solution &, ,, ..., n of FI =F2 = ... = Fk = Q therecorresponds the solution , .2 , ..., $n of J\=fz= =fk-^> and viceversa, where = ii + M24 + + un H .

Definition. The complete resolvent DUD^ ... DJn-v (=Ftl ) of(fit fz> ---ifk) obtained by eliminating xz , x-A , ..., .r,, in succession iscalled the complete u-resolvent of (Flt F^, ..., Fk}. .

Since Fu = mod (/,/2 , ...,/*), by 15, we have(Fu}x^ UlXl+ ... +UnXn =.() mod (Flt F, ..., Fk}.Fu is a whole function of x, x2 , ..., xn , HI, u.,, ..., UH which resolves

into linear factors when regarded as a function of x only. The linearfactors of rank r, that is, the linear factors of Du(r~ l) , are of the type#-!&-...- Ur r~ Ur+l Xr+l - ... - Un Xn


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II] THE RESOLVENT 25where ft, ..., ft, av+n , x* is a solution of Fl =F2 = ...=Fk = 0.For if a- - is any linear factor of Dv(r~ 1} then is a root ofDJ

r-v = Q to which corresponds a solution ft ft, ..., ft, *r+ i, -, &* of/=/;=... =/i = ( 14) and a solution ft, ft, ..., ft, av +i , .. ,- xn ofFl = F=... =Fk = 0, where - ?^ft + . . . + ,.ft + ur+l av+i + ... + a,ar, .

The linear factors of .FK expressed in the above form supply all thesolutions of /i=/2 = ... =/t = 0, viz. ft 4, -, &, av.n, -, #, and allthe solutions of J^ = F2 = ... =-Ft - 0, viz. ft, ft, ..., ft, arr+1 , .... #, ofthe several ranks ;= 1, 2, ..., n ; but it is only when ft, ft, ..., ft. areindependent of u lt u.,, ..., u n that we know the solution from merelyknowing the factor.

19. A linear factor of Fu of rank r such as the above will becalled a true linear factor if ft, ft, ..., ft are independent of MI, u, ,#,that is, if it is linear in a; -MI, u, ..., u n .

If a linear factor of /% /> not a true linear factor the solutionsupplied by it is an imbedded one.

Let x $ or x - z^ft - ... - w,,ft - ui+lxg^ - ... - unxH be a non-truelinear factor of Fv , so that ft, ft, .... ft depend on ult M 2 , ..., UH . Thenft, ft> -, ft, *+i> -, &n is a solution of F1-FZ = ... =^ = 0, and soalso is i?!, r).,, ..., rig , ayn, ..., xn where ^, 772, ..., ^ are obtained fromft, ft, ..., ft by changing lf -2 , ..., MK to t'a , v2 , ..., vn . Hence 77, 17,,..., T/,, ^+1 , ..., xn (where r) = u 1 r) l +... + us r}il +ul>+lxg+1 +...+unxn)is a solution of/j =/o = . . . =/t = 0, and therefore makesFu vanish. Butit does not make Du (g~ l]...Du{n ~ l} vanish since this does not involvex, ..., xg , and cannot have a factor x yj, where rj involves i'i, r2 , ..., vn .Hence it makes some factor Du (r~ l) of Fu of rank r < s vanish. ThenDu(r~ l) vanishes when x, .r,.+1 , ..., xs are put equal to *?, 77r+1 , ..., ijt \and by putting i\, r,, ..., VH (of which Dlt(r~ l) is independent) equal toM!, o, ..., M B it follows that Du(r~ l) vanishes when x, zr+i, -, xt areput equal to ft ft+1 , ..., ft. Hence the solution ft ft, ..., ft, xi+l , ...,xnis an imbedded one ( 16).

It follows that all the solutions of FI F = ... -FK = are obtainablefrom true linear factors of Fu ; and that all the linear factors of thefir*t complete partial u-resolrent (different from l) are true linearfactor*.

It also foUoics that if t/tm- As a spread of rank s which is not im-l>r,l


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26 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [llWe have not proved that all linear factors of .Fn are true linear

factors*, and whether this is so or not must be considered doubtful.20. If an irreducible factor Ru of Fn considered as a whole

function of all the quantities #, #2 , -, xn , ult u.,, ..., un has a truelinear factor all its linear factors are true linear factors.

