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Cambridge Tracts in Mathematics and Mathematical Physics
General Editors
J. G. LEATHEM, M.A. E. T. WHITTAKER, M.A., F.R.S.
No. 6 Algebraic Equations,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
C. F. CLAY, Manager.
Hontion: FETTER LANE, E.G.
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[All rights reserved]
ALGEBRAIC EQUATIONS
by G? Bf MATHEWS, M.A., F.R.S.
Fellow of St John's College.
Cambridge:
at the University Press
1907
.l^h\i^
(STambriljge:
PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS.
A 111
PEEFACE.
nnniS tract is intended to give an account of the theory of equations -*- according to the ideas of Galois. The conspicuous merit of this method is that it analyses, so far as exact algebraical processes permit, the set of roots possessed by any given numerical equation. To appreciate it properly it is necessary to bear constantly in mind the difference between equalities in value and identities or equivalences in form ; I hope that this has been made sufficiently clear in the text. The method of Abel has not been discussed, because it is neither so clear nor so precise as that of Galois, and the space thus gained has been filled up with examples and illustrations.
More than to any other treatise, I feel indebted to Professor H. Weber's invaluable Algebra, where students who are interested in the arithmetical branch of the subject will find a discussion of various types of equations, which, for lack of space, I have been compelled to omit.
I am obliged to Mr Morris Owen, a student of the University College of North Wales, for helping me by verifying some long cal- culations which had to be made in connexion with Art. 52.
G. B. M.
Bangor, August, 1907.
CONTENTS.
CHAP. PAGE
I. Galoisian Groups and Resolvents 1
II. Cyclical Equations 30
III. Abelian Equations 41
IV. Metacyclic Equations 45
V. Solution by Standard Forms 55
Notes and References 62
CORRIGENDUM.
p. 9, line 11 from bottom the text should be
"because v,i + i is derived from v^ + i [7iot v^]...'
CHAPTER I.
GALOISIAN GROUPS AND RESOLVENTS.
1. Suppose that Ci, Ca, ... c« form a set of assigned algebraic quantities, and that
/(^) - ir" + Ci^**-^ + ...+Cr a;''-'' + ... + c„. If we can find another set of algebraic quantities cci, x^^ ... x^ such that
2^j = -Ci, ^XiXj = C^, ..., XxX^ ... Xn^^-^Cn (l)
we shall have identically
fix) = (^ - X^ (X-X2) ... (X- Xn).
Under these circumstances (supposing that th^ algebra we are using is the ordinary one)
for x = Xi, X2, ... Xn and for no other values of x.
Thus every solution of (1) leads to the complete solution of the equation /(^) = 0. Conversely the complete solution oif{x) = 0 in the form x = ^i, ^2, ••• ^n leads to the complete solution of (1), considered as a system of simultaneous equations, in the form
Xi, ^2> ••• '^n ~ taj fcb) ... ^l
where $a, ib, •" ii represents, in turn, every permutation of
tl) %2) ••• %«•
If the values ^1, 4, ••• in are all distinct, /(x) = 0 has no multiple roots, and the solutions of the simultaneous equations are all distinct, and are nl in number.
If f(x) = 0 has multiple roots, its solution may be made to depend upon an equation without multiple roots. Suppose, for example, that f(x) has a root r of multiplicity a ; then the first derived function /i (x), that is to say df/dx, has a root r of multiplicity (a - 1). Hence
M. 1
2 GROUPS AND RESOLVENTS [CH. I
if ^ = dv(y; /i), the highest common factor of/ and /i, the equation //<t> = 0 has coefficients which are rational functions of Ci, C2, ••• c„, and its roots are the distinct roots of /(.^), each occumng only once. Moreover, if /i = d'/lda^, we can, by finding dv(/i, /a), dv(/2, /g) and so on, determine by rational operations the exact multiplicity of any repeated root of /= 0 : hence the complete solution of //<{> = 0 leads to that of /= 0. In all that follows it will be assumed that / has no multiple roots.
2. It has been proved in various ways that the roots of /(^r) = 0 actually exist ; that is to say, if real or complex values be assigned, at pleasure, to the coefficients, then there are exactly n determinate real or complex numbers a^u x^-, ... Xn such that
f{x) = U{x-x^
for all values of x. Another theorem which will be assumed throughout is that every rational symmetric function of the roots can be expressed as a rational function of the coefficients.
3. What gives special interest to the subject in hand is that the actual determination of the roots of a given equation is a problem which differs in complexity according to the assumptions made with regard to the coefficients, and the value of n. Thus, if n < 5, and the coefficients are left arbitrary, it is possible to construct an explicit algebraic function of the coefficients which is a root of the equation. For 72 > 4, this is no longer the case ; a fact first proved by Abel, who also perceived the real reason for the limitation, namely, the special properties of the group of permutations of n different things when 71 < 5.
When the coefficients are numerically given, the rational roots, if any exist, can be found by trial, and the values of the irrational ones can be found by approximation. With these processes of approximation, however, we shall not be concerned ; our main problem is, in fact, the following :
Given a particular equation with numerical coefficients, it is re- quired to find the simplest set of irrational quantities such that all the roots of the given equation can be expressed as finite rational functions, in an explicit form, of the set of irrationals. What is to be understood by the simplest set of auxiliary irrationals will appear as we proceed.
4. Before entering upon the general theory, it will be useful to consider the case of a cubic equation with arbitrary coefficients, and
1-4] GROUPS AND RESOLVENTS 3
roots a, (3, y. Since the value of a + ^ + y is known, it will be sufficient if we can find the values of two other independent linear functions of the roots. If we take an arbitrary linear function a + 1/3 + my, this will, in general, assume six values by the permutation of a, f3, y: these values will be the roots of an equation
2/^ + m^y^ + ... + me = 0
the coefficients of which are rational in I, m and known quantities. Let us try to make this a quadratic in ^. Then if w is a complex cube root of unity, there will be six roots of the form
Assuming, as an identity independent of a, y8, y, a + 1/3 + my = oi {P + ly + ma) we have l^o), m^oy^ : so that we obtain a function
^1 = a + wy8 + w^y
the values of which, when a, /?, y are interchanged, become
3^2 = a + a)2^4-coy,
y3 = Q)2a + (u^ + y-(o2y2,
3^4 = (oa + 0)2/3 + y = 0)3/1,
^5 = o)a + ^ + o)2y = o)3/2, ^Q = (o^a + p + <ay = in^yi.
Consequently
Vi + y^ =(a+o)/3 + a)2y)3 + (a + w^/? + o)y)3 = A,
a quantity symmetrical in a, ^, y, and therefore rational in the coefficients of the given cubic ; in fact,
A = 22a3 - Sta'fi + 12af3y = - 2Ci^ + dc^C^ - 21Cs.
Similarly y^y^ = ^a^ - :Sa/3 = c^^ -Sc^^B
another rational function of the coefficients: so that y^^, yi are the roots of the rational equation
f-Af + B^ = 0.
Let ,^|^ + V(^--4^y
with a fixed determination of the radicals involved. Then we may put
a + /3 + y = -Ci, a + 0);8 + o>2y = 9, a + 0)2/8 + o>y = BjO,
1—2
4 GROUPS AND RESOLVENTS [CH. I
and hence
By giving ^ all its six values, we obtain all the six permutations of
o, A y-
It will be noticed that the success of this method depends on finding a power of a linear function of the roots which is a two-valued function of the coefficients ; this has been done with the help of an auxiliary number <o which is a root of the rational quadratic w^ + o> + 1 = 0.
In a similar way for the general quartic
(a-P + y-Sy
is a three-valued function of the coefficients, and may be explicitly found by means of an auxiliary rational cubic ; after this the solution of the quartic may be completed.
6. If, after the manner of Lagrange, we try to extend this process to a quintic, we take «, a complex fifth root of unity, and form the rational equation satisfied by
The degree of this is 24, and it is only in special cases that it can be solved in a manner similar to that which is applicable in the foregoing examples. Thus the method breaks down ; at the same time, a generalisation of the process, due to Galois, is of the highest importance in the whole of the theory.
6. Galois begins by considering the rational equation satisfied by the most general linear function of the roots. Let m,, i/a, •• «^n be a set of absolutely undetermined symbols, subject merely to the ordinary algebraic laws of combination ; and for the sake of brevity let n\ = fi. If we pat
i = n «, = ttia?! + UtXt + ... + UnXn = 2 UiXt,
where 4^, «^, ... at. are the roots (all different) of/{x) = 0, we can obtain from 1^, by interchanging the roots in all possible ways, fi essentially different expressions v, , v,, . . . tv. The product
4-7] GROUPS AND RESOLVENTS 5
where 'y is a new indeterminate, is an integral function of v with coefficients which are integral and rational in C], Ca, ... c^ as well as in Ui, Uo, ... Un because F(v) is a symmetrical function of the roots of/.
The equation F(v) = 0 is called the complete Galoisian resolvent of f{x) = 0. Its discriminant is a rational integral function of Ci, c.2,...Cn, th, U2,...Un, which does not vanish identically: so that we may, if we please, assign numerical values to the parameters til, U2, ...Un without making any two roots of the resolvent equal to each other. In particular, these numerical values may be ordinary real integers.
7. The most important property of F is that any rational function of the roots off can he expressed as a rational function of any one of the roots of F.
Let the given rational function be <;^(iz?i, ^2» ••• ^«), and let
be the expressions obtained from <ji by applying the substitutions which derive '^i, v^, v^, ...v^ from Vi . These expressions cf>i are not necessarily all different in form ; and two which have different forms may have the same value. But it must be remembered that (f>i is derived from <^i by the same permutation which changes Vi to Vi. Consider the expression
^ ^ ^ [V-Vi V-V.2 V-Vy.) ^ ^ '
\l/{v) is an integral function of v, in general of degree (/x-1), but possibly lower, and it is a symmetric function of ccx, x^, ... Xn. Hence the coefficients of ypiv) can be expressed as rational functions of Ci, C.2, ...Cn', and if, after doing this, we put v = v-^, it follows from the above identity that
or <f>^ = ^p^ = B(Vi; Ci,C2,...Cn; Ui,U2,...Un)
where B denotes a rational function of the quantities in the bracket. This equality reduces to an absolute identity if on the right-hand side we replace '^i, Ci, ...Cn by their expressions in terms of x^, ^2, •••^n, Ui, U2, ...Un.
The discriminant of F is
^ = F'(vOF'(vd"-F'M,
6 GROUPS AND RESOLVENTS [CH. I
and the quotient A/F'(vi) is expressible as a rational integral function of Vi : hence we may also put <^ into the form
^ A - A
where J(vi) is a rational integral function of Vi.
It should be observed also that <^,- can be expressed as the same function of Vi that <^i is of v^.
Finally, <^i is expressible as a rational function of any root of F(v). Thus if we choose Vi, all we have to do is to replace, in the foregoing proof,
Vu V2, ... v^
by 5i(^i)» ^iW, ■.. Si(v^),
where Si is the perfectly definite substitution which converts -^i to Vf.
In general, <^ is not the same rational function of Vi as it is of v^.
8. Several important consequences immediately follow from the theorem just proved. In the first place, we may put <f> = Vi, and thus infer that
All the roots of the Galoisian resolvent may be expressed as rational functions of any one of them.
An equation having this property is called a normal equation ; the Galoisian resolvent is accordingly a normal equation. It must be remembered that the same equation may be normal from one point of view and not from another, if, in the definition, we understand "rational function" to mean "rational function with rational coefficients." By a field of rationality we shall understand the aggregate of all the expressions obtainable from a finite set of symbols ^1, ^2j ••• ^m by a finite set of rational operations ; that is to say, all the expressions which can be reduced to the form
^(tut„ ...tj'
where </>, if/ are finite polynomials with ordinary whole numbers for their coefficients. The elements ti, t<i, ••■ tm may be partly undetermined parameters, or umbrce, partly determinate numbers ; those which are numerical may be irrational arithmetically, but are here considered rational in the sense of being given or determined. The simplest field of rationality is that of ordinary rational numbers; this is contained in every other field.
If tm + i is any algebraic number or symbol not contained in the field (^1, ti, ... tm), the field (^i, ti, ... t^, tm + i) is said to be obtained from
7-10] GROUPS AND RESOLVENTS 7
the former field by the adjunction of tm+\'. this term is specially- employed when tm+\ is a numerical quantity.
In the case of the Galoisian resolvent we may say, then, that it is a normal equation in the field
(Ci, C^i '•• Cn\ Ihi '^2) ••• U-n)'
9. If, in the theorem of Art. 7, we put <f> = cci, we arrive at the proposition that
Every root of an eqttation witJwut multiple roots can be expressed as a rational function of any one root of its Galoisian resolvent.
If rational values are given to the parameters Ui, u^, ... %, the resolvent equation becomes normal in the field (ci, c^, ... c„). More- over if Ci, Ca, ... Cn are given, not as symbols, but as actual numbers, the resolvent becomes a definite numerical equation. Unless this equation has multiple roots, it is still true that the knowledge of the value of any one root of the resolvent leads to the complete solution of /= 0 ; because to calculate the function xj/ (y) of Art. 7 in its rational form it is sufficient to know the values of the elementary symmetric functions of ^i, iTg, ... x^^ and these are given by/.
10. The total resolvent F(v) may or may not be reducible without adjunction ; in the second case/(iy) = 0 is said to be an equation with- out affectimi.
The irreducible factors of the resolvent oj an affected equation are all of the same degree.
Let i/^i {v)y xl/2 (v) be any two such factors : let '^i be any root of ^^ (i;) = 0, and V2 any root of xf/2 (v) = 0. Then (Art. 7) v^ can be expressed as an integral function, / ('^i), of Vi . If the Tschirnhausen transformation y--=J(x) is applied to xj/i (x) = 0, we obtain an equation ^(^) = 0 of the same degree as i/'i = 0 which has a solution y^Vz in common with if/^ (y) = 0 : hence x (3/) is divisible by xj/^ {y\ and the degree of i/^i cannot be less than that of 1/^2. By a similar argument, the degree of j/^o cannot be less than that of »/^i ; therefore the degrees must b^e equal.
