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

ABELIAN EQUATIONS

215

whole unchanged, it follows that, after the operation, each
cycle still has the same letters occurring in it and in the
same cyclic order, though the cycles may have interchanged
positions. Since s may be any substitution in the group C(,
except 1, we conclude that the group is imprimitLie, whenever
t > 1, ~ 103.
Let M be a cyclic function of the roots c, a, *.., a**, Zc1 a
cyclic function of the roots A, f,, /,-l, and so on. We have
then
XM- (A 1.l,...-l).,il - a, a, ' o o,
There will be t such conjugate cyclic functions, Jf, M1,,
M?,.., iM'~-l
Let Q represent the aggregate of all the substitutions in the
group G which do not replace a cycle by another, but simply
interchange the elements in each cycle. This aggregate of
substitutions is a group; the product of any two of them gives
a substitution belonging to G, which does not interchange the
cycles. Thus, Q is a sub-group of G.
As no substitution in Q can change ac into any element not
belonging to the cycle c, Q is an intransitive group.
The function M is readily seen to admit the substitutions
in Q and those only; hence, if we adjoin Mil to the domain 2,
the group of f(x) = 0 reduces to Q, ~ 163.
As Q is intransitive, the equation f(x) = 0 is reducible in the
domain Q(.), ~ 156.
Let f(x, M) be a function of x, defined thus:
f(x, M) (x - a) (x - a,)... (x - a,_).
We proceed to show that this is one of the factors of f(x) in
the domain Q(). Since Q is intransitive and permutes the
roots in each cycle among themselves only, the coefficients of
f(x, M) admit all the substitutions of Q. Therefore f(x, M)