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

~154] RATIONAL FUNCTIONS BELONGING TO A GROUP 297

But, if s is a substitution of G not in H, then As is not identical
with 4 as to the variable p, since Vs is different from VI, Va,... V. We may therefore choose an integer p such that
/ belongs to H. This proves
THEOREM 1. Every rational function 4 with coefficients
in R of the roots of an equation with the group G for the domain
R belongs to a definite subgroup of G. There exist such functions
4 belonging to any assigned subgroup of G.
We next prove the important supplementary
THEOREM 2. If a rational function 4, with coefficients in a
domain R, of the roots of an equation with the group G for R, belongs to a subgroup H of index v under G, then the substitutions
of G replace 4, by exactly v distinct functions; they are the roots
of an equation
(13) g(y) —(Y-1)(Y-12) ~. ~ (y-4,)
with coefficients in R and irreducible in R.
As in ~ 10, let
(14) G= -+Hg2+Hg3+...+Hgvo
Let h be any substitution of H. Then
(h)e = = ('), == 'V. -
Thus P takes at most v values under G. But, if
X = g(j<i),
then 4g~-1=4, so that gigj-~ is a substitution h of G leaving 4
unaltered and hence is in H. Then gi=hgj, contrary to (14).
Thus 1i,,2,..., 4 are distinct, where a has been written
for by. They are called the conjugates to f-4/1 under G.
Any substitution s of G merely permutes 4i,..., A
amongst themselves. For, any product gis may be written in
the form hgj where h is in H; then
(0=)s =., =.gj = 4j = j.
Hence the coefficients of (13) are unaltered by every substitution of G and therefore equal quantities in R.
If g(y) has a factor with coefficients in R, it has a factor