If h is the degree of each irreducible factor, we have an identity
with mh = ix,
so that m and h are conjugate factors of /x.
Every one of the equations \\/i (v) = 0 is normal, and they are all Tschirnhausen transformations of any one of them. Each may be
8 GROUPS AND RESOLVENTS [CH. I
called a pi'imary resolvent of f{x) - 0. The knowledge of any one root of a primary resolvent leads to the complete solution of f(x) = 0.
11. A simple example will help to illustrate the results so far obtained. Let the given equation be
and let a, 6, c be used instead of Ui, Wa, ^s-
The complete resolvent is F= <f>x^i where
<^=(v-a)2 + (6-c)^ x = (^-^)'+(c-a)% ^ = (:v-cy + (ia-by. One root of <^ = 0 is a- bi + ci, and from this the roots 1, i, - i of the original equation are obtained. If w^e put
Vi = a- hi + ci,
then ±-j , 1,
b-c
give the roots of/=0 as rational functions of Vi.
12. The reducibility of F shows the existence of asymmetrical functions of Wj, ^2, ••• ^n which nevertheless have rational values. The coefficients of the terms of a primary resolvent if/ (v), considered as a polynomial in -y, «^, Wg, ... ^«, are all rational ; but when expressed in terms of ^2^1,572, ... cPn they cannot all be symmetrical, otherwise every permutation of the roots of / would leave ^ (v) unaltered, and this is not the case.
13. Consider now a primary resolvent
^i{v) = (v-Vi){v-V2) ... (v-vn). Any one of its roots, say Vi, can be derived from Vi by a perfectly definite permutation of a?i, a^2> ••• ^n : let this be called Si. Including the identical substitution Si, we have in connection with \f/i just k substitutions Si, S2, .-. Sh. It is a most important theorem that these substitutions form a group ; that is to say, for every pair of substitu- tions Sa, Sh (the same or different) we have SaSb = Sc, where Sc is a definite substitution of the same set.
It follows from Art. 7 that since Vu and «i are both roots of F(v)=0, there is an integral function J{v) such that
Moreover it appears from the same article that
J(Va) = Sa (Vb) = Sa {Sb (Vi)}.
10-15] GROUPS AND RESOLVENTS 9
But since the equations
have a common root v^, and the first is irreducible, while both are rational, each root of the first is a root of the second, and in particular
that is to say, Sa {su (v^)} is a root of i/^i (v) = 0, and is therefore equal in value to Sc (vi), where Sc is a substitution of the set 5i, Sg, ... s^. But this equality in value must also be a coincidence in form, on account of the arbitrary nature of the parameters u^, u^, ... w„. Hence
it being understood that SuSa means the result of first applying Sj, and then applying §«. In a similar way SaSb = Sa; but Sa is, in general, different from Sc.
14. If xf/2 is any other of the primary resolvents, there will, in the same way, be a group of substitutions connected with it. This is, in fact, the same group as the one associated with i/^i. For suppose that
x}/.2(v) = (v-Vn+i)(v-Vh + 2) "• (.v-V2k): then v,i+i can be expressed in the form
and by the usual argument it follows that
^2 = {v-J M) {v - J(y,)} ...{v- J{vn)]. The notation may be so arranged that
J(Vi) = Vu+i (i=l, 2, ... h),
and this being so, we conclude that
because Vu+i is derived from 'y,i by the change of Vi into Vi, and the only substitution which does this is 5^.
The group (si, S2, ... s^) is called the Galoisian group of the equation f{x) = 0. If the complete resolvent is irreducible without adjunction, h-n\ and the Galoisian group consists of all the permutations of
15. We will now select any one of the primary resolvents, denote it by ^\l (v), and call it simply, for the present, t/ie resolvent of /(x). Assuming nothing about /(a;) except that its coefficients are actually given, F(v) and subsequently i{/ (v) can be found by rational operations. The degree of ij/ (v) in v at once gives the order of the Galoisian group.
10 GROUPS AND RESOLVENTS [CH. I
But we can go further than this, and determine, from an examination of if/ J the elements 5i, 5.2, ... 5^ which form the group. The notation may- be so arranged that
ij/ = (v-vi)(v-n2) ... (^-%),
Vi = Ui^i + u^^ + . . . + M„ir„.
Now the change of v^ into Vo effected by the substitution 52 may also be
effected by a substitution o-g operating on the parameters Wi, ^/2, ... w,j.
For instance, if
l?i = UxXx + U<]X^ + U^z + ^4^74 + U^^ + 7^6^6j then 52 = (^1^-2^4) (^s^e^o), 0-2 = (U1U4U2) (u^UaUe).
In general, if 5,- contains the cycle (iVa'^h ••• ^t^Oj ^i contains the cycle (uiUk ■ . • UiUa) and there is a one-one correspondence between the substitutions St and the substitutions o-^. If o-j is applied to if/ (v) in its rational form, the result is a function x ('^) of the same order, which has a root Vt, and therefore coincides with ilf(v). Thus there are at least h distinct permutations cr, forming a group, which leave il/(v) formally unaltered. The same argument applies to the other primary resolvents obtained from F, and since there are only hm substitutions o- altogether, it follows that there are precisely k substitutions o- which leave \f/ formally unaltered ; from each of these we can deduce uniquely a substitution s belonging to the Galoisian group.
For instance, in the example of Art. 11, if we take «/- as the resolvent,
oTi = 1, 0-2 = (ab),
and the corresponding Galoisian group is
5i = 1, Si= {sciX^. After obtaining the elements of the Galoisian group
G={Su 52, ... S/0,
its properties, as a group of substitutions, or more generally as an abstract group, may be investigated. These are, in themselves, wholly independent of the values of ^1, x^, ... ar„.
16. It will now be supposed that the coefficients of / are numerical ; and, as explained in Art. 8, any quantity in the field (<;,, C2, ... c„) will be considered rational, no matter whether the coeffi- cients Ci are arithmetically rational or not. It will now be proved that
Every rational function of the roots of f which is unchanged in numerical value by the substitutions of the Galoisian group has
15-17] GROUPS AND RESOLVENTS 11
a value which can he exjyressed in a rational form : that is to say, it is equal in value to a certain rational function of the coefficients off
Let the given function be <^ (^i, ^2, ••• ^«) and letvi,^^,... Vn be the roots of the resolvent if/ (v). Then (Art 7) there is an integral function J(v) such that
<f> = <t>l = J(Vi) <I>2 = J (^2), <A3 = •^('^3), • • . </>A = /(%),
where <^2, ^2, ••• i>h are derived from <^ by applying the Galoisian substitutions Sa, S3, ... s^. Hence
<t>i + <f>2+ •..+ff>h = JM + J{V2)+...+ J(Vh)
— ^ [plj (^2, ••• (^n } Ui, U2) •-• Unj,
where /S' is a rational function, because '^J(Vi) is a symmetrical function of v^, v^, ... Vu and the coefficients of \l/{v) are rational. If, now^, ^i means the value of <^i, we have, by hypothesis,
7 ^ \Ci, C2, ... Cji ', Ui, U2, ... Un)
where 8 means the value of the rational function >S'.
If the coefficients Ci are represented symbolically, the function S, even in its lowest terms, may contain the parameters explicitly ; in this case the value of <^ is expressible as the quotient of any numerical coefficient in the numerator of S by the corresponding coefficient in the denominator. The fact that we thus have alternative rational equivalents for <^ implies one or more rational relations connecting the coefficients Ci. If, on the other hand, the coefficients Ci are actually given as numbers in a definite field (for instance, if they are all of the form a + /3j2, with a, ft rational numbers in the ordinary sense) the parameters^ at the last stage of the process, disappear of themselves, and we obtain the value of <^ as a definite number in the field. The point of the proof is then that the value in question is expressible as a quantity in that particular field.
17. Conversely, eve?y 7'ational function of the roots which has a rational value keeps that valus when any substitution of the Galoisian group is applied to it.
Let <^ be the rational function, and A its rational value. Express- ing <^ as a rational integral function of Vi, we have
12 GROUPS AND RESOLVENTS [CH. I
and hence the rational equation
J(v)-A=0
is satisfied by v^ and consequently by -i^i, Vg, ... v^.
Thus J(Vi) = A;
that is to say, A = s, /(vi) = Si<f>,
which proves the theorem. It must be remembered, of course, that Si<f> may or may not be formally different from <^. Moreover, in any actual case, if we reduce J{vi) to a degree lower than h by means of ^ (t'l) = 0 we shall in the end obtain A explicitly, if the value of <fi is actually rational : so the process of Art. 7, applied to a particular function <f> and a particular equation /, decides whether the value of <f> is rational or not.
Finally, there are rational functions of the roots which have rational values, but change these values when substitutions other than those of G are applied to them.
To show this, let 6 be an undetermined rational quantity ; then
^l,{6) = {e-v,){e-v,)...{6-v,) = A,
where A is rational in (6 ; c^, c^, ... c„ ; Ui, u^, ... Un). If t is any substitution not contained in the Galoisian group, tif/ (6) = \f/i (6), where if/i is a primary resolvent distinct from if/. Considered as an equation in^,
cannot have more than (^ - 1) roots, even when the parameters have fixed numerical values (subject to the usual restriction A #= 0). Since there are (m-1) conjugate resolvents into which xf/ can be transformed, we have to exclude at most {h-l){m-l) values of 6. For any other rational value of 0, it is the substitutions of G, and these alone, which leave the value of if/ (0) unaffected.
Every coefficient of «/', considered as a polynomial in B^u^^u^, ... Un, is unaffected in value by the substitutions of (r ; it not unfrequently happens that some one of these coefficients, or a simple linear combination of them, can be seen to have its value changed by all substitutions not belonging to G ; in this case it may be taken instead of xj/ (0). For an example, see Art. 29 below.
As a result of the three theorems last proved we may define the Galoisian group of / as the aggregate of those pennutations of a*!, ^2, ••• ^n which leave unaltered in value eve7'i/ rational function of the roots which has a rational value.
17, 18] GROUPS AND RESOLVENTS 13
18. If <f> is any rational function of the roots of / it has been proved that <^ can be expressed as an integral rational function of Vi, and it has been observed that in virtue of i}/ (vi) = 0, this integral function can be reduced so that its degree does not exceed (A- 1). An independent proof of this affords a little more information. If, with the usual notation,
[V-Vi V-V2 V-Vu) ^ ^
xiv) is an integral function of v which is also rational, because it is unaltered by any substitution of G. Consequently
a rational function of v-^ , which may also be reduced to the form
S ' where 8 is the discriminant of i/', and j {v-^ is an integral function, which in virtue of \p (v-^ = 0 may be supposed put into its reduced form, so that its degree is not greater than {h - 1). If ff> is an integral function of the roots, the coefficients of j will be integral in
Cj, C2j ••• Cm Ml, U^t ••• Un-
Similarly, <^^ =j (vi) /8. • (i = l, 2, S, ... k)
The quantity 8 is not zero, because it is a factor of A.
The substitutions of G give to <^ the different forms <^i , </>2, ... <f>h' these, however, need not be all different in value. Those substitutions of the Galoisian group which leave <^ unaltered in value form a subgroup, or factor, oi G which may be called the invariant group of<f>. ^
In fact, if Sa, Sb are any two such substitutions,
numerically: hence 5a<^-<^ = 0,
and since the expression on the left hand is a rational function of the roots which has the rational value 0, we may, by Art. 17, apply the substitution s^ to it, and conclude that
Sb(sa<t>-<}>) = 0; that is, Sb (sa<f>) = Sb<f> = <i>
numerically. Hence SaSb leaves the value of <f> unaltered, and the
14 GROUPS AND RESOLVENTS [CH. I
substitutions in question form a group, because SaSt, is identical with a substitution of Gy and it has been shown that it leaves <^ unaltered in value.
It must be carefully remembered that the invariant group of <^ consists exclusively of substitutions which belong to G. There may be other substitutions which leave <f> unaltered in value, or even in form, but if they are not in the Galoisian group they are not to be included. The fact is that we cannot infer for certain that if Sai>-^ = Oi then Sb(Sa<t> - <f>) = Oj unless Si, belongs to the Galoisian group (cf. Art. 17, end).
Writing, as usual, Si<f> = <f>i, the function </> is a root of the rS-tional equation
But if the invariant group of <^ is of order ^ > 1, the roots of this equation are repeated each k times : hence if we put k/k = I, which is necessarily an integer, <^ is a root of a rational equation
^l^h,^'-'-¥ ... + bi = 0.
19. If fix) is rediicible without adjunction, its Galoisian group is intransitive^ and conversely.
First suppose that G is intransitive: this means that a certain number of roots
iTi, iTa, ••• scr (r<n)
are only interchanged among themselves by the substitutions of G. Consequently (Art. 16)
(X - Xi) {X- CC^...{X- Xr)
being unaltered by any substitution of G has rational coefficients, and f(jt!) is reducible without adjunction.
Conversely, suppose th.&tf(x) has a rational factor
fi(x) -{x~ x^ (x - X2) ...(x-Xr) (r < n)
then, if G is transitive, it must contain a substitution s, which converts some one of the roots x^ x^^ ... Xr, say Xi, into a root Xr+i, formally different from Xi, x^, ...Xr. Hence sf contains the factor (x-Xr+i): but since f is rational ^1 =/, , and consequently
0 =/i (Xr+i) = {Xr^i - Xi) (Xr.,1 - X^) . . . (^r+1 " ^r) ;
implying ihsitf(x) = 0 has equal roots, contrary to hypothesis. Hence if /(a:) is reducible, G is intransitive. The example of Art. 1 1 gives a simple illustration.
It is possible to resolve /(^) into its irreducible factors by means of rational operations, even when the coefficients are connected by known
18-20] GROUPS AND RESOLVENTS 15
algebraic relations. Unless the contrary is expressed, it will be assumed henceforth that / is irreducible without adjunction.
20. Suppose that by the adjunction of a quantity 0 the resolvent if/ becomes reducible in the field (0 : Ci, Ca, ... c^ : Ui, ?^2, ... Un). If we have
it follows by comparing coefficients that 0 satisfies one or more rational equations in the original field. These must be consistent with each other, so that 0 must satisfy a definite irreducible equation
a(e)= 0^ + a.e^-'' + ... + ai = 0
with rational coefficients, which we may suppose integral because, if necessary, 0 may be replaced by z6, where z is any rational quantity.
