Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

284

GROUP OF AN ALGEBRAIC EQUATION [CH. XIV

and hence * are integral rational functions of ml,..., m,
cl,..., c, with integral coefficients, and therefore are numbers of the domain R. Thus the coefficients of the polynomial
in V, given by the expansion of
(3) F(V):(V- VI)(V- VS)... (V- VF,),
are numbers of the domain R.
If F(V) is reducible in R, let G(V) be that irreducible
factor in R for which G(Vi)=0. If F(V) is irreducible in R,
take G(V) to be F(V) itself. In either case, G(V)=0 is an
irreducible equation in R, having the root Vi; it is called a
Galoisian resolvent of equation (1) for the domain R.
The corresponding resolvent for equation (2) in R(1) is
G(V) - (V- V1)(V- VC) - V2-2V +2=O.
For the domain R(i), the resolvent is V-V1= 0.
147. Theorem. Let (xl,..., xn) be any rational integral
function, with coefficients in a domain R, of the roots of an equation with coefficients in R. Let s be any substitution on the roots
and let it replace ~ by As, and an n!-valued linear function V1,
with coefficients in R, by Vs. Then
(4) =~ (V)X(Vs)
(4)
where X is a polynomial with coefficients in R, while F' is the
derivative of the polynomial (3) with coefficients in R, so that
F'(Vs)ZO. Thus qs is the same rational function p(Vs) of V,
that = —1 is of V1.
If sjs = s, then sk replaces 0s by k,. Thus Sk permutes
*,..., Sn! in the same manner that it permutes V1,..
Vs! (~ 146). Hence the terms of
(5) X(V)q5 (L + F( F(V) _ __ _ F(V)
V-VI 2VVs f!V -V
* Several detailed proofs of this fundamental theorem on symmetric functions,
which is frequently applied below, are given in Dickson's Elcientary Theory of
Equations.