Let Ru be of rank r. Then Ru is independent of #1} %2 , ..., xr andthere is a one-one correspondence between its true linear factors andthe sets of values 1} 2 , , r of #1, ,r2 , ...,xr (not involving MI, M2 , > *Ofor which (./?)*= Wia;i+...+* vanishes. Let

where pi, p2 , , p^ are different power products of ult u, , un andRI, RZ, , RU are whole functions of Xi, xt , ..., xn independent ofMj, u2 , ..., un . Then the sets of values &, .,, ..., ,. required are thesolutions ofRI=Rz - = -Btt = regarded as equations for xlt x.2 , ..., xr .These come from the solutions &, >, ..., ,., xr+l , ..., xn of rank r ofthe same equations in x-l} x2 , , x-n . Now there is at least one solu-tion of rank r, since Ru has a true linear factor ; and only a finitenumber of such solutions altogether, since Ru has only a finite numberof such factors. Hence the first complete partial ? xni and having a true linear factor, has all its linear factorstrue linear factors, and is a whole function of ult u.2 , ..., un .

* Kronecker states this as a fact without proving it. Konig's proof contains anerror (K, p. 210). It is not correct to say as he does that E t (h) X tW vanisheswhen x = ( , but only when x, j, 2 , ...,,, are put equal to t , i', s ', ... , ,,'.


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II] THE RESOLVENT 2721. The irreducible spreads of a module. Let RH be any

irreducible factor of Fu of rank / having a true linear factor. Weknow thati=d

-tfu ~ A. II (./' lli-l'i; ... M,-cTri llr^i .rr+! ... UnXH).i=dHence ()***,*+..,+** = ^ n i u i fa ~ fii) + + r fa- - #)}1=1

To RH corresponds what is called an irreducible spread, viz. thespread of all points .rlf , ..., xri , xr+1 , ..., x* in which xr+l , ..., x* takeall finite values, and xis , ..., xri the d sets of values supplied by thelinear factors of 7?,,, which vary as .zv+i, ..., XH vary.The degree d of Ru is called the order of the irreducible spread.From the two identities above several useful results can be deduced.It must be remembered that Ru is a known polynomial in x, xr+1 , ...,^m MI> u, -, u,i- No linear factor of Ru can be repeated, unlessdRzv+i, -, *' are given special values; for otherwise J?H and ^ wouldhave an H.C.F. involving x, and Ru would be the product of two factors.Whatever set of values xr+ i, ..., x* have, whether general or special,the d sets of corresponding values of .**!, .r, ..., xr , viz. Xn, x# t ..., xriare definite and finite, because RH is regular in .r.From the second identity it is seen that (Ru)x=u lxl+...+u1&tt is inde-pendent of M,.+I , ..., M,O and vanishes identically (i.e. irrespective ofMJ, MO, ..., un) at every point of the spread and no other point. Hencethe whole coefficients* of the poirer products of MI, t^, ..., M,. in(Ru)x=H lx l +...+u H?H fill vanish at every point of the spread and do notnil ranifh at any other point. These coefficients equated to zero give asystem of equations for the spread ; but it is not necessary to takethem all, and some are simpler than others. The coefficient of urdgives an equation (xr , xr+l , ...,xn) -AH (xr - xri} = for xr , whoseroots are the d values of xr corresponding to given arbitrary values ofxr+l , ..., xn . The coefficient of iiiiird~ l gives an equation

where


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28 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [llSimilarly we have x^' 92 = 0, . . ., xr-$ r-i = 0. The equations

^-A _^l _^? _*r-l

* T/i > * 1 ~~7T"99 9are called more particularly the equations of the spread, the first givingthe different values of xr as functions of xr+l , ..., xn , and the othersgiving #,, #2 , ..., #,.-i as rational functions of xr , xr+l , ...,#. Ifav, #r+i, >} # have such values that 9 = 9' = then 91, 92 , ..., 9r_xall vanish and the expressions above for #,, #2 , ..., xr^ become inde-terminate. In such a case the values of x^ x.2 , ..., .T,.^ may be foundby taking other equations from (Ru}x=u tx l+...+unxn for them.

22. Geometrical property of an irreducible spread.An algebraic spread in general is one which is determined by anyfinite system of algebraic equations, and consists of all points whosecoordinates satisfy the equations and no other points. Such a spreadhas already been shown to consist of a finite number of irreduciblespreads 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 theirreducible spread, contains the whole of it.