If, by any means, this irreducible equation has been found, it is possible to actually resolve ^ into its irreducible factors in the new field; and this resolution is unique. We shall have
'/' = XlX2•••X^
and Vi will be a root of one of the irreducible equations Xt = 0- Arranging the notation so that xi (^i) = 0, and for convenience putting Xi = Xj we have an equation
xW = o,
which, in the new field, will serve as a primary resolvent of /= 0. This is clear, because x('^) is only a transformation of a product
SO that (Art. 1) x(y)-^ is a normal equation ; and every rational function of ^i, ^2, •••^w can be expressed, in the new field, as an integral function of Vj , the degree of which is less than that of x^ and which is not of higher degree than (I- 1) in 6. As in Art. 10 it can be proved that the functions Xij Xa? ••• X^ ^^^ ^11 of the same degree in v, and are Tschirnhausen transformations of each other.
In expressing any rational function of ^j, . . . Xn as a reduced function of Vi in the new field, we may proceed as follows. In the original field let <^ =j(vi) be the reduced expression for <f> (Art. 18) ; divide j(v) by X (v) until the remainder is of degree lower than that of x- We thus obtain an identity
j(v)=Q(v)x(v) + k(v),
and by putting v=Vii we have
<^ =i(^i) = K^^O,
16 GROUPS AND RESOLVENTS [CH. I
because xC^O'^O. The coefficients of k, which are integral functions of B, may be reduced to their lowest degree by dividing them by a{6).
It will be noticed that x(^') must contain 6 explicitly, because it is a factor of i/'(t'), which is irreducible in the old field.
21. We are now approaching the culminating point of Galois's theory. Unless (r is a simple group, it will contain self-conjugate factors distinct from the identical substitution : and among these there will be a certain number of maximum self-conjugate factors. Let r be a maximum self-conjugate factor of (r, of order k and of index l{ = k/k) with respect to G. The notation may be so arranged that
Let z be an undetermined rational number, and
e = <^(;ri, Xi,...Xn) = (Z- V^) (Z-V2)...(Z- Vk),
where Vi, V2,--'Vk are the roots of the resolvent 1/^ = 0, which correspond to the substitutions of r. Then the value of 0 is unaltered by any substitution of F, and by choosing z properly (Art. 17) we can make sure that the value of 6 is altered by every substitution of G which is not contained in T.
Consequently ^ is a function of which T is the invariant group, and is a root of a rational equation
a{e) =6^ + «i^'-i + a^e^-"" + ...+ a^ - 0.
So long as 2;, Ui, U2,,.,Un remain undetermined, the coefficients in this equation are integral in the field {z\ u^, u^^,...Un\ Ci, C2,...Cn): it is possible to give fixed rational integral values to z, u^, ?^2 , • • • w„ so as to make the coefficients rational in (ci, Cg, ...c„).
22. It is important to determine the Galoisian group of the equation satisfied by 0. To do this, it is necessary to use a lemma, derived from the elements of the theory of groups. All the substitu- tions of G may be arranged in the form
Sl, Siy ... Sjc
tiSu tiS^y ... tiSk
where t^ ts, ... ti are distinct elements suitably chosen from G.
If any substitution s of G he applied by premultiplication to the elements of a row in this scheme it will produce a new row which con- sists either of the elements of the same row, usually in a different order,
20-22] GROUPS AND RESOLVENTS 17
or else the elements of another row, usually in a different order. In no case can elements of the same row be changed into elements of two different rows.
To prove this, suppose, if possible, that, for instance,
where a, h are different. Then, since 5i, Si,...Sk form a group,
therefore taSi = hSj, ta = hSjSf^ = t^Sr
which is impossible, because tiyS,. is in the ^th row, and (on account of the way in which t^, U,-..ti are chosen) is distinct from #„, which is in the ath row.
Hence we may say that the application of any substitution of G produces a permutation of the rows of the table. These permutations form a group, denoted by (r/r, and called the complementary group (or factor-group) of G with respect to r. The only substitutions of G which leave the first row in its place are the elements of r, and these leave every other row in its place, because
SitjSy, = tjSiSk = tjSm
for all values of i, j, k, since r is self-conjugate.
Moreover any substitution which converts the first row into the ith. must be of the form ttSa. Applying this to any element tjSb of the ^*th row, we obtain
tiSa . tjSf).
Now because r is self-conjugate, we may put
Sjj = tjSc,
and hence tiSatjSb = titjScSb = titjSc.
Finally titj = tkSa, where h is a definite substitution determined by ti, tj alone: hence
titjSc = tjcSdSc = tkSe
and the substitution tiSa converts the ^th row into the /?;th. Conversely, the only substitutions which change the Jth row into the ^th are those which change the first row into the ^th. Consequently G/T, considered as a group of permutations of rows, may be represented in the form
(ti, T2,...Tj)
where n is the definite substitution of G/V which changes the first row into the *th.
2
18 GROUPS AND RESOLVENTS [CH. I
The substitutions t^, t2, ...ti do not, as a rule, form a group : but they behave like a group when considered as operations on the rows of the table.
23. It will now be shown that the Galoisian group of the equation a(0) = 0 is holoedrically isomorphic with G/T. The values of 0 are all different, and we may denote them in such a way that
e,^^A=t.sA. (;:;;t::l)
This being so, every permutation of rows in G/T corresponds to a permutation of (^i, 0^, ... 6^, and every substitution of G produces on (^1, 62, ... Oi) the same permutation as it does in the rows. Now let Q(6i, 0^, ... Oi) be any rational function of the roots of a(0) = 0 which has a rational value. Then
where B is another rational function. Since the value of R is rational, it is unchanged numerically by any substitution of G. This substitu- tion applied to Q produces a permutation of ^1, ^2> ••• ^i corresponding to an element of G/T. If, then, H is the group of permutations of ^1 , ^2 , • • • ^i which is holoedrically isomorphic with G/F, considered as a permutation of rows, every substitution of H must leave Q (^i, B^, ... 0^ unaltered in value. Conversely, if $ is unaltered in value by every substitution of H it must be rational, because in this case every substitution of G leaves it unaltered in value. Therefore (Art. 17) H is the Galoisian group of a (^) = 0; and we may put H= G/T, in the sense that these two groups are holoedrically isomorphic.
Since GIT is transitive, IT is so too, and hence the equation in 6 is irreducible (Art. 19). Moreover, we can prove, as in Art 18, that it is a normal equation, by taking the function
lA^A^-^e^y"^
where <^ is any rational function of ^1, ^2, ••• ^j and <^, <^2, ••• <^i are the functions derived from it by applying the substitutions of G/T.
24. Consider, now, the effect of adjoining 6^ to the field of rationality : this means that every function Ii{6i; Cj, Ca, ... Cn) which is rational in form is to be considered rational in value. The group T is the largest group in G which leaves the values of all such functions unaffected, and it is, in fact, the Galoisian group oi f{pc) in the new
22-24] GROUPS AND RESOLVENTS 19
field. To prove this it has to be shown that every rational function R{^u ^2, ••• ^n) which has a rational value A in the new field is expressible as an explicit rational function of 0-^. To prove this, take the function
6 being arbitrary, and apply to it all the substitutions of G. Then the function
can be expressed in a rational form
S(0: Ci, Ca, ... c„) (Arts. 7, 17). Now if
so that a(0) = 0 is the irreducible rational equation satisfied by ^i in the old field, we have
where k = h/l. Moreover, among the denominators
(e-e,),(0-e,),...(e-e,)
only I are distinct, namely. Hence it follows that
where T (6) is a rational integral function of 0. If the value of B is unaltered by each substitution of r, all the fractions with the de- nominator 0 — 0-^ must have the same value B in the numerator, and we may write, as an arithmetical equality,
true for all values of 0. The quantities L2, L^, ... Li are all rational functions oi x^, x^, ... Xn. By putting 6 = 6^, we obtain
and this may, if we please, be replaced by an equivalent integral function of degree not exceeding (/-I).
2—2
20 GROUPS AND RESOLVENTS [CH. I
The theorem proved amounts to this : —
If B is a rational function of the roots of f{x) = 0, which has for its inva7'iant group a self -conjugate factor y F, of G, the effect of adjoining 6 to the field of rationality is to reduce the Galoisian group of fix) = 0 from G to V,
25. In the new field we can construct a new total resolvent for f{x). In fact, if (v — v^) is any factor of the old resolvent i// {v\ and if the substitutions of V give Vy the values Vj, v^y ...Vji, then the new
total resolvent is
F-y{v) = (v-Vi) (v - V2) ... (v - Vk)
= 7^+Pi1^~'^+ ... +p]c
where the coefficients are rational in the new field. In one, at least, of these coefficients 0^ must occur explicitly, because il/(v) is irreducible in the original field. Moreover
il^{v) = F,(v)F,(v)...Fi(v) where Fi (v) is obtained from Fi (v) by changing ^1 to ^f, then expressing Oi and its powers in terms of ^1, and finally reducing the coefficients by means of a (6^) = 0.
If Fi (v) is reducible in the new field, all its irreducible factors must be of the same degree (cf Art. 10), and any one of these may be taken as a new primary resolvent. Every root of / may be expressed as a rational function of -i^i, ^1, Ci, Cg, ... c„, ^^, ^^2, ••• «^n> where Vi is any root of the new primary resolvent.
26. The equation a(0) = 0 satisfied by the adjoined irrationality 61 is usually called a Galoisian resolvent of /(x) = 0 : but we shall find it convenient to call it a Galoisian auxiliary equation, or simply an auxiliary equation when there is no risk of mistake. On the other hand the equation Fi(v) = Oy obtained in the last article, may be properly called a resolvent.
If we form the auxiliary equation according to the general method of Art. 21, its coefficients will contain the parameters Mi, e^, ... m„ in a complicated manner. In any practical case we at once simplify the auxiliary equation as far as we can by giving definite values to the parameters, thus making 6^ a definite numerical irrationality to be adjoined to the field. It may or may not be convenient to give definite numerical values to the parameters as they occur in Fi (v) : for some purposes, even in a practical case, it may be convenient to leave them umbral. This is one of the main reasons for distinguishing between an auxiliary and a resolvent equation : in other respects they
24-27] GROUPS AND RESOLVENTS 21
are similar, for example they are both normal. The real service rendered by an auxiliary equation is to define a new field of rationality in which the Galoisian group of /(^) = 0 is of lower order than it was originally, while at the same time the Galoisian group of the auxiliary equation in the miginal field is of lower cyrder than that off(x) = 0. Unless this last condition is satisfied, we do not gain anything by the construction of an equation a (0) = 0, even though the adjunction of one of its roots lowers the order of the Galoisian group of/; because in this case the Galoisian group of a(0)=0 is, in its abstract form, just the same as that of f{cc) = 0, and we are confronted with the original problem in another shape.
If, however, as we have supposed, ^ is a rational function of the roots of / which has for its invariant group a proper self-conjugate factor of G (that is, one which is not merely the identical substitution), the problem is really simplified by being made to depend upon two equations
a(^)-0,
'/'i(^, ^i) = 0, where the first is of order /, a proper factor of h, and has a Galoisian group of order I in the old field ; while the second is rational in the field obtained by the adjunction of ^i, any root of the first, and has a Galoisian group in the new field the order of which is either hjl^ or a factor thereof, and is equal in any case to the degree of \l/x in i?, if we suppose, as we may do, that i/'i is irreducible in the new field.
27. As soon as the original Galoisian group of / has been determined, we can construct what is called a composition-series for G in the form
G^ Gx, G2, ••' Gp, 1,
where Gi is a maximum self-conjugate factor of G, G^ a maximum self-conjugate factor of Gi, and so on. Using the conventions Gq=G, (tp+i = 1, we have a set of indices
^1 > ^2 ) • • • ^pi ^p + ii such that ei is the index of Gi with respect to Gi-i. The group Gp is simple and its order is ep + i.
We have seen that if we construct a quantity a, which is a rational function of ^1, ^2, ••• ^« and which has Gi for its invariant group, a will satisfy an equation
a (a) = a^i + aitt^'-^ + . . . + tte, = 0 which is rational and irreducible and normal in the field (cj, Ca, ... c„).
22 GROUPS AND RESOLVENTS [CH. I
By the adjunction of any one of its roots, we obtain a new field of rationality, which we may denote by (a, c), and in this field the group oi/isG,.
We can now construct a function for which G2 is the invariant group in the new Jleld. Let ^1, L, ... tm (where m^hjeie^ be the elements of 6^2, and let 6 be an undetermined rational quantity of the new field. We may arrange our notation so that
Vyy Vs, ... 'Om
are the expressions obtained from -^i by applying the substitutions of Gi ; and then, if we put
P is invariant for G^ in the field (a, c). By choosing 6 properly, as a rational function of a, it will be possible to secure that no other substitution of Gi leaves 13 numerically unaltered (cf. Art. 17). Employing a notation which is now usual, we may write
Gi = S1G2 + S.2G2 + . . . + Sefi2>
as an equivalent for a tabular arrangement such as that of Art. 22. Hence we see that the effect of applying all the substitutions of Gi to p is to produce me2 expressions which have only 62 different values, each repeated m times. They are the roots of an equation rational in the new field, and of degree mSi : but since all its roots are of multiplicity m, it is of the form {b (/S)}"" = 0, where b (/3) is also rational, and of degree e^.
Consequently ^ is a root of an auxiliary equation
b(l3) = P''+b,/3'^--' + ...+be,= 0 with coefficients which are rational in the new field.
