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

298

GROUP OF AN ALGEBRAIC EQUATION [CH. XIV

7y(y) which is zero for y= 4l and hence (by B of ~ 149) for
y=2,..., y=4,. Thus y=g, so that g(y) is irreducible
in R.
EXAMPLE 1. The group G of x+x3 +x+1=0 for R(1) is 1, (x2x3)},
if x1=-1 denotes the real root. The conjugates to V1=X2-X1 under G
are 1 and p2=x3-x1; they are the roots of y2-2y+2=0.
EXAMPLE 2. The group G of x4+l=0 for R(i) is {1, (X1X3)(X2X4)} if
X1=E, X2=iE, X3= —, x4= -i, where e=(1+i)/v/2, so that e2=i. The
conjugates x1 and X3 to xi under G are the roots of y2-i=O, which is irreducible in R(i), to which e does not belong.
155. Galois' Generalization of Lagrange's Theorem. If a
rational function 0, with coefficients in a domain R, of the roots
of an equation f(x) =0, with the group G for R, remains unaltered
by all those substitutions of G which leave unaltered another rational
function 4, of the roots with coefficients in R, then ~ equals a rational
function of 4, with coefficients in R.
In case i is an n!-valued function V1, the only substitution leaving (
unaltered is the identity, and this leaves any a unaltered. For this case,
the theorem states that any rational function p with coefficients in R
equals a rational function of V1 with coefficients in R. This follows from
the like result in ~ 147 for the rational integral numerator and denominator
of p.
Let H be the subgroup of G of index v to which 4 belongs.
By means of (14), we obtain the v distinct conjugates f1,...,
4' to 4 — l4 under G. Since every substitution h of H leaves
f unaltered, each product hg, replaces 0 by 0=i b. Any substitution s of G replaces X by a certain 4j (end of ~ 154) and
likewise he by j. Thus, for g(y) defined by (13),
X(y) _g(y)( - + +2... - )
\y-Ai y-2 Y-AV
is an integral function of y each of whose coefficients is unaltered
by every substitution of G and hence is in R. Taking 41q 4
as y, we get ~=X(A). g'(4).
The theorem will be shown to be a generalization of
LAGRANGE'S THEOREM. If a rational function qf of the
independent variables xi,... remains unaltered by all
those substitutions on Xi,..., Xn which leave unaltered another