Let FI =Fz = ... = Fj: = be the equations determining any algebraicspread, and jFV = F% = ... - F'k > =0 the equations determining an irre-ducible spread. The spread they have in common is determined bythe combined system of equations Fl F% = ... = Fk -F - . . . = F'k > = 0,and is contained in the irreducible spread and has the same or lessdimensions. If it is of the same dimensions as the irreducible spreadthe complete w-resolvent of F^ - . . ' =Fk = J&Y = . . . = F'v = will havean irreducible factor Ru" of the same rank as the irreducible factor Jiu'of the complete w-resolvent of FI = F^= ... =F'k > = Q correspondingto the spread of the same. Also all the roots of R ti" - regarded asan equation for x are roots of Hu

' - 0. Hence jRu' is divisible by fiu",and since they are both irreducible they must be identical. Hence the

spread of F= ... -Fk =F = ... =F'k ' = Q contains the whole of thespread of FI =Fs=...=F'e = 0, and the spread of Fl =F2 = ...=Fk =contains the same. This proves the property stated above.


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Ill] GENERAL PROPERTIES OF MODULES 29

III. GENERAL PROPERTIES OF MODULES23. Several arithmetical terms are used in connection with

modules suggesting an analogy between the properties of polynomialsand 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 sumor difference. Also a prime module answers to a prime number and aprimary module to a power of a prime number. Such terms must notbe used for making deductions by analogy.

Definitions. Any member F of a module 31 is said to contain M.Also the module (F) contains 31. It is immaterial in this statementas in many others whether we regard .F as a polynomial or a module.The term contains is used as an extension and generalisation of thephrase is divisible by.

More generally a module M is said to contain another M' if everymember of M contains M' ; and this will be the case if every memberof the basis of M contains 31'. Thus (Fi, F, ..., Fk) contains(F-L, F, ..., Fk+l ), and a module becomes less by adding new membersto it.

If M contains 31' and 31' contains 31 we say that 31, M' are thesame module, or 31= M '.

If M contains 31' the spread of M contains the spread of 31', butthe converse is not true in general.

If in a given finite or infinite set of modules there is one which iscontained in every other one, that one is called the least module of theset ; or if there is one which contains every other one, that one iscalled the greatest module of the set. Two modules cannot be com-pared as to greater or less unless one contains the other.

There is a module which is contained in all modules, the unitmodule (1). Also (0) may be conceived of as a module which containsall modules ; but it seldom comes into consideration and will not bementioned again. These two modules are called non-proper modules,and all others are proper modules. In general by a module a propermodule is to be understood.

The G.C.M. of k given modules J/i, M2 , ..., Mk is the greatest of allmodules 31 contained in J/j and J/>... and 3fh , and is denoted by(Mi, 31,, ..., 31k). In order that 31 may be contained in each ofJ/i, 3L, .... 3fh , or that each of 3/1; 3L, ..., 3fb may contain M, it is


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Ill] GENERAL PROPERTIES OF MODULES 31Nevertkele M J7, 37') is not called the quotient of 37 by (37, 37')because it is not true in general that the product of (M,M'} andMI(M, M') is M.

If 37, M', M" are three modules such that M'M" contains M it isclear that M' contains MM" and M" contains 37/37'. Since 3737'contains M, M contains 37/37'. The module 37/37' is a module con-tained in 37 having a special relation to M independently of what M'may be ( 26 (i)).

There is a least module which can be substituted for 37' withoutchanging 37/37', viz. 37/(37/37'), 26 (ii). This module is containedin 'M, 37'), for (37, 37') can be substituted for 37' without changingM 37', but is in general different from (M, 37').

24. Comment on the definitions. The non-proper unitmodule (1) has no spread. Conversely a module which has no spreadis the module (1), since the complete resolvent is 1 and is a member ofthe module. The unit module is of importance from the fact that itoften 'comes at the end of a series of modules derived by some processfrom a given module.

(J/i, Af.2 , ...,37t) and [J/i, 372 , ..., 37c] obey the associative law[37,, 37.2 , 373] - [[M,, 37,], 371] = [37n [3/s , 373]], and the commutativelaw (3/i, 372) = (372 , 3/J. Also (3/j, 37o, . . . , 37t) obeys the distributivelaw M (37! , J73) -P/3/,, 3/3/2) ; but [J/lf 3/2 , ..., 3/J does not

Example. As an example of the last statement we have

while [(^ , #,) (^2, a-22), (ar, , x^ fa a-,)] - (a?i a?) (^i , ^2).Given the bases of M1} M, ..., J/, 37, 3/' we know at once a

basis for (JS/i, J/,, ...,3/t) and for Ml M.2 ...Mt ; but it may beextremely difficult to find a basis for [J/i,3/2 , ... , J/J or for MjM'.Hilbert (H, pp. 492-4, 517) has given a process for finding a basis of[3/s . J/j. ..., 3/J ; and the same process can be applied for findinga basis for J//3/'. This process is chiefly of theoretical value in sofar as it has any value.We can have (i) 3/3/' = 3/37", or MjM' = M/M", without M=M":ii 3/ 37'= 37" without 3/3/"= 3/'; (iii) 3//3/'= 3/" and 3//3/"= J/'without 3/=3/'3/"; and (iv) 3/=3/'3/" without 37 3/' -- 37" or37 37" = 37'.