This equation is normal, because (ri/(r2 is a simple and simply transitive group; hence by the adjunction of any one of its roots, all the others become rational, and the Galoisian group of / becomes G2 in the new field (a, /?, c). Moreover we have
F^(v) = {v- V,) (v - V2) .-.(v- v„,) =v'^ + q^v'^-'' + ... + qm
a total resolvent for / in tlie field (a, /?, c) with coefficients which are rational in that field. This process may be continued until the Galoisian group of / is reduced to Gp ; and finally, by forming an auxiliary equation of degree e^ + i, G is reduced to unity, and each root of / is expressible as a rational function of the field (a, p, ... \, c), where o, /8, ... X are roots of the (p + I) auxiliary equations. If
27, 28]
GROUPS AND RESOLVENTS
23
desirable, this rational function may be transformed so as to be integral in the adjoined irrationalities. The ^th auxiliary equation is of the form
^^• + ri^*~^ + ... +re. = 0,
with coefficients rational in Ci, c^, ... c^ and the selected roots of the preceding auxiliary equations.
28. It will be well to illustrate these very important results by a special example. Let the given equation be
Then if r is any one of its roots, r^ = l, and the other roots are r^, r^ r*, r^, r^. Thus we have a very simple case of a normal equation. It may be proved that fix) is irreducible without adjunction : this will, indeed, appear incidentally from what follows. If we put
v-y = ar + br^ + cr^ + dr^ + e7^ +fr^, V2 = ar^ + br* + cr^ + dr + ei^ + /?*^, V3 = ar" + br^ + cr^ + dr^ + er +fr^, V4 = ar^ + br + c/-^ + dr"^ + er^ +/^, V5 = ar^ + br^ + cr + dr^ + er^ +/r^, Vq = ar^ + br^ + cr^ + dt^ + ei^ -^fr, then Vi is derived from v^ by changing r to /**, and Vi, Va, ••• '^6 are the roots of a primary resolvent xp {v) = 0. Expressing the operation of changing Vi into Vi as a permutation of the roots of / we have Si = l, .^, = (124) (365), 53 = (132645)
s, = (142) (356), 55- (154623), s, = (16) (25) (34). These are the elements of the Galoisian group of /, and combine according to the multiplication table
|
1 |
S2 |
53 |
Si |
S5 |
s. |
|
^2 |
S4 |
Se |
1 |
S3 |
S5 |
|
S3 |
So |
^2 |
S5 |
1 |
54 |
|
S4 |
1 |
^5 |
S2 |
Se |
Ss |
|
s. |
Ss |
1 |
Se |
S4 |
52 |
|
Se |
S5 |
§4 |
s^ |
S2 |
1 |
24 GROUPS AND RESOLVENTS [CH. I
which is to be read s./ = S4, 8283 = 8^, etc. It appears from the table that SaSi, = SbSay SO that G is Abeliaii, and every one of its factors is self-conjugate. As a matter of fact, if we put S3 = s, the elements of G are
X f Sf o J a f S y 8 f
and the group is cyclical. It is also transitive, so that /(w) is irreducible without adjunction.
One factor of G is (1, ^2, 84), and from this we can derive an auxiliary quadratic. To find a function of which (1, Sa, S4) is the invariant group, we start with
(t + v,)(t + V2){t + v;); in this expression the coefficient of fa is
and this is, in fact, a function such as we require, because (83, S5, ^e) each convert it into
which has a different value because/ is irreducible. If, now, we put
then ^1 + ^2 = - 1 J and 3/1^2 = 2, in virtue of /(r) = 0. Consequently yi is a root of the auxiliary equation
3/^ + ^ + 2 = 0 (1).
Let us take 3/1 = ,j
and adjoin it to the field of rationality, which thus becomes (3/1). The Galoisian group of / reduces to (1, ^2, 54), of which the only self- conjugate factor is unity. Hence r must be the root of an auxiliary cubic, and since r is changed by Sa, ^4 into ?'^, r* respectively, this auxiliary cubic is
(z-r)(z-r^)(z-r*) = 0; or, on multiplying out, and expressing the coefficients in the new field, this is
ii^-i/iz'-(y, + l)z-l=0 (2).
If Zi is any root of this equation, the others are Zi', Zx\ finally the roots of the original equation may be expressed in the form n = 2^1, ^2 = 2^1', r3 = z^ = y^z^ + Oi + 1) z^ + 1, n = z^^ = -z^-z^^ yu n = z^^ = - (1 +yx)z^^ -z,-l, U = z^ = z^ - yi^^i - (1 + 3^1). If we solve (2) by the method of Art. 4, we find that
(zi + uizi^ + ii?z^y
28] GROUPS AND RESOLVENTS 25
is a root of the quadratic
^2 + (2yi-13)#-7(2yi + l) = 0, one root of which may be put into the form
. _ -2^1 + 13 + 3721 _ 14 + 3^21 - ^V7 *- 2 - 2 •
Let a definite cube root of this be extracted, and called 6 ; then since
{z^ + oiZ^ + ta^zf) {z^ + lo^Zx^ + <^Zi^) = 2l/i + 1 = ijl,
we may write
2^1 + Zi + Zi^ = —^ , Zi + (o% + <si%^ = 0,
Zx + (a'^z-^ + mzi = iJl/6 ; whence, by addition,
Szx- + ^ + -o~
^ - 1 + ij7 - 14 + 3^21 + ^V7 ^
2 "^ 14
The quantity 0 is of the form a + ySi, with a, /3 real ; and the question might be asked, whether a and p admit of representation by means of real radicals. This is not the case, because a is the root of a cubic with all its roots real, so that the formula expressing it again involves cube roots of complex quantities*.
By the adjunction of 1/1 the resolvent if/ (v) can be expressed as the product of two rational factors ; one of these is
F,(v) = (v- Vx) (« - V,) (v-v^) = 'i^- Pv^ + Qv-E, where
P = (a + b + d)y,- (c + e +f) (1 +3/1),
Q =-(a' + h^^(P) (1 +3/0 + (c^ + ^+/Oyi
+ (ac + hf+ de) (2 - 3/1) + {ae + ftc + df) (3 + y^
- (bd + da + ab + ef+/c + ce + af+ be + cd\ E = a^ + b^ + c^ + d^ + e^ +/3
+ (a^b + a^c + a'e + b'c + b'^d + by+ cH + &e
■\-d^a + d^e + d^f+ e'b + e^f+fa +/^c) y,
- {aH + «y + b'^a + b^e + c^a + c'^^ + cy + c?'^> + c?'^
+ e^a + e^c + eH +Pb +pd ^fh) (1 + y^) + (a6c? + aef-\- bee + cc?/*) (2 - y^ + (a6/+ ac^ + bde + c^) (3 + 3/1)
- (a^c + abe + ac^ + ac/+ ode + ac?/*
+ ^c^ + 6c/+ 6<?/'+ ^^/+ cc?e; + def).
* Holder, Mathematische Annalen, xxxviii, 307.
26 GROUPS AND RESOLVENTS [CH. I
The other rational factor may be obtained from this by changing y^ into - (1 + y,).
This example affords a verification of the theory of Art. 15. The permutations of the parameters which leave i/^ (v) foi-mally unaltered are
o-i = 1, 0-2 = {dha) {efc\ o-g = {edfhca\
0-4 = Q)da) (Jec\ 0-5 = {chfdea\ <r^ = (fa) {eh) {dc\
and these could have been found by experiment from ^(v\ without assuming any special relations among the roots of f{x). We should then infer the Galoisian group oi f(ai) from the permutations o-^, and hence finally discover the relations connecting the roots. The per- mutations o- which leave i^i(v) unaltered are 1, org, 0-4, as may easily be verified; while 0-3, 0-5, a-^ each convert Fi(v) into the other rational factor of if/ (v).
Instead of starting with the factor (1, Sa, S4) we might start with the factor (1, Se). This leads to the auxiliary equations
f + f-2y-l = 0 (3),
z'-y,z+l = 0 (4),
where we may suppose
Zi = r, yi = r + r\
With the notation of Art. 4 we find that A = B -1, ^ 7-H2UV3 2 '
and the reduced forms for the roots are
ri = z^, ri = Zi^ = yiZ^-ly r3 = Zi^ = (yi^- l)z,-yi; r, = z,' = -(i/,'-\)z,-y,'+l = -W-l)(z,^l),
n = 2^1' = - y^ z^^y^-\, ra = z^ = ^z^+y
29. In general, a composition-series for G may be constructed in more ways than one ; but in every case the indices e^, e.^, ... ep^i are the same in number and value, and only differ in the order in which they occur*; moreover, the factor-groups Gi/Gi+i are the same, except for the order in which they occur, and all of them are simple. Thus the number and the degrees of the auxiliary equations are the same in every case, and however they are formed, the problem of solving them has just the same degree of difficulty. This shows very clearly how deeply the theory of Galois penetrates into the special nature of any given equation.
* Burnside, Theory of Groups, pp. 118-123.
28-30] GROUPS AND RESOLVENTS 27
A few words may be said as to the effect of adjoining a rational function of the roots, which has for its invariant group r, a factor of G which is not self-conjugate. If the order of V is k, and we put hjk = /, it can be proved, as in Art. 24, that the adjoined function </> satisfies a rational equation of degree /, that its Galoisian group is simply isomorphic with the permutations of (r, t^^^ ...tiV) arising from pre- multiplication by substitutions of (r, and that the adjunction of <^ reduces the Galoisian group of / from G to r. If we adjoin all the roots of the equation satisfied by <^, the group of / sinks to that factor of G which leaves each element of (r, tj^, ...tiT) unaltered. This factor is the group consisting of all the substitutions common to r and its conjugate groups ^^r^^"^ ; a group which is self-conjugate in r. Consequently, the adjunction of all the roots of the auxiliary equation a (0) = 0 is equivalent to the adjunction of any rational function for which the self-conjugate group last referred to is the invariant group ; hence it is unnecessary to adjoin any irrationalities except those of which the invariant groups are self-conjugate in G.
To avoid misunderstanding, it may be remarked that a group Gi of the composition-series is not necessarily self-conjugate in G ; but before constructing the ^th auxiliary equation, we have reduced the Galoisian group of/ from G to Gi-i, and in this group Gi is self- conjugate. The advantage of choosing Gi as a maximum self-conjugate factor of Gi_i is that in this case Gi-i/Gi is a simple and simply tran- sitive group*; hence the ^th auxiliary equation is normal, and, subject to this condition, of the lowest possible degree.
From what has been said it follows that the natural classification of equations is according to the properties of their Galoisian groups. Equations of quite different degrees are solvable by processes of just the same complexity, provided that their Galoisian groups, in their abstract form, are identical.
30. There is an important theorem which, to a certain extent, forms the converse of that stated in Art. 24, and more generally in Art. 29. It is as follows : —
Suppose that <t>(^) = 0 is any rational equation such that the adjunction of one of its roots makes a primary resolvent ^{v) re- ducible: then this same reduction may he effected by means of one of the Galoisian auxiliary equations constructed after the manner which has been explained.
* Burnside, pp. 29, 38-40, and Art. 22 above.
28 GROUPS AND RESOLVENTS [CH. I
We may suppose that <t>(y) = 0 is irreducible. By hypothesis, if/ (v) becomes reducible in the field (i/i) : let the new irreducible factor which has the root i\ be x (% Vi), a function which must contain ?/i explicitly.
With a proper arrangement of the notation, we have identically
X(^. ^i) = (^- '^i) {'e-v^)...(v- Vk). Th£ substitutions (s^ 52) ••• ^fc) of G which are associated in the usual way with Vi, v^^ ... Vu, must form a group V. To see this, we observe that by Art. 7 we may write
where Ja, •••i* denote rational functions. Hence the equation
x{iaW}-o
Has a root Vi in common with x (^) = 0, and consequently
for ft = 1, 2, ... A:. But since Sa and Si, belong to the Galoisian group, we can infer from
that S^{SaV^)=ja{v^)\
hence xK(5aVi)} = 0
and SaSi must be one of the set 5i, ^2, ... 5*.
Now let u{xx, ^2, ••• ^») be a rational function of the roots of/ for which r is the invariant group ; this will satisfy a rational irreducible equation
a{u) = 0
of degree hjk. We shall have a resolution
xl/{v) = xj/i (v, u) «A2 (y, u) ... xlfi (v, u) with / = hik; and we may suppose that i/^i (i?i, u) = 0.
Whatever value the rational quantity t may have, the function
{t-v,){t-v,)...{t-v^)
is invariable for the substitutions of F: hence it may be expressed (Art. 24) as an integral function of u and t, say J{t, u). But the function is also x (^, Vi) '• so that the rational equation in u h(u) = Jit,u)-x{t,y,) = 0
has a root u-Ux in common with a (ii) = 0. By giving t a suitable value we can make u^ the only common root. The process of finding the highest common factor of a {u) and b {u) leads to an identity
Pa + Qb = Ru- S,
30] GROUPS AND RESOLVENTS 29
where R, S are integral functions of i/i ; and since a, h have a linear factor in common, we must have
Wi = SIE, • a rational function of yi which may be reduced to an integral form
by means of <^ (3/1) = 0.
Hence i/^i (y, u,) = if/^ {v, 0 {y^)],
a rational factor of ^ (v) which vanishes for v = Vi, and must therefore coincide with x (v, y^ because x is irreducible, and the degrees of both factors are the same. This proves that any new irreducible factor of \l/ obtained by the adjunction of 3/1 can also be obtained by the adjunction of a quantity u^ which can be expressed as a rational function of the roots of/
Rational functions of the roots of /have been called by Kronecker natural irrationalities (in the case when their values are not rational, of course) : thus we may express the theorem by saying that every possible resolution of the Galoisian resolvent of an equation hy means of algebraic operations can be effected by the adjunction of natural irrationalities.
The roots of a chain of normal Galoisian auxiliary equations are natural irrationalities : in a certain sense they form a " simplest " set of irrationalities in terms of which all the roots of the given equation can be rationally expressed.
CHAPTER 11.
CYCLICAL EQUATIONS.
31. The only irreducible equations which have unity for their Galoisian group are linear, and require no discussion. The next simplest irreducible equation is one of which the Galoisian group is cyclical, so that
with s"=l.
This is called a cyclical equation The necessary and sufficient condition that a rational function of its roots should have a rational value is that its value remains unaltered when the substitution s is applied to it.
The group G must be transitive, since/ is supposed to be irreducible : hence s must consist of a single cycle which, with a suitable notation for the roots, may be written in the forms
s = {xxX^ . . . ^„) = (12 . . . w).
