An introduction to the modern theory of equations, by Florian Cajori.

222

THEORY OF EQUATIONS

Let s be a substitution in G whtich changes X to X1. That
same substitution transforms the sub-group Q into the conjugate
sub-group s-iQs_ Q1. Now the substitutions in the sub-group
Q, leave X1 unchanged. For, to operate with the substitutions
in Q1 is the same as to operate with s-lQs, where s-1 changes
X1 to X, and X remains unaltered by the substitutions in Q,
while s changes X back to X1. But X and Xi are roots of a
cyclic equation; hence X1 is a function in 0 of X, and X is a
function in 2 of X, so that X and X1 belong to the same group
Q. Therefore, Q == Q.
Since the same reasoning applies to X and any one of the
other roots X2, *.., X_,,_j, it follows that Q is identical with all
of its conjugate groups; that is, Q is a normal sub-group of G,
having the index m.
194. The Converse Theorem. If the group G of the equation
f(x) = 0 has a normeal sucb-group Q, whose index is a prime number m, then, by adjunction f a root of a cyclic equation of the
degree m, the group G is reduced to Q.
If the group G has a normal sub-group Q of the prime index
m, and if we select a function Mi which belongs to the subgroup Q, the conjugate functions all belong to the same group
Q. By ~ 162, each function lM[, M *, M, _, is contained in
the domain Q(,). Hence this domain is a normal domain, ~ 132,
and Mis the root of a normal equation, ~ 139. In the domain
2(M) we have Q as the group of the equation f()V) = 0, ~ 163.
But, if m is a prime number, the normal equation is also a
cyclic equation; for, the degree m of the normal equation is
also the order of the Galois group, ~~149, 150. Take any
substitution s (not the identical substitution) in the Galois
group. The different powers of s constitute a sub-group, the
order of which is a factor of the order of the Galois group.
As m is prime, the order of s must be m and the sub-group is
S, S2, s3, *.., Sm. The Galois group and its sub-group, being of
8~ 8 I' ' I 1i.I