/&*. (i) (arlf a^) (arlf ^)2 = (a?lf(a?,, a:2)3/(^i, x^ = fa,


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32 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill(ii) (>! , ^/(X2, x?) = (xj. , #2), whil e (^ , #2) :'/(>i , x2 ~) = (#, , ztf ;(iii) (a-j2, -o2)/(^i , a?2) - (X , #2)2 and (a^8, ^/(^ , #2)2 = (^ , ,r2) ,

while (X2,#22) =1= (ar, , a-g) (^ , #2)

2;

(i v) (a?! , ^2)6 = (X3, #i2 a?a , #23) (X3, ^ ^22 , ara8),while (xi,x^fl(x*,x?Xz,z.?} and (X,#2)6/(X3> #i#22 , #23) are bothequal to (#! , #2)3.

25. The product of the o. C.M. and L. c. M. of two modules containsthe product of the modules.

Let M=(F1,F2,...,Fk) and M' = (F1',FZ', ...,F',,) be the twomodules and let FL be any member of the basis of their L.C.M. Then,since FL = QmodM, FIFL = mod MM' ; and since FL = mod M',Ft FL = mod MM' ; i.e. the product of any member of the basis of(M, M'} with any member of the basis of [M, M'] contains MM' , or(M, M'} [M, M'] contains MM'.When M, M' have no point in common (M, M') = (1) and con-sequently [M, M'] contains MM', i.e. [M, M'] = MM'. This case isproved by Konig (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 M/M' and MI(M/M") are mutually residualwith respect to M, i.e. each is the residual of the other with respect to M.

Let M/M'= M" and M/(M/M') = M'" then we have M'"= M/M",and we have to prove that M" = M/M'". Let M/M'" = J/ iv . NowM'M" contains M; therefore M' contains M/M" or M'". AlsoM"M'" contains M ; therefore M" contains M/M'" or J/h . Again,since M' contains M'" (proved) and M'"Miv contains M, M'Mivcontains M, i.e. Miv contains M/M' or M". But M" contains M~lv(proved). Hence M" = Miv - M/M'".

Two results follow from this :(i) M/M' is a module contained in M of a partirular ti//> ; for

MjM' and its residual with respect to M are mutually residual withrespect to M, and this is not true in general of any module containedin M and the residual module (Ex. ii, 24).

(ii) The least module which can be substituted for M' withoutchanging M/M' is M/(M/M'~). Let M be any module such thatM/Miv = M/M' ; then the product of Miv and M/M' contains M, andM contains M/(M/M'\ Also M/(M/M'} is one of the modules J/ il ;


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Ill] GENERAL PROPERTIES OF MODULES 33for if M. (MM , = M'" then M/M'" = M/M1, by the theorem. HenceM/(H/M') is the least of the modules J/~IV which can be substitutedfor M' without changing M/M'.

27. //' J/, M" r//? mutually residual with respect to any modulethey arc mutually residual with respect to M'M".

Suppose M', M" are mutually residual with respect to 31. ThenM'M" contains 31; and if M'M" 31' = 31'", 31'M'" contains M'M"which contains M; hence M'" contains MM' or M". Also M"contains M'M' M' or M'". Hence M" = M'" =MM M'. SimilarlyM'=M'M\ M" (cf. statement iv, 24).

Any module M with respect to which 31', M" are mutuallyresidual contains [J/', M"] and is contained in MM".

28. // M, Ml , 3L, ..., Mk are any modules, then311(31,, M2 , ... , 3/t) = [M/M,, M/M,, .., M/3It],

and [M,, 3L, ...,Mk]M= [31,, M, M,.:M, ..., J/t J/].For J/'J/,,, J/,, ...,Mk) contains 3//3T, and therefore contains

[MM,, MM,, ..., M/Mk }. Also M^MJM,, ..., 3//3/J contains3/, x M Mi which contains J/; hence (37, , ..., J/t) [3//J/1, ... , 3/ J/t]contains M, and [J/'J/!, ..., 3/ J/J contains 3/;3/); ..., J/t). Thisproves the first part.