If/? is any prime factor of w, and n = mp^ the group
is self- conjugate in Gy and we can form an auxiliary equation
a(a) = 0, of degree p, which reduces the group of / to Gx- If q is any prime factor of w, and m = Iq, the group
is self-conjugate in Gi, and we can form another auxiliary equation
ft(/3) = 0 of degree q, with coefficients rational in the field (a), which reduces the group of /to (t2 : and so on. It thus appears that if
n=p^q''...z'
81, 32] CYCLICAL EQUATIONS 31
whereat?, q^ ... 2; are different primes, the complete solution of /= 0 can be obtained from (^ + ^ + . . . + ^) auxiliary equations : h of these are of degree p, k of degree q, ... t oi degree z.
Each of the auxiliary equations is cyclical. For example, the group of b((3) is G1/G2, and this is cyclical, because if we break up Gi into parts (or rows) with respect to G^ we have
G,= G, + s^G, + s'PG, + ... + 5(«-i)^(^2,
and hence s'^^ = s'^G^ + s^'+^^^G^ +...+ s^^-'^+'^^G^
a cyclical permutation of the parts. In other words, the group of b is of the form (1, o-, o-^, ... o-^~^) with a-^^ 1, and so for any other auxiliary.
32. Thus the solution of any cyclical equation may be made to depend upon the solution of auxiliary cyclical equations of prime degrees. In the first place, however, we shall explain a process of solution which is applicable to the original equation as well as to its auxiliaries. This solution expresses the roots of / rationally in terms of its coefficients, a primitive nth. root of unity c, and the nth. root of a quantity which is rational when e is adjoined to the original field.
Let Oi = iCi + €0^2+ ... + €^~^a^n '-
then s^i = ^2 + €^3 + • • • + ^'^~^^n + c*""^^! = €"^^1 :
and similarly s% = €-% :
hence s* (^/) = c-^'^^i'^ = ^A
and 61^ must be a rational quantity in the new field, because its value is unaffected by any substitution of G, and the group of / in the new field must be either G itself, or a factor thereof Consequently we may put
where X/R denotes some one definite nth root of the rational quantity jB, for instance the real root, if it exist. B may, and in general will, explicitly contain the auxiliary quantity e.
Now consider the expression obtained from 0^ by changing c to €*, where k is any positive integer. Calling it 0,^, we have
and hence s ( ^J = p^^^, = ^ .
32 CYCLICAL EQUATIONS [CH. II
Assuming that the value of ^, is not zero, it follows that
eic-B^e,' (/: = 2,3,...7i-l),
where Bk is a rational quantity in the new field. Finally nxi = -Ci + B^-\-$2+ .,. + O^-i
= -ci + e, + BA' +Bse,'+... + Bn.ie.^'-K
By changing 0^ into t-% we obtain a similar expression for wi+i . As an illustration, take the example of Art. 28. In the first mode of solution, after the adjunction of ^i,
Oi' = ^ ^ = 5 - 42^1 - (3 + 63^1) o),
In the second mode of solution and the roots of the first auxiliary equation are given by
t
33. The method above explained breaks down when ^1 = 0 for each primitive root €. To avoid this difficulty, Weber * has put the expression for Xx into a slightly different form as follows.
We have identically
Wiri + Ci = 2^i (e = 1, 2, ... w - 1)
and hence n {x^ - ^1) = 2 (e-** -i)6i.
Now let h = n/p, where p is any prime factor of n ; the coefficient (c-**-l) vanishes whenever i is a multiple of j», while on the other
* Algebra, i, 689.
32-34] CYCLICAL EQUATIONS 33
hand (a^h-^i) is not zero, because /is irreducible. Consequently there must be one integer i at least such that Oi does not vanish, and i is prime to p.
If, therefore n=p'^q^ry ...
where p, q, r ... are different primes, we can find integers A., fx, v ... prime to p, q, r ... respectively, such that 0^, 0^, 0^,, etc. are all different from zero.
Taking any positive integers t, w, ^, z, ... and denoting, as before, the generating substitution of G by s, we have
where u= -t + \w + fxi/ + vz+ ....
The greatest common measure of A., //, v, etc. is prime to n: con- sequently there are positive integers $, rj, ^, etc. such that
X$ + fjLy) + v^+ ...= I (mod n)
and if we put O^O^^O^^O,^ ...
^ is a quantity which does not vanish and is such that
s(o,6-') = e,e-K
Consequently Ot = Bt6' (jJ = 1, 2, . . . ?^^
where Bt is rational ; and
with ^" = -^,
where ^ is a non-vanishing quantity, rational in the new field.
34. Since s (Ox'') = €-^^^a% the lowest power of 0^ which is rational is determined by the congruence
X^ = 0 (mod n)
or ^ = 0 (mod n/d)
where <^ = dv (n, A). On account of A. being prime to p, d is also prime
to p, and we may write
njd^pHx
where k is an integer. If we put
then <Aa^* = Tx
3
34 CYCLICAL EQUATIONS [CH. II
where Tx is rational, and
We may in the same way derive from ^,x, ^^, etc. quantities <^M> ^vi etc. such that
and so on.
The integer /i is prime to p, mi is prime to q, and so on : hence we can find integers $, rj, ^, etc. such that
li\i+ mi fji.7j + 72 iv^+ ...= 1. (mod n)
Now s(0,<i>x--<f>^-y <!>,-' ...) = €^e,<i>x-<f>^-y<f>,-^
where u = -t + liXic + mifiy + nivz + ... ;
so that u = 0 (mod n) if
x,y,z,...=t^, trj, tC, .... Consequently, if we put
then ^, = /S',<^*
where /Si is rational :
na;i + Ci = Si<t> + S,<ly'+...+Sn<i>''-'' (1),
and <^A, <f>fi., etc. are determined by the binomial equations
<}>r=T,, v^^t;,...
the degrees of which are the powers of primes which occur in n. By giving <^x, <^^, etc. all their different values, <^ assumes ?^ different values, and if these are substituted in (1), we get all the roots of the given equation. Of course the adjunction of the quantities <^a, <^^, etc., is equivalent to the adjunction of the single quantity 6 which is determined by a binomial equation of degree n ; but the equations which determine </>a, etc. are all lower than the one which determines 0. In this 7'espect the last form of the solution may be considered the simpler one. All this illustrates the fact that what is to be called the " simplest " solution of an equation is partly a matter of convention.
Thus, again, if, in the present case, we solve the eciuation by a cliain of Galoisian auxiliaries, they will all be of prime degree, and for each of them one at least of the quantities 6i must be different from zero, so that Weber's supplementary transformation is unnecessary. In these respects the solution is the simplest of all: on the other hand, just because the expressions for the roots are more explicit, they are more complicated in appearance.
34, 35] CYCLICAL EQUATIONS 35
35. In the solution of the general cyclic equation complex roots of unity appear as auxiliary irrationalities. These roots of unity are themselves the roots of cyclic (or Abelian) equations, and it is natural to inquire how far the solution of these special equations can be carried.
If n=p^p^...p^
where ^i,;?2, etc. are powers of different primes, the complex roots of ^" = 1 may all be expressed in the form
where a, /3, ... \ are roots of
so that it is sufficient to consider the case in which n is a> power of a prime.
We shall begin by supposing that n=p,&n odd prime ; the equation to be solved is therefore
/(;r) = a;P-^ + af-'^ + ... + ^ + 1 =: 0.
If r is any one of its roots, the others are r^, i^, ... r^~^. These may be expressed in a more convenient form as follows. Let ^ be a primitive root oi p ; that is to say, a primitive root of the congruence
^P-'El. (modjo)
Then 1, g, g^^ ... g^~^ form a complete set of residues of jp, and if we write
the roots oifQr) will be denoted by suffixes in such a way that
In this notation, every integral function of the roots which is unaltered in value by the substitution
5 = (nr2...rp_i) is rational.
The function in question can be reduced to the form <^ (n), where <^ is a rational polynomial. If the substitution s is applied to the original form of the function, its effect is the same as changing i\ into r/ in <^ (ri). Hence if A is the value of the function, which by hypothesis is unaltered,
A=<f>{r,) = 4>{r,^)-<^(rf) = ...
p 1 a rational quantity, because symmetrical in the roots of /.
3—2
36 CYCLICAL EQUATIONS [CH. II
If (v - Vi) is a factor of the total resolvent of / and we put s*(vi) = Va+i, the factor
xf, (v) = (v- Vi) (v-V2)...(v- Vp.,)
will be rational, and moreover it will be irreducible, because otherwise there would be an identity
v^-i^-'^(aui + ...) + ••• = (^-'i'i) (v-Va) (v-vp) ...
= 'y*-'y^"M(n + ^a+i + r^+i + ...) wi + ...} + ... leading to n + ra+i + rp+i + ...=a
with a rational, and less than (p-l) terms on the left-hand side. This is impossible, because /(.-r) = 0 is an irreducible equation*.
Hence if/ (v) is a primary resolvent of/, and the Galoisian group of/ is (1, s, ^, ...s^~^), so that / is a cyclical equation. We may proceed to solve it, either by forming a chain of auxiliary equations, the degrees of which are the prime factors of (p - 1), or else by adjoining a primitive (p— l)th root of unity, and proceeding as in Arts. 32, 33.
36. An example of the first method (for p = l) has been com- pletely worked out in Art. 28. In the general case, let jt? — 1 = ^, where e is a prime. Putting
a will be a root of an auxiliary equation
a(a) = 0,
with rational integral coefficients and of degree e. If /= gk, where ^ is a prime, we put
P = ri + rge+i + r^2ge+l + . . . + Tp-gey
and now )8 is a root of an auxiliary equation
i(/8) = 0
of degree ^, with coefficients which are rational polynomials in a. We proceed in this way until all the prime factors of (jt? — 1) are exhausted. A case of historical interest is when p = n. The auxiliary equations are (taking 3 as the primitive root of 1 7)
a24-a-4 = 0,
2/-2/Sy + (a^-a + )8-3) = 0,
• Weber, Algebra, i. 596 ; or my Theory of Numbers, p. 186.
35-38] CYCLICAL EQUATIONS 37
All these equations, except the last, have real roots, and a, )8, y, 8 can all be obtained explicitly in forms containing real arithmetical surds : thus we may put
^^-1 + V17 ^_-l + V17 + V(34-2V17)
but the expressions for y and 8 are too complicated to be worth writing down.
37. To solve the equation considered in Art. 28 by the method of
Art. 32, we put
0-1 = r + lo^r^ - (o/^ + oir^ — wV - r^,
(- to being a primitive sixth root of unity, and the cyclical order of the
roots of / being r, r', r^, r^, r"^, r' when we take 3 as the primitive root
of 7). It is found by actual multiplication that
e^ = (5 _ 3a,) (r + 7^ + r* - 7-=^ - r' - 1^\ ^i" = - 7 (16 - 39(o) = (1 + 3a)) (2 + 3a>)^
where it may be observed that in the field {i^^ the norm of Ox is 7^ It
may also be verified that
2-0, 8 + 30) 18 + 19(0^4
0^0., _55 + 39o).5 ^^~l-2o,- 2401 ' ' so that finally
1 Ox 2-0). 2 8 + 3o).3 18 + 19o) 55 + 39o)
r = -Y + Y + -^^i +-^^^1 ^4 — ^1 + — ^i — ^1-
38. The simplest way of calculating the quantities Oi is the following. If h is any one of the numbers 1, 2, 3, ... (jo- 3), the product O^Oy, is not rational, and its quotient by ^a+i is equal to the coefficient of r in the product O^^h after reducing it by first replacing all powers of r higher than r^'^ according to the formula ^.ap+b^^b^ and then replacing any rational term a by its equivalent value
Now ^^^^ = 5€^°«i«+fti^d6^+6_ {a,h = \, 2, ...io^)
The only pairs (a, h) which contribute to the coefficient which we wish to find are those for which
a-vh =p,
38 CYCLICAL EQUATIONS [CH. II
The second set contributes, after the first reduction, a coefficient of r which is
2ginda + Aind(p + l-«)^ (« = 2, ... JO^l)
the other set, after the second reduction, contributes
Since ind (j>-a) = ind (- a)
= i (/?-!) + ind a,
the sum last written
o = l
Q 0
and hence ij^^ ;^^nda+Aind(p+i-a)^ (a = 2, 3, ...p-1)
'h+l
On the other hand, if ^ =^ - 2, then 6l6}^ is rational. Its value may be written in the form
$$ _ — ^gind a+(p- 2) ind 6 ^a + &
since €^ ^ = 1. Now if we put a= tb (mod p), we obtain the equivalent expression
^^dt^a + t)b^ (^>, ^-1, 2, ...^1)
The terms for which t=p-l contribute for any other value of t
hence the value of all the remaining terms
< = p— 2
--=-1 2 £^<*' = -l; t=i
and finally ^i^p-2 = -p-
This, together with
-l-^="*°'2 jindm + fcind (p+l-7rt) f^ = 1 2 . . . » — 31
^;»+l m = 2
enables us to find the values of 0^, 6^, ... Op_^ with great facility. Of course the indices of the powers of « are reduced, at the first opportunity, to their least residues, mod (p-1).
38, 39]
CYCLICAL EQUATIONS
39
^ As an example, when p = 7,we construct the table of indices for the primitive root 3 : —
|
m |
12 3 4 5 6 |
|
ind m ind (8 - m) |
6 2 14 5 3 3 5 4 12 |
and hence find
"•2
6 0
1'^3 _ n
0,
0.
+ C^ + €^ + e^ + 6=^ = - 1 + 2o>,
= e^ + 6=^ + €V C^ + €^ = - 3 - <o,
By multiplication we find that
^i« = -7(l-2a))2(3 + <o)2
= -7(16-390))
as before; and all the results of Art. 37 may now be obtained with ease.
