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

318

EQUATIONS SOLVABLE BY RADICALS [CH. XVI

an algebraic equation Fi(x)=0 be reduced to G'1 by the adjunction of all of the roots of a second equation F2(x) =0, and let the
group G2 for R of the second equation be reduced to G'2 by the
adjunction of all of the roots of the first equation. Then G'1 and
G'2 are invariant subgroups of G1 and G2, respectively, of equal
indices, and * the quotient-groups G1/G'l and G2/G'2 are simply
isomorphic.
By ~ 154 there exists a rational function 11 with coefficients
in R of the roots ~1,..., of the first equation, such that
/1i belongs to the subgroup G'1 of G1. Since the adjunction
of the roots 71,..., rm of the second equation reduces G1 to
G'1, property A of G'1 (~ 149) shows that {1 lies in the enlarged
domain:
(3) V(1,. ~, & n) =(1,.... 7, rm)
where 01 is a rational function with coefficients in R.
Let 1i, 42,..., ok denote all of the numerically distinct
values which Ai can take under the substitutions (on I,...
in) of G1. Then G'1 is of index k under G1 (~ 154, Theorem 2).
Let oPi,..., /I denote all of the numerically distinct values
which li can take under the substitutions (on l7,..., rm)
of G2. The k quantities f/ are the roots of an equation irreducible in R; likewise for the I quantities ~. Since these two
irreducible equations have a common root Al =, they are
identical (~ 144). Hence the 4's coincide in some order with
the ~'s; in particular k=Z.
If s, is a substitution of GI which replaces ~1 by X/z, then
st transforms the group G'i of 4l into the group of Xi of the
same order as G'1. Since X equals a ~, it is in the domain
R'=(R, rl,.., 'rm) and hence is unaltered by the substitutions of the group G'1 of F(x) =0 for that domain R' (~ 149,
property B). Hence the group of Xt contains all of the substitutions of GC' and, being of the same order, is identical with
G'I. Thus G'1 is invariant in G1. The group for R of the
irreducible equation satisfied by Al is therefore the quotient
group G1/G'1 (~ 159).
* This supplement and the proof here employed are due to Holder, Mathematische Annalen, vol. 34, (1889), p. 47.