Again [3/1; ..., J/,..] J/ contains Mf . :M and therefore contains[MJM, ..., J/* J/j. Also J/[J/i/J/, ...,Jfi/lf] contains J^ andtherefore contains [J/j, ..., J/t]; hence [J!/i J/, . . . , Mk/M] contains[J/!, ..., J/t] 'M. This proves the second part.

29. Prime and Primary Modules. Definitions. Amodule is defined by the property that no product of two modulescontains it without one of them containing it.A primary module is defined by the property that no product oftwo modules contains it without one of them containing it or bothcontaining its spread. Hence if one does not contain the spread theother contains the module.

Primary modules will be understood to include prime modules.Lasker introduced and defined the term primary (L, p. 51), thoughnot in the same words as given here. The conception of a primarymodule is a fundamental one in the theory of modular systems.Any irreducible spread determines a prime module, viz. the modulewhose members consist of all polynomials containing the spread. Thatthis module is prime follows from the fact that no product of two


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34 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [illpolynomials can contain the spread without one of them containingit ( 22) and the module ; and the same is true if for polynomials wewrite modules.

If M= (F1} F.2 , ..., Fk) is the prime module of rank r determinedby an irreducible spread of dimensions n-r, and if the origin bemoved to a general point of the spread, the constant terms ofFi, F.2 , ..., Fk will vanish, and the linear terms will be equivalentto r independent linear polynomials, i.e. the sub-determinants oforder > r of the matrix

S\ SF, Sf\

dFk dFwill vanish, while those of order r will not vanish, at the origin. Thiswill be equally true for any general point of the spread withoutmoving the origin to it. Any point of the spread for which thesub-determinants of order r of this matrix vanish is called a singularpoint of the spread, and the aggregate of such points the singularspread contained in the given spread. The singular spread (if anyexists) is therefore the spread determined by F1} F2 , ..., Fk and thesub-determinants of order r of the above matrix.

If M--(Fi, F.>, ..., Fk ~) is the L.C.M. of the prime modules deter-mined by any finite number of irreducible spreads of the samedimensions n - r, the same definition holds concerning singularitiesof the whole spread. In this case the singular spread consists of theintersections of all pairs of the irreducible spreads, together with allthe singular spreads contained in the irreducible spreads consideredindividually.

30. The spread of any prime or primary module is irreducible.For if not the complete w-resolvent has at least two factors corre-sponding to two different irreducible spreads of the module neither ofwhich contains the other, and is the product of two polynomialsneither of which contains the whole spread of the module, i.e. themodule is neither prime nor primary.

31. There is only one prime module with a given (irreduciblespread, viz. the module whose members consist of all polynomialscontaining the spread.


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Ill] GENERAL PROPERTIES OF MODULES : 35Let M-(Fi,Fz , -, Fk ) be any prime module of rank r. It will

be sufficient to prove that every polynomial which contains the spreadofM contains the module J/. The first complete partial ?/-resolventofM other than 1 will be a power Rum of an irreducible polynomial RHin j-, zr+i, ...,#. Also the complete ^-resolvent is a member of(/u/2, ---j/fc)? 18, which is prime ; and every factor except Bum isof too high rank to contain the spread of (f\,f, -,ft)- Hence Rum ,and therefore R u itself, is a member of 0/i,/3 , -,./*) Hence(Ru)x=u i x1+...+un^n ls a niember of M, and also the whole coefficient ofany power product of tfl5 ->, ...,un in (Ru)x=i, l xl+...+*nxH ' We haveproved (21) that

(#)*=,*,+...+**= + Wr"'1 (ti Ai + -. + Ur-i ' ; then .P becomes a rationalfunction of j*r , a*r+1 , ..., xn of which the denominator is '', where / isthe degree of F. This rational function vanishes for all points of thespread at which ' does not vanish, and its numerator is thereforedivisible by &', i.e.-*l --** -^1 j-" - " '"' r~ l ' ' r+lt '"'

or tf'Ffa , x.2 , . . . , -rn ) = mod (ft , . . . , /v-i ,) = mod J7.Hence ^=OmodJ/, which proves the theorem.

It follows that a module which is the L.C.M. of a finite number ofprime modules, whether of the same rank or not, is uniquely deter-mined by its spread, and any polynomial containing the spreadcontains the module.