39. Suppose now that n=p'^, a power of a prime. The primitive wth roots of unity in this case are the roots of the equation
/w=
X
pa _ J
= cc^''-^ (^-') + ^^""' (^'-2) + . . . + ^^"-' +1=0,
which is irreducible, and of degree jt?«"-^ {p — 1). It is also cycHcal, because there are primitive roots of p'^ which can be used, as in the case when a = 1, to fix a cyclical order of the roots, and the arguments of Art. 35 may be repeated. The indices of the composition-series will be the prime factors of (jo-1) and also the prime p repeated (a-1) times. Hence if we solve the equation /(^) = 0 by a chain of Galoisian auxiliaries, (« — 1) of these will be of degree p, and (Art. 30) no purely algebraical solution can replace these auxiliaries by others of lower degree.
40 CYCLICAL EQUATIONS [CH. II
Finally, if
n=p''q^ry ,., <l> (n) = p''-'^ (p - 1) qP-'^ (q - I) ...
the primitive nth roots of unity are <^ (n) in number, and they may be determined by as many chains of auxiliary equations as there are different prime factors of n. Tlie degrees of the auxiliary equations are the prime factors of <^ (w). It should be observed that the primitive nth. roots satisfy an irreducible equation of degree <f>(n), but this equation is not cyclical.
A specially interesting case is when the auxiliary equations are all quadratics. The necessary and sufficient condition for this is that <^ (n) should be a power of 2 ; this is equivalent to saying that
n = 2'^pqr . . .
where JO, q, r, etc. are different primes, each of the form 2™+ 1. When n is of this form, and then only, a regular polygon of w slides can be inscribed in a circle by means of the rule and compass; because the complete solution of ^" = 1 leads to the determination of cos 2'rr/?i and sin 27r/w, and conversely, while every construction with rule and compass can be put into an analytical form which involves only linear and quadratic equations. This remarkable connexion between geometry and analysis was discovered by Gauss.
The values of n, below 100, which are of this special form are
3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96.
Of these the only ones which are not considered in Euclid's Elements, or at least easily brought into connexion with the cases (w = 3, 4, 5, 6, 15) which he does consider, are 17, 34, 51, 68 and 85.
CHAPTER III.
ABELIAN EQUATIONS.
40. A GROUP is said to be Abelian when its elements satisfy the commutative law of multiplication : that is to say when ss = s's, s and s' denoting any two elements of the group. An Abelian equa- tion is one of which the Galoisian group is xlbelian. Cyclical equations form the simplest class of Abelian equations : it will be shown in this chapter that every Abelian equation may be solved by means of auxiliary cyclical equations.
It will be supposed, in the first place, that the given Abelian equation is irreducible. This being so, its Galoisian group G is transitive, and will contain a substitution Si which converts ^i into any other assigned root iPi.
The substitutions of G which leave Xi unaltered form a subgroup of G. Let o- be any one of these : then since ^r^ changes a^i to Xi ,
Si -^ a-Si {Xi) = a-Si (^i) = Si {a- (^i)} = {Vi, that is to say, Si~^(TSi leaves a^i unaltered. But since G is Abelian, g.-^aSi^Si'^SiO-^a ; consequently o- leaves every root unaltered, and is the identical substitution. It follows from this that G is simply transitive, and that if ^i, cc2, ••• ^n are the roots of the given equation
G = (l, 52, .93, ...Sn)
where Si is the definite substitution which changes ^1 into iVi .
Moreover the adjunction of ^1 reduces G to unity : consequently ^2, -■■ .'Pn are expressible as rational functions of ^1, and /(^r) = 0 is a normal equation.
Let the rational expressi >ns of the other roots in terms of ^1 be
To these equations (Art. 17) we may apply any substitution of G : thus from ^i = ^i(^i), ^j = ^j(^i)
we deduce Sjo^i = 6i (xj), siXj = Oj {x^ .
42 ABELIAN EQUATIONS [CH. Ill
But SjXi = sj {siX^ \ = Si {sjXj } = Si Xj :
consequently Oi {xj) = Oj (xi) ,
that is, Oi{ej(x,)} = ej{di(x,)}.
By applying a Galoisian substitution to this we infer that
«. {»j W } = »j {Si W } U, j,k=l,2,...n]
with the convention that 0^ (x^^ = x^. In other words, the rational function
ei\ei{x)]-e,{ei{x)]
must either vanish identically, or have a numerator which is divisible by f{x). In general, it is the latter case that occurs ; so we may write, to express this fact,
6i {6^ {x) } - 6j {e, (x) ] = 0. (mod f(x))
Conversely if the roots of a normal equation /(x) = 0 can be ex- pressed in a form Xi = 6i (x^ such that these congruences are satisfied, the Galoisian group is Abelian. For we have arithmetically
that is 6i (xj) = 6j (xi) :
but since SiXi = Xi^ Oi (xi), and SjXi = Xj^6j (xi),
it follows that Sj (siXi) = Oi (xj) , Si (sjXi) = Oj (x!) ;
consequently Sj (SiXi) = Si (sjXi) ,
and in this we may change Xi to x^. Finally, then, SiSj = SjSi identically, and the group of the equation is Abelian. It will be observed that this converse theorem is true whether /(;r) is irreducible or not.
41. The simplest way of expressing the elements of an Abelian group is by what is called a basis*. The elements 5i, Sg, ... s^ form a basis of G when every element of G can be expressed in one and only one way in the form
s{^s.J^ ...Sh {x^mi, i/^7n2, ...t^ m,,)
with Xj y, ... t positive integers, and mi, /Wg, ... m^ the least positive integers such that
If desirable, the base may be so chosen that mi, m2,...mh are powers of primes : of course their product is equal to n, the order of G.
* Weber, Algebra, ii, 38-45.
40-43 ABELIAN EQUATIONS 43
42. No generality will be lost, and the notation will be much simplified if we suppose that the basis of G consists of three elements s, t, u, of order a, b, c respectively, so that ahc = n, and all the elements of G are expressed by
Let p be any prime factor of a ; then the substitutions for which, in their basic form, i is divisible by p form a self-conjugate sub-group of G, the index of which, with respect to G, is p. Since p is prime, this is a maximum sub-group, which we may denote by Gi, and a rational function of the roots for which G-i is the invariant group will satisfy a rational cyclic equation of degree p. By adjoining one root of this equation, the Galoisian group of/ sinks from G to G-^.
Suppo«5e, now, that ^ is a prime factor of ajj) : then the substi- tutions of G which, in their basic form, are such that i is divisible by pq, form a maximum self-conjugate factor of G^, which we may call G2. A function for which G2 is the invariant group in the enlarged field will satisfy a rational cyclical equation of order q, and the adjunction of one of its roots reduces the group of/ from Gy to G^. By proceeding in this way, we can exhaust all the prime factors of a and reduce the group of / to those substitutions of which the basic forms are t^u^. If p is any prime factor of h we have a group (^%*) withy divisible by y, and a corresponding cyclic auxiliary of degree y, and so on. The group of /is finally reduced to unity by a chain of auxiliary cyclic equations, the degrees of which are the prime factors of n : that is to say, if n ^p'^q^r'^ . . . , there will be a auxiliary equations of degree jo, ^ of degree q^ y of degree r, etc.
43. As a simple illustration, we will take
the roots of which are the primitive 20th roots of unity. If we arrange the roots so that
|
Xi = r, X2 - r\ |
Xs = r\ X4 = r\ |
|
x, = r'\ Xe = r'\ |
X7 = r^\ Xs = r'^ |
|
substitutions of G are 1 |
|
|
S2 = (1243) (5687), |
S3 = (1342) (5786), |
|
^4 -(14) (23) (58) (G7), |
S5 = (15)(26)(37)(48), |
|
§6 = (1647) (2835), |
S7 = (1746) (2538), |
|
.s = (18) (27) (36) (45). |
44 ABELIAN EQUATIONS [CH. Ill
If we apply these to the function given by the only new function arising is
^2 = ^6 + ^6 + ^7 + ^8 = —y\'
Hence yi is a root of a rational quadratic. To find it, we have, with the help of/(r) =^ 0,
y^ = r + 'i^ + r' + r^ = 2r'' - r° + 2r*, y,^ = (2r^ -r^ + 2r«)^ = 5r^o [mod/(r) ]
= -5, and the first auxiliary equation is
3^1' + 5 = 0. If we now put
Zi = Xi + a;^ = r + 7^, Z2=£C2 + cc^^'i^ -^r^
we find that 2^1 + 2^2 = 3^1 > ^1 2^2 = — ^ so that the second auxiliary equation is
Zi-yiZ^-l = Q. Finally x^ and ^4 are the roots of
cc^-ZiXi- 1=0. By actually solving the auxiliaries we see that we may take
„-;/5 --'(V5 + 1) ^ V(10-2V5)+,-(>/5 + l). yi - * V 0, *i - 2~~^ ' ^1 ~ 4 '
and as a verification we observe that the expression last written is exp (67^^720), one of the primitive roots required. The group G is in this case dibasic : if we put
8 = S2j t = Sg,
then (s, t) is a basis, and the basic representation of G is
Si = lj Si = S, 83 = 8^, 84 = 8^, S5 = 8^tj 8q = 8Hy 87 = 8t, Ss = t,
with s^ = t^ = i.
It is a very remarkable fact, discovered by Kronecker, that if the coefficients of an Abelian equation are ordinary real integers, its roots can be expressed as rational functions of roots of unity, with real rational coefficients. Proofs of tiiis theorem have been given by Weber and Hilbert, but they are too long and difficult to be re- produced here.
CHAPTER IV.
METACYCLIC EQUATIONS.
44. Suppose that p h a prime number, and that g is any one of its primitive roots. The numbers (1, 2, S, ... p) form a complete system of residues to the modulus p, and we can form a group of permutations of these numbers in the following manner.
Let s denote the operation of changing any residue z mto z+ 1, and reducing the result to its least positive residue, mod p. Thus
s(ip-l)=p, s{p)=l, s(l) = 2, etc.,
and we may write
s(l,2,...p) = (2,S,...p, 1).
Let t denote the operation of changing z into gz, and reducing the result to its least positive residue, mod p. Thus
t(l,2...p) = (g, 2g, ..•p^'g,p).
Evidently 5 is a cyclical permutation of order p ; since
t'(l,2,..,p)=(g\2g\...p),
and ^ = 1 (mod p) only when A is a multiple of (p - 1), it follows that t is of order (jo - 1). It will be observed that t does not displace p, and that like 5 it is a cyclical substitution.
It will now be proved that ihep(j)-l) operations
' ^ ln=l,2,,..(p-l)J
form a group.
We have t^s' (z) = s" (g^z) ^g^'z + c
= g'(z + l)
provided that / = cgP-'-\
46 METACYCLIC EQUATIONS [CH. IV
Giving / its least positive value we infer that
and ^lf.^f = ^.1^^.lf = ^^H''^^.
Since rt, h, c, d may be any four integers, this proves that the operations form a group. For convenience, it will be called the metacyclical group, mod jp, and the reference to p may be omitted when no mistake is likely to arise.
45. There is another way of regarding the group, more convenient for some purposes, and representing the group as a set of linear substitutions. We have
^ti'{z) = gy{z + x)
= lz + m,
provided that g^ = /, g^a^ = m. (mod p)
If X and y are given, the last two congruences determine /, m uniquely to the modulus p. Conversely if I, m are given and / is prime to p, X and y are uniquely determined to the moduli /?, (/?-!) respectively. Thus the group may be represented by the substitutions
and in this form may be called the integral linear group.
The group is doubly transitive : that is to say, there is a definite
substitution which converts any two given residues a, /? into any two
other given residues y, 8. This follows from the fact that the
congruences
la + m = y, ip + m = 8 (mod p)
admit of one and only one solution, because
and (a - p), (y - 8) are both prime to p.
As an example, \et p= 1, g=' 3, and let it be recjuired to find the operation of the group which interchanges 1 and 2. The congruences
l+m = 2, 2l + m= 1 (mod 7)
lead to / = 6, w = 3, and the required operation is (z, 6z + 3), or, in the other notation, s*^. As a verification
5^(1, 2, ...7) =(5, 6, 7, 1,2,3,4), ^(5, 6, 7, 1,2, 3, 4) = (2, 1,7,6,5,4,3).
44-47] METACYCLIC EQUATIONS 47
46. It has been shown that
where / is different from c, while the index b remains unaltered. It follows from this that if d is any factor of (p - 1), including unity and Qo - 1) itself, and iip-l^de, the operations
gm^nd rm=l,2,...p-]
U = l, 2, ... eJ form a group of order pe.
This group is self- conjugate in the metacyclic group, because there
is an integer i such that
Let US put
p(p-l)=h, p-l=Piqi=PiP^2='-=^PiP-2-"Pr,
where Pi, p^, ---Pr are the prime factors of (p-1). Then we have a composition-series
with indices jt?i, p^, "■ Pr,p\
tbe notation being such that Grpq. means the group of which the operations are
Lw = l, 2, ... gj In particular, G^ means the cyclical group (1, ^", ^^ ... s^~^).
47. Suppose now that we have an equation of prime degree, and that its roots are x-^^x^^ ... x^. We obtain a group of permutations of its roots by applying to their suffixes the operations of the metacyclic group. If this is the Galoisian group of the equation, the equation is said to he metacyclic. An equation of this kind can be solved by a chain of auxiliaries, each cyclical and of prime degree. That the auxiliaries may be taken of prime degree follows from the composition- series just given for Gn : that they are cyclical may be inferred from the fact that they are normal as well as of prime degree, or again from the fact that Gpq._^ ^ Gpq. is holoedrically isomorphic with the cycHcal
ffroup
(^, ?{^ ... ^(^»-^''^),
where d =piP2 • - - pi-i (cf Art. 23).
48 METACYCLIC EQUATIONS [CH. IV
48. Kronecker has put the solution of a metacyclic equation of prime degree into a very interesting form, which is analogous to that given for cyclical equations in Arts. 32-4. Before reproducing it, a few explanations and lemmas will be necessary.
As in Art. 32, we take e, a primitive jt?th root of unity, and write
^fc = ^1 + €*^2 + ... + e(^-i)%. [k = l,2 ...(p-l)]
If s, t are the generators of the metacyclic group,
as before : to find the effect of t, we observe that
i i
where h is determined by the congruence
gh = 1, (mod p)
leading to h = g^'^ (mod p)
With this value of h t {0^} - €*<'^-i)(9fcft.
It is convenient now (cf. Art. 35) to introduce a slight change of notation. We shall write
^, = e^ ^'-0, 1, 2, ...(;?- 2)]
on the understanding that Bgi means B^, where r is the least positive residue of g^ to the modulus p. We also make the convention that for any positive integers rriy n,
provided that m = n. (mod p-l)
Thus there are only (p - 1) distinct quantities ^i, and these are the same as the quantities 6i in a different order : in particular,
Ul—^Q= ^p-l , ^l=Bg.
The effects of s and t upon ^< can be found from previous formulae : thus
Let us now write
f^ = ^i^o"^ /l = -^A'^ • • • /i= -^i + l^r") • • • fp-2 = ^p-l^p-2'
Then 5 (/)=/<,
and t (ft) = /-^'^'^i . €-^^'-'-^')^;f,
with the special case
t(/o)=/p-2'
48, 49] METACYCLIC EQUATIONS 49
Consequently any rational cyclical function of /«, /, /a, ... /p_2 is unaltered by i? and t : the quantities fi are therefore the roots of a ratio aal cyclic equation of degree (/> - 1). The change of c to €^ converts fi into /i+i ; hence it follows that when the cyclical equation aforesaid is reduced by means of the equation satisfied by €, the imaginary root of unity will disappear. In other words we have identically, after this reduction,
(/-/o) (/-/i) - (/-/p-2) -f'-' + mj^-^ -f ... 4- m^_„
where Ml, m^y ... rrip-i are/orw2a%metacyclic functions oi x^, x^, ... Xp, and have rational values when the given equation is metacyclic.
Suppose that we have a set of quantities ^o, </>i, ... <^p_2, each of which is rational in e^ Xi^ x^, ... Xp and which also satisfy the following conditions : —
(1) 5(</>o, <^i, ... <t>p-2)^^o, <^i, ••• <^p-2;
(3) the change of c into €^ produces the same cyclical permutation as t~'^ ;
(4) cychcal functions of <^0 5 <l>u ••• ^p-2 are metacyclical functions of ;2?i, X2, ... Xp, and can be expressed in a form which is free from €.
Then by arguments precisely similar to those employed in Arts. 7, 24 it may be proved that
<l>i = B{A\ (z = 0,l, 2, ...jo^)
where i2 is a rational function free from e, and the coefficients of the powers of/i are metacyclic functions oi x^, x^, ••• Xp. i
49. From the equations which express the quantities fi in terms of the quantities ^i we can eliminate all the ^s except ^o in the following manner. Raise the first equation to the power g^'^, the second to the power /"^ etc., and multiply all the results together : observing that ^p-i = ^Q, we have
^^-.^-^ ^ff-^^ff-\..ff-'-\..fUfp_, (1).
The primitive root g may always be chosen in such a way that where ^ is a positive integer. Supposing this done,
50 METACYCLIC EQUATIONS [CH. IV
Now the quantities V^, V^ ••• -^^-a satisfy all the conditions enumerated in the latter part of Art. 48, so that we may put
%'"' = ii(/«) (2),
where M is & rational function of the nature explained above
The positive integers ri, n, ... r^-a can he iiui(iuely determined so that
g^-^ = qp-^p + rp_2, f-^ = qp-zP + ^i,-3, etc.
with 0<ri<p (^ = l, 2, ...p-2)
and the quantities qi positive integers or zeros. If, now, we write, as an abbreviation,
K, = R{f,)f,^v-^^A^^-^...flU (3),
we obtain from (1), after multiplying both sides by V"^'j
V = ^oVo''^-^//^-^ -Al-Jp-. (4).
From this it follows that
^^^=K^fi^V-^rP^' •••/i+p-2 (5),
where Ki is derived from K^ by changing/,, /i, / ... into /, /•+!, /•+2, ... respectively.
The relations (1), (2), (4), (5) are all reducible to identities, whatever .^1, Xzi ••• scp may be, solely in virtue of the equation satisfied by €, and the definitions of -^f, /», etc. If a^i, x^, •■• Xp are the roots of a metacyclic equation with numerical coefficients, /o, /i, ... /_2 are the roots of an auxiliary cyclical equation with rational coefficients. By the adjunction of /o the other roots become rational, and finally, if we put
a definite pih. root of/, we have
pX^ = -C, + ^^i = -C, + %KiirP-^Trp-^ ... Tj^ 3r,+^_2. i
If, in the expression on the right, we give to each quantity t^ any one of its p different values, we only obtain p different expressions on the whole : thus the formula may be used to determine any root of the given equation, and it does not lead to any value of Xi which is not a root.
50. When jt? = 3, the metacyclic group consists of all the per- mutations of three things : hence the general cubic equation is metacyclic. To solve it by Kronecker's method we take ^ = 5,
.&0 = a + wy8 + o)2y, .^i = a + w^^ + o>y,
/o = '^1^0~°J /l = ^Q^l'^'
49-51] METACYCLIC EQUATIONS 51
With the notation of Art. 4, we find that/o,/ are the roots of
B'P-{A'-2&)f+B = 0 (1).
Moreover f^^f^ = ^^-2^
v-/oyxW)^ .....(2),
and we have now to express V in terms of/;. To do this by the general method is a good exercise ; but it is simpler to proceed as follows. We have
(V-.V)(V-V) A
hence A (V - ^/) - B^ {/, -/o) = ^^ (/o +/i - 2/o)
= A^-2B^-2B'f„
and y- i (V - V + ^) - ^^^^~ -^'-^^ (3).
If we write /o = r/, /i-tj^
we obtain from (2) and (3)
_MA^-B^-BV,y A(A^-B^-B^f,y ,
^0 - j-s To Tl, ^1 = ^1 Tj^To.
To put the solution into its simplest form, we must express the multipliers of t^^t^ and t^^tq as linear functions of/o and/i respectively. The final result is
^n -= -1 Vtj,
^1
B(BVo-A' + B') A
B(B'A-A' + B') ,
A
This gives the solution in a definite form whenever the values of A and B are both diff'erent from zero. When A =0, the expressions for ^0 and -3^1 assume the indeterminate form 0/0 : in this special case the cubic has the rational root - cJS, and the others are the roots of a rational quadratic. When ^ = 0 the cubic may be written
(3^ + Ci)'4-Ci'-27c3, and is cyclical. Finally, when A = B = 0 the cubic has three equal roots.
51. It is an interesting problem to find the most general form of a metacyclic equation of the fifth degree. To do this, we must first find the most general form of a cyclic quartic. There will be no real loss of generality if we suppose the sum of the roots of the quartic to
4—2
52 METACYCLIC EQUATIONS [CH. IV
be zero ; assuming this, there will be four rational quantities 6, c, d, e such that, X being a root of the required quartic
The elimination of 0 leads to the required equation in the form
a'-2(2bd + c') ex^ - 4 (6' + <^^) cex
- [6^ - (26'^e^ - ^hc'd + c')e + d'e'] e = 0.
Since (bO + dO^f = 2bde + (b' + dJ'e) B^
we may write
Xx= cje+ J(2bde + {b^ + d^e) Je\
x^ = - cje + J{Wd6 - (b^ + d'e)Je\ x^ = cje - J(2bde + {b"" + d^e)^e), ^4 = - cJe - J(2bde - (6^ + d^e)^e).
By a change of notation, these expressions may be put into other equivalent forms. To make the formula absolutely general, an arbitrary quantity a may be added on the right-hand side, and the quartic modified by changing x into (x - a). The quartic is then cyclical in the field (a, b, c, d, e).
Now let /, rriy n, p^ q be any rational quantities ; and let
Xi = ri\ (^ = l,2,3,4)
f{x) = la^ + mx^ + nx +p^
Then Xi, x^, x^, x^ being the roots of a cyclic quartic as previously constructed, $ will be a root of a rational quintic which is metacyclic in the field (a, 6, c, d, e, /, m, w, p, q).
It is supposed here that the notation for the roots of the quartic is 80 arranged that its Galoisian group consists of the cyclical permuta- tion (xiX.2X-^4) and its powers. This having been done we may give each of the quantities Tj all its five values, without obtaining more than five values for $. There will generally be five different values : but there may be repetitions for particular values of (a, b, ... q).
52. The general quintic can be transformed, with the help of solvable equations, to the standard form
x^ + ax + P = 0 (1),
and if this is metacyclic its roots can be actually found in the following manner.
^1. 52] METACYCLIC EQUATIONS 53
The generators of the metacyclic group may be taken to be 5 = (12345), ?f = (1243)(5); and if we put c = e^'^*/^,
it is found by actual calculation that, in virtue of 2^^ = %xi' = %xi = 0,
eA=4>J^ = -eA (2), '
e,% + e.^e,^eie, + e,A = o (3)^
Ox, O2, etc. having the same meaning as in Art. 48 and elsewhere. If we write, for simplicity,
<^^5 =u
and eliminate 0^, 0^ from (2) and (3), the result may be written in the form
u%'' + u'' (o,%y + (e,%y - uo,'' (e,%) = o.
This is satisfied identically, and in the most general manner, by putting
o,' = i'(i+iy{i-iyA (4),
I and t representing two independent parameters. Now one root of the quintic is given by
~^'^ e} i\i'-i)f^ 6, ^^■^^
by means of (2) and (4). Eliminating ^i from this and the second of equations (4), we find that
+ (P + 1) (P + 22P- 6P - 22/ + 1) ^] = 0 (6).
It will now be supposed that / and t have values such that the equations (6) and (1) are equivalent : thus
/(/^-l)(/^ + /-l)(/^-4/-l)^^+125a = 0 (7),
/(/^-l)(/^ + 22^-6/' -22/ +1)2^ +3125/8 = 0 (8).
It remains to make use of the fact that (1) is metacyclic. The sub- stitution s makes no change in </>, and in virtue of ^iCiOJj = 0 the substitution t converts <^ into - <^ : consequently <^^ is a metacyclic function, and its value is rational. Denoting it by y, we deduce from the first of equations (4)
P(P-iyt* = 5y (9),
54 METACYCLIC EQUATIONS [CH. IV
and from tliis and (7)
7 (r + / - 1) (/^ - 4/ - 1 ) + 25a (/2 - 1) / = 0. The solution of this is given by
')^ + (25a-3y)2^-4y = 0|
■(11),
From (7) and (8)
25(r+/-i)y-4/-i)/?
(/^+l)(/^+22P-6/2-22/+l)a and from (4)
5^o(p + i-iy(P-4i-ir(uini-iyi*p^
(I' + If (i' + 22/^-6/^-22/+ Ifa' "'^^'^^^
Equations (10), (11), (12) and (5) contain the complete solution of the problem, supposing that the value of y is known ; and it will be observed that, in accordance with theory, the degrees of the auxiliary equations are 2, 2 and 5, the prime factors of the order of the meta- cyclic group.
The quantity y is a root of the equation *
(y-a)*(y2-6ay + 25a2)^5^;8^y (13),
SO that the quintic is, or is not, metacyclic in any given field according as (13) has or has not a rational root in that field. If the field is (a, ^), we must have rational quantities X, /w, such that
whence
syx 5yx
(X - 1)* (X^ - 6X + 25) ' "^ (A - 1)4 (X2 _ 6X + 25) *
It may be observed that the solution of (6) assumes a very elegant form if we put
where ^ (z) is a lemniscate function of 2; ; that is to say, one for which
5^3 = 0.
* Weber, Algebra, i, 675.
CHAPTER V.
SOLUTION BY STANDARD FORMS.
53. As explained in Chap, i (Art. 27), the first step towards the solution of an equation, after determining its Galoisian group, is to construct a series of Galoisian auxiliaries. If the degree of each auxiliary is prime, the equation is solvable by radicals, because each auxiliary is cyclical ; and it can be proved that in no other case is the original equation solvable by radicals. The group of each auxiliary is simple ; hence the only outstanding difficulty is the discussion of non-cyclical equations, of which the Galoisian groups are simple. The reason why the general equation of order n cannot be solved algebraically when w > 4 is that the group of even permutations of n things is simple* except when n^L The cases 7^ = 2 and ?^ = 3 are also exceptional, because in the first case there are no even permuta- tions, and in the second they form a cyclical group of order 3.
The most effective way of attacking an equation of which the group is non-cyclical and simple is to transform it, if possible, into another equation of standard form, for which the solution is known or has been tabulated. The spirit of the method may be illustrated, in the first place, by considering the cubic equation
where a, h denote redl positive quantities. If we put
a) ^ %, 3F = 4a, c = 4:hl¥ the equation becomes
and by properly choosing the sign of k, we can make this
4/ ± 3^ - c = 0, with c> 0. If the coefficient of i/ is - 3, and c is a proper fraction, we may find a real quantity 6 such that cos 3^ = c, and then
y = coBO, cosfo + Yp ^^^ (^ "*" y) ' * Burnside, Theory of Groups, p. 153.
56 SOLUTION BY STANDARD FORMS [CH. V
while if c> 1 we find 6 such that cosh 3^ = c, and then
y = cosh 0, cosh (6 + -— j , cosh (0+ ^ ).
On the other hand, if the coefficient of ?/ is + 3, we may find 0 such that sinh 30 = c, and then
y^sinliO, sinh(^ + -^), sinh[^+— ^j.
Thus in every case the equation is solved with the help of a table of trigonometrical or hyperbolic functions.
54. Several methods of this kind, all indeed ultimately equivalent, have been applied to the general quintic. One of these, the solution by means of the icosahedral irrationality, will now be given in outline ; for further details the reader is referred to Klein's lectures on the icosahedron, and to the treatise on modular functions by Klein and Fricke.