32. //' J/ /.s '( primary module and J/, the prime module deter-mined by 'tt* spread tomeJmtie power of J/j contains M.This theorem, in conjunction with Lasker's theorem ( 39), is

equivalent to the Hilbert-Xetto theorem ( 46). The proofs of thetheorem by Lasker and Konig are both wrong. Lasker first assumesthe theorem (L, p. 51) and then proves it (L, p. 56) ; and Konigmakes an absurdly false assumption concerning divisibility (K, p. 399).32


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36 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [illBy the same reasoning as in the last theorem it follows that .Rum

(but not .#) is a member of 0/i,/2 , ,/*), and(Ru

m}x^u,xl +...+unxn

={... + MT^CMI^, + ... + Mr-i^r-i) + ur

d$}

m= mod M.Picking out the coefficients of urdm and urdm-m u1m, we havem = 0modlf, and W = JT modM ; .\ fr"*= modM ;

and similarly , = . . . =^ x = mod M. Also if F is any member ofMI, then, by the last theorem,

Hence the product of any rw* polynomials ^ and


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Ill] GENERAL PROPERTIES OF MODULES 37This can be proved by considering residuation on the cone. The

original i /(/- 1) generators have a residual on the cone of |/(7 1)generators, which again have a residual of \l(l 1) generators, ofwhich /-I can be chosen at will. This last set of generators is residualto the irreducible curve and together they make the whole intersectionof the cone with a surface of order / having an (/- l)-point at O.Hence there are / surfaces of order / containing the irreducible curvewhich have an (/- l)-point at and in which the terms of degree /-Iare linearly independent, while there is no surface containing the curvewith less than an (/-l)-point at 0. The prime module determinedby the curve must therefore have at least / members in its basis. Themodule has in fact a basis of / + 1 members, the / + 1 linearly inde-pendent surfaces of order / containing it (including the cone) ; andthese can be reduced to / members.

In the case n = 2 the curve is an ordinary space cubic determininga prime module

C/i i/ > fa) = (vw> - v'w, wu>~ w'u, w' ~ u'v\where it, c, w, u', v', w are linear. The basis of three members can bereduced to two/i af^f^ bfs provided constants a, 6, X, X' and linearfunctions a, ft can be chosen so that

or l-aor 1 - a - ft \u + XV, aa = Xv + XV, bft - Xzr + X'w' ;and this can be done.

35. The L.C.M. of any number of primary modules irith the samespread is a primary module with the same spre


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38 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [illcontains M'M" which containsM' and ifMl does not contain

the spread of M (that is of M"*) M'MZ contains M, and M2 containsM/M' or M"; i.e. M" is primary.

37. Hilbert's Theorem (H, p. 474). If F,, F2 , F3 , ... is aninfinite series of homogeneous polynomials there exists a finite numberk such that Ft, - mod (F1 , F2 , ...,Fk} when h > k.

The following proof is substantially Konig's (K, p. 362). It mustbe clearly understood that Fl} F2 , F3 , ... are given in a definite order.In the case of a single variable the series Fly F2 , F3 , ... consists ofpowers of the variable, and if Fk is the least power then Fh = mod Fkwhen h > k. Hence the theorem is true in this case. We shall assumeit for n 1 variables and prove it for n variables.The series F-, 1 , F2 , F3 ', ... is called a modified form of the seriesF,, F.2 , F3 , ... if F^F, and F- =^mod (F} , F2 , ..., ^_0 for i>l.Thus the modules (F^ F2 , ..., F{ *) and (Fi, F2', ..., F-} are the same.The theorem will be proved if we show that the series FI, F2\ ... canbe so chosen that all its terms after a certain finite number becomezero. We assume that F1 is regular in xn , and we choose the modifiedseries so that each of its terms Fl after the first is of as low degree aspossible in xin and therefore of lower degree in xn than FI. The termsof the series FI, F.2', ... of degree zero in xn will be polynomials inxl , x%> > xn-\ and these can be modified so that all after a certainfinite number become zero, since the theorem is assumed true for n 1variables. Let F'^, F\2 , F\3 , ... be all the terms of FI, F2 ', F3 ', ...,taken in order, which are of one and the same degree I > in xn ; andlet fi^f'i.,, be the whole coefficients of #ul in them. Thenfi^f'^yfij, are polynomials in n 1 variables; and we cannothave /'jf = mod (/',i ,/',2 , .. j/'^j) for any value of i ; for if

is of less degree than I in #H , which cannot be. Hence the number ofthe polynomials /',l? /'/ , ..., or the number of terms F'^, F'i2 , ... inthe series FI, F2 , . . . , is finite. And the number of values of / isalso finite, the greatest value of / being the value it has in F^.Hence the theorem is proved.

The theorem can be extended at once to an infinite series Fi,F3 , ...of non-homogeneous polynomials since they can all be made homo-geneous by introducing a variable ;r of homogeneity.

The following is an immediate consequence of the theorem :


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Ill] GENERAL PROPERTIES OF MODULES 39Any mcxlule of polynomials has a basis consisting of a finite number

To prove this it is only necessary to show that a complete linearlyindependent set of members of any module can be arranged in a definiteorder in an infinite series. If / is the lowest degree of any member wecan first take any complete linearly independent set of members ofdegree /, then any complete set of members of degree 1+1 whose termsof degree / + 1 are linearly independent, then a similar set of membersof degree / - '2, and so on. In this way a complete linearly independentset of members is obtained in a definite order. It does not matter inwhat order the members of a set are taken, nor is it necessary to knowhow to find the members of a set. It is sufficient to know that thereis 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 .uivenmodule 37, not an //-module, arranged in a series in the order describedabove ; and make all the members homogeneous by introducing a newvariable # . We then have a series of homogeneous polynomialsbelonging to an //-module 3/ , whose basis consists of a finite numberof members of the series. The module J/ is called the H-modukequivalent to 3/, and a basis of 37 obtained from any basis of 37(1 byp'utting .r = 1 is called an H-basis of 37. The distinctive property of an//-basis (Fn F.2 , ..., Ft) of M is that any member F of 37 can be putin the form A^ ->- A 2F + ... + AbFk where ^4,Fr (/= 1, 2, ..., k) is notof greater degree than F. Every module has an H-basis, which maynecessarily consist of more members than would suffice for a basis ingeneral.

The following relations exist between J/ and its equivalent 77-module 37 : (i) to any memberF of J/ corresponds a member F of 37of the same degree as F, and an infinity of members xF of higherdegree ; (ii) to any member F of 37 corresponds one and onlyone member of M, viz. (Fo^^i; (iii) there is a one-one correspondencebetween the members of 37 of degree / and the members of M idegree ^ /.

If x F = mod 3/o, then (/**)., =1 = mod 3/, and F = mod M.by (i), i.e. there is no member x F of 3/ such that/^o is not a memberof 3/,,, and 3/( , (.r ) =3/,,. Conversely an H-module M in n variables.i\, x.>, ..., .vn is equivalent to the module 3/J)|=1 if MI{x^) Al, an


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40 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [illIn any basis (Fi, F2 , ..., Fk ~) of an //-module in which no member

is irrelevant, i.e. no Ft = mod (F1 , ..., F^ , Fi+l , ..., Fk), the numberof members of each degree is fixed ; as can be easily seen by arrangingF1} F2 , ... , Fk in order of degree. Hence in any H-basis of a modulein which no member is irrelevant the number of members of each degreeis fixed. On account of this and the other properties of an //-basismentioned above an //-basis gives a simpler and clearer representationof a module than a basis which is not an //-basis.

Example. Find an //-basis of the module (x^x2 + xl x^).Take the //-module (x?t x^x^ + x^^) and solve the equationx X = mod (xi2, x2x + x1x3"),

x X = XiXi + (x.2xQ + XtXs)X2 .we haveXl = xsX,

orPuttingi.e.

i.e.Hence

- X (x? Yl -i.e. X = Q mod

Again, if we solve the equationx Y = mod (xi

we find Y = mod (x^and if we solve

Q = mod (xi2= mod (xf,

=0 = 0,= -x-i X, when x - 0,

= x3X+ XQ YI , X2 = xlX + x Y2 .- xlX +

(xf x2and

is the /T-module equivalentis an //-basis of

we findHence

to (xf, x(KI , x2 + XiXs).The extra members XiX2 ,more quickly by multiplying x2 + XiX3 first by xl and then by x2 .method given is a general one.

39. Lasker's Theorem (L, p. 51). Any given module M isthe L.C.M. of a finite number of primary modules.

Let M be of rank r. Express its first complete partial ^-resolvent])u(t'~ l) in irreducible factors, viz.

might of course have been foundThe


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Ill] GENERAL PROPERTIES OF MODULES 41and let Ci, C3 , ..., C5 denote the irreducible spreads, of dimensionsn - r, corresponding to Hi , H2 , , Rj respectively.