A point on a sphere may be determined by its north polar distance 6 and longitude <t>. If we put
n
Zx : Z2 = tan - (cos <f> + i sin <^),
Zx , Zi may be taken as homogeneous coordinates defining the position of the point. Suppose, now, tliat we have a regular icosahedron inscribed in the sphere, with one vertex at the point ^ = 0 and another on the great circle <^ = 0. If we put
f=z,z.,(zx^'+\Wz^'-z^'') the roots of /= 0 correspond to the twelve vertices of the solid. The binary form / has two covariants
H^- {zx"" + ;^2^) + 228 {zx^'z^' - z^'z^^) - 494 z^'z^\ T = (z,"^ + z^r) + 522 {zx^'z^' - z^z^) - 10005 {z^z.}'' + z^^'z^), and the three forms are connected by the identity
ir»+ 7^2= 1728/^. The roots o^ H=0 correspond to the centres of the equilateral triangles into which the surface of the sphere is divided by the great circle arcs into which the edges of the icosahedron are projected from the centre of the sphere ; and the roots of 7^= 0 correspond to the middle points of the sides of these triangles. H is the Hessian of /, and T is the Jacobian of H and /
Let ^ J9 be a side of any one of the 20 triangles, and CD any other of the remaining 29 sides. Then there is a definite rotation about a
53, 54] SOLUTION BY STANDARD FORMS 57
diameter of the sphere which brings AB into coincidence with CD. Similarly there is a definite rotation which brings AB into coincidence with DC. We thus obtain 58 rotations, each of which, applied to the icosahedron, brings it into a new position in which it occupies the same space as before. Besides these, there is the rotation about the diameter bisecting AB, which brings AB into coincidence with BA. Altogether, there are sixty different positions of the icosahedron, and if we include, as the identical operation, that of leaving the icosahedron alone, we have a group of 60 rotations which form a group. Each rotation may be associated with a linear substitution applied to Zi and z^. If we put
g27ri/5 = €,
then /=1, ?^ = 1,
and 5, t generate a group of 120 homogeneous substitutions, with which the group of rotations is hemihedrically isomorphic ; because if {azi + fiz2, yZi + 82^2) is any one of the substitutions,
f(aZi + j3z2, 'YZi + Sz2)=-(-oiZi-(3z2, -yZi-Sz^^ which corresponds to the same rotation. Every one of the homogeneous substitutions leaves/ H, T absolutely unaltered, but produces a certain permutation among their roots.
Consider, now, the function T. It evidently has the rational
factor
<^i - Zi^ + z^ ;
and if we apply to this the substitution t, we find that t(<fi^) = - <t>i. Now the roots of <^i = 0 are the ends of a diameter of the sphere : hence t must correspond to a rotation through an angle tt about a perpendicular diameter, the extremities of which are unaltered by t, so that they are given by
(€+€^)Zi + Z2_ Zi Zi-(€ + €^)Z2~ Z2*
or </»2 = z^^-2(€ + €') z,Z2 - z^ = 0.
If we put <^3 = z^^-2 («" + €«) z^Z2 - z^^
it is easily proved that the roots of <^3 = 0 are at the ends of a diameter perpendicular to each of the two others : hence, writing
r = <t>i<t>2i>s = Z^ + '^ZxZ^^ - hz^'z.^ - Wz^t " ^Z^Z.^ + Z^
r is a factor of T and the roots of t = 0 are the vertices of a regular octahedron. Since this has 12 edges, there are 24 rotations which bring it into coincidence with itself; of these 12 belong to the icosahedral group, and form a factor of it.
58 SOLUTION BY STANDARD FORMS [CH. V
By applying all the icosabedral substitutions to t we obtain five different sextics Ti(=t), Tg, xg, t^, xg the product of which is T. If, now, we form the equation
(r - Tj) (t- To) ... (t - T5) = T'+p,r* +...+p, = 0
the coefficients are binary forms which are invariable for the icosabedral group and of degrees 6, 12, 18, 24, 30 respectively. Each coefficient equated to zero must give an invariant set of points on the sphere; and since there are no sets of 6 or 18 points, and the only sets of 12 and 24 are given by /= 0,/^ = 0, the equation must reduce to the form
r' + afT^ + b/h-T=0 where a, b are numerical. By a comparison of coefficients it is found that a = - 10, i = 45, so that finally
T^-lO/T^ + 45/V-r-O. Putting t2//= r
we find that r satisfies the equation
r(/'^-10r + 45)2-7^//=. The Hessian of t is given by
K = - (;^« + zi) + {z^z., - z,z.I) - 7 {z,'z^^ + z^^zi) - 7 {z.'zi - z^zi) ;
like T this has five conjugate values and is invariant for the same group as T.
Suppose, now, that /, m are arbitrary numerical quantities, and let
I/k mPTK .
y=Ti^-HT (^)-
This is a function of the ratio Zijz^ which assumes only five values when the icosabedral substitutions are applied to it. The invariant quintic of which it is a root can be found by a process similar to that by which the equation satisfied by r was constructed. The result is, that if we write
-p=J^ 7^=^1 = 1728-^
aj = 8f + Pm+ ;
bj = -l^+ +
Ji j\
<'J = l'- 7-+ 7a
y satisfies the equation
^ + (iay'' + ^by-\-c = 0 (3).
•(2)
54, 55] SOLUTION BY STANDARD FORMS 59
55. Conversely, suppose a quintic given in the form (3) : if we can find, in terms of a, 6, c quantities /, m, j, j^ such that j +j\ = 1728, and the last three of equations (2) are satisfied, the roots of the given quintic will be expressible as rational functions of any one root of the normal equation
By combining equations (2) we find
jiilb + c) -- m'a (4),
'('■-x)^ ('■-¥)■ <«•
./la + Sb\ ,„ 21QPm dlm^ 21Qm^ ji- ) = P + — ^— + "V- + .„ (6).
From (2) and (6), by squaring,
27ay = l728/« + 432/^m + 2lfl + ^^^\ fm'^
4320.3 3 _^/2 72'^ ,2 4 18.216/w^ 27w« Ji \Ji h / 3x Jx
jif { =Jil + 432f w + ( 18 + —7— l^m"^ + —T—Pm^
\ m J \ Ji J Jx
/81 2.216^,2 4 18.216, , 27.1728 »
\Jx Jx / Jx Jx
On subtracting the last equation from the one before, we find that (1728 -ji) is a factor of the right-hand side ; since j +j\ = 1728, this cancels with the factor j on the left hand, and we thus obtain
•fo^ 2 • /^» + 86X2-1 , 3^^2x3
Comparing this with (5), we infer that
27 a'-^.ila + Sby^lc-K^ (7),
and by eliminating m'Yii from this and (4) it is found that I satisfies the equation
{a' + abc-b')P-(na^b-ac'' + 2b^c)l-(27a'c-Ua^b^ + bc') = 0
(8).
If D is the discriminant of (3), that of (8) is a^Dl5^, so that / is rational in the field (a, b, c, JD, J 6). The adjunction of JD reduces the group of the quintic from the symmetrical group to the alternate group of order 60 ; the quantity Jb is what is called an auxiliary irrationality, and does not affect the group.
60 SOLUTION BY STANDARD FORMS [CH. V
Having determined /, equations (4) and (5) give
._ {aP-m-Zcf . .
^~a%axi'-lf)l-bc} ^^^'
a rational function of / ; and since
\ JiJ 3x
we find, after substituting for j and m^\jx from (4) and (9), that
^ = \a^f^l-bc •••^^')-
Thus m can also be expressed as a rational function of /: of course, the above expression, like that obtained for J, can be transformed in various ways by making use of the equation satisfied by /.
To make this method actually useful for solving numerical quintics, we require a table giving the roots of the icosahedral equation
for difierent numerical values of/ When D is positive, /, m, j are real ; but when D is negative, j is in general complex, so that a complete table would have to include imaginary values oi j.
56. When a = 0, the foregoing results require modification, because in this case lb + c = 0, and the formulae (9) and (10) become in- determinate. Starting afresh with equations (2), after putting a = 0, it is found that if ^ is a determinate root of
b^t^ + c''$-QW = 0 (11)
we may put
bl = -c ¥m = nb^c-(?^
¥j = b' (1 7286» + 63c0 - c"" (c' + Sib') $ tPj\ = - 63^>V + c" (c* + 8160 ^1
and the formula (1), combined with H^-jf = 0, will give the roots of the equation
y + 5by + c = 0.
Another special case that requires examination is when a > 0, and equation (8) of last article is satisfied by putting
(ac-b')l = bc.
•(12),
55, 56] SOLUTION BY STANDARD FORMS 61
This leads to Sac = 46^, whence also, supposing that c does not vanish, I = ib/a = 3c/b. It is found that the equations (2) of Art. 54 reduce to
i+ii-1728,
. 2"6^ Ub^m
|
J- |
a' ' Sa' |
J |
|
|
r^ |
166^ |
||
|
. |
Ji ' |
3a'' |
|
|
The elimination of i and |
j\ leads to |
||
|
9a'm''+2 |
^}'ab'm + S.2''(32b' |
-27a')b'' = 0, |
|
|
the roots of which are |
|||
|
966 |
S2b (27a' - |
32b') |
|
|
a |
da' |
||
|
and the corresponding |
values of j are |
||
|
0127,3 |
|||
|
0, |
%'^.(^la'-m^). |
Now, if we take i = 0 the auxiliary equation is H -0. Referring back to equation (1), Art. 54, we see that this must be rejected, because it introduces a zero factor into the denominator of the expression for y. Thus the solution is
4J)fK 32b {21a'- 32b') f\K y~ aH^ MHT
with 21a^H' - 2^'b' (21a' - Ub')/' = 0.
This may be simplified by putting
166^ _ 27^^~'''
thus 2, = g{l + 24(l-2.^)•^},
with H'-2\3'n{n-l)r = 0.
If a = 2p', b = 3j^?^ this solution fails : but the equation is then
f + lOp'y'' + Ihp'y + 6j9« = 0 ; that is to say,
the roots of which are obvious.
NOTES AND REFERENCES.
8. The very important idea of a field of rationality has been made precise by Dedekind (Dirichlet-Dedekind, Vorlesungen liber Zahlen- theorie, Suppl. xi.) and Kronecker (Grundziige einer arithmetischen Theorie der algebraischen Grossen : Journ. f. Math. 92 = Werke 2).
15. On the problem of finding the irreducible factors of a poly- nomial, see Kronecker (Grundz.) and K. Runge (/. /. Math. 99). Special devices often shorten the work in particular cases.
Another way of finding the Galoisian group is explained by 0. Holder {Encycl. d. math. Wiss. I., p. 486). Except theoretically, the problem is not of much interest.
40. It will be observed that the definition of Abelian equations includes cyclical equations as a particular case ; it is, however, con- venient to retain both terms.
43, end. For the proof referred to, see Weber's Algebra^ ii. pp. 736-821 (or Acta Math. 8), and Hilbert, Die Theorie der algebraischen Zahlkorper, chap. 23 {Jahresb. d. deutschen Math.- Ver. 1894-5).
47. Weber applies the term metacyclic to all groups for which the indices ei (p. 21) are primes, and calls the corresponding equations metacyclic. Another (perhaps preferable) term is soluble. The defini- tion of a metacyclic group given in the text agrees with that of Kronecker {Berl Ber. 1879).
55. The algebraical eliminations contained in this article appear to have been first carried out in this way by Gordan (see Klein, Ikos, p. 192, note).
NOTES AND REFERENCES 63
The following list contains references to a selection of treatises and memoirs relating to the subject of this tract and its applications. Many- more will be found in the Encyclopddie der mathematiscJien Wissen- schaften, vol. i., sections B la, c, 3b, c, d.
L. Bianchi. Lezioni sulla teoria del gruppi di sostituzioni e delle equazioni algebrlche secondo Galois; Pisa, 1900.
C. Jordan. Traite des substitutions et des equations algebriques ; Paris, 1870.
E. Netto. Substitutionstheorie und ihre Anwendung auf die Algebra; Leipzig 1882 (trans. F. N. Cole, Ann Arbor, 1892): Vorlesungen ilber Algebra; Leipzig, 1896-9.
J. A. Serret Cours d'algebre superieure; Paris (5th edition), 1885.
H. Weber. Lehrbuch der Algebra; Braunschweig, 1895-6 (2nd edition, 1898-9).
F. Klein. Vorlesungen ilber das Ikosaeder und die Auflosung der Gleichungen mm funften Grade; Leipzig, 1884.
F. Klein and R. Fricke. Vorlesungen uber die Theorie der elliptischen Modulfunctionen ; Leipzig, 1890-2.
H. Weber. Elliptische Functionen und algebraische Zahlen ; Braunschweig, 1891.
E. Galois. (Euvres mathsmatiques ; ed. E. Picard, Paris, 1897 (also in Liouv. (1) xi.).
N. H. Abel. (Euvres; ed. Sylow et Lie, Christiania, 1881.
P. Bachmann. Die Lehre von der Kreistheilung ; Leipzig, 1872.
J. L. Lagrange {nouv. fnem. de Vacad. ray. de Berlin, 1770, 1771 : or collected works, iii.).
C. Jordan {Math Ann. i. 145, 583) ; L. Kronecker {Grelle, xciv. 344, Berl Ber. 1853, 1856); 0. Holder {Math. Ann. xxxiv. 26); E. Heine {Crelle, XLvm. 237).
Full references to the literature of cyclotomy will be found in Bachmann's Kreistheilung : other special applications are
(1) The solution of the quintic, for which see R. F. A. Clebsch {Math. Ann. iv. 284) ; F. Brioschi {Math. Ann. xiii. 109) ; P. Gordan {Math Ann. xiii. 375, xxviii. 152) ; C. Hermite {C. R. xlvi. 508, and LXi.-n. passim); L. Kiepert {Crelle, lxxxvii. 114); L. Kronecker {Crelle, Lix. 306).
64 NOTES AND REFERENCES
(2) Polyhedral equations: H. A. Schwarz (Crelle, lxxxvit. 139).
(3) Equations connected with elliptic and modular functions : W. Dyck {Math. Ann, xviii. 507) ; L. Kiepert {Crelle, lxxv. 255, and Math. Ann. xxvi., xxxii., etc.) ; Klein {Math. Ann. xii., xv.) ; A. G. Greenhill {Proc. Lond. Math. Sc).
(4) Equation of seventh degree with simple group of order 168. F. Klein {Math. Ann. xiv. 428) ; P. Gordan {Math. Ann. xx. 515, XXV. 459).
(5) Determination of the inflexions of a plane cubic. 0. Hease {Crelle, xxxiv. 193).
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