Consider the whole aggregate Mt of polynomials F for each ofwhich there exists a polynomial F', not containing Cit such thatFF' = mod M. We shall prove first that Mt is a primary modulewhose spread is 67, (i = 1, 2, ... , f).

Let Ft , F.2 be any two members of J/f . Then since F^F-! = mod M,andF2F2r - mod J/, where neither jFY nor F2' contains C,, we have(A 1Fl + A zF^)F1'F2' = ()modM, where I'V/y does not contain Ct .Hence A 1F1 + A ZF.2 belongs to the aggregate J/i, i.e. Mi is a module.Again, since FF' = Omod J7, .F contains


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42 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [illWe have F= mod (M,


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Ill] GENERAL PROPERTIES OF MODULES 4oa unique module J/(1) . The isolated spreads of J/ J/(1) are the relevantspreads of J/ imbedded to the second degree : and the L.C.M. of thecorresponding (indeterminate) primary modules and M(l) is a uniquemodule J/(i) . The process is continued until a module J/K) is obtainedsuch that M J/ l/l = (l), when there will be no more relevant primarymodules to find.

41. An unmixed module is usually understood to be one whoseis'Jnted irreducible spreads are all of the same dimensions ; but it isclear from the above that this cannot be regarded as a satisfactoryview. It should be defined as follow^ :

Definition. An unmixed module is one whose relevant spreads,both isolated and imbedded, are all of the same dimensions ; and amixed module is one having at least two relevant spreads of differentdimensions.

An unmixed module cannot have any relevant imbedded spreads.A primary module is an unmixed module whose spread isirreducible. This cannot be taken as a definition because the meaningof unmixed depends on the meaning of primary.

Condition that a module may be unmixed. In order that a moduleJ/ of rank / may be unmixed it is necessary and sufficient that itshould have no relevant spread of rank > r. This condition may beexpressed by saying that F=Q mod J/ requires F= modM where< is any polynomial involving xr+1 , ..., XH only. For if M contains arelevant primary module of rank > r a can be chosen which containsit, and an F which does not contain it but contains all the otherrelevant primary modules of M, so that < .F= mod J/ does notrequire .f7 = mod J/ ; while if J/ contains no relevant primary %module of rank > r there is no < containing a relevant spread of J/and F=0 mod J/ requires F= mod M 6 ; = modM ( 42).A primary module Q has a certain multiplicity (S 68). To a givenprimary module Q (fi) of multiplicity t*. corresponds a series of primarymodules Q (l\ Q (-\ ..., QW of multiplicities 1, 2, ..., p. all having thesame spread as Q (* } and such that Q (p} contains Q, Q, ..., QMmay be regarded as successive stages in constructing Q^. Tvprimary modules with the same spread and the same multiplii />//that one contains the other must be the same module.


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44 THE ALGEBRAIC THEORY OF MODULAR SYSTEMS [ill42. Deductions from Lasker's Theorem. A module of

rank n resolves into simple (primary) modules of which it is the pro-duct (25).

If M' does not contain any relevant spread of M then M\M' = M.Let M/M' = M". Then since M' M" contains M, and M' does notcontain any relevant spread of M, M" contains all the relevantprimary modules into which M resolves, i.e. M" = M.

It follows that ifM/M' 4= M, M' must contain a relevant spread ofM.Thus ifapolynomialF exists such that (x^a^F, (x2 a2) F, . . . , (xu -an ~)Fare all members of M, while F is not, M contains a relevant simplemodule whose spread is the point P(a^, a2 , -, an) ; for M/P^M.

Example. The module M = (as*, x^, x* + x x4 + x^xzx^ has arelevant simple module at the origin; for XiX*x* is a member ofM(i-l, 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 otherrelevant primary modules of M, is (x^, x, x3 , #4) ; cf. Ex. iii, 17.This example shows that it is possible for a mixed moduleM to containa relevant primary module of higher rank than the number of membersin a basis ofM. For the rank of (xi\ x23 , xs , #4) is 4.

IfM is an //-module not having a relevant simple module at theorigin the variables can be subjected to such a linear homogeneoussubstitution that xn will not contain any relevant spread of M, and weshall then have M/(xn) = M, and AT will be equivalent to MXn =\ ( 38).Thus the only condition (remaining permanent under a linear substitu-tion) that an H-module M may be equivalent to the module MXn=\ isthat M should not contain a relevant simple module.A simple //-module M is not equivalent to MXn=\ ; in fact Mj-n =i

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