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

ABELIAN EQUATION S

213

adljunction of x, the group of the Galois domain is reduced
to 1, ~ 163.
The Galois domain of g(x) = 0 is a, a a...,,_ ~ 143. Each
of the roots cu,, a,...,, a,,_ is a number in the Galois domnain and
each of the roots admits of the substitutions of the sub-group
Q= 1; hence each root is contained in the domain 2,(a) ~ 162,
and each root can be expressed as a function in 0l of one of
them. Therefore, g(x) = 0 is a normal equation and the domain
Qi) is a normal domain, ~ 132. We have then
(tk = ^k( )
and the Galois group of g(x)= 0 consists of the substitutions,
~149, pk= (=, 't(,)).
We have, ~ 148, PhPk = (a, = hk(k)),
PkPh = (a, <Pk^(ft))As the group is assumed to be commutative, we must have,
OfhC(a) = 0 q(a)1
i.e. g(x) =0 is an Abelian equation.
188. Theorem. In a substitution belonging to a transitive
Abelian group all the cycles consist of the same zember of
elements.
Let the substitution s be resolved into its cycles, and let r
be the least number of elements in any cycle. The substitution
s', applied to the elements in that cycle, leaves the elements
unchanged. Since, ~ 187, in a transitive Abelian group no substitution, except the identical one, leaves an element unaltered,
s' must be the identical substitution. But this can only be the
case when all other cycles (if there are others) consist of r
elements.
Ex. 1. Name the Abelian group of degree five, in which the cycles in
one and the same substitution do not have the same number of elements.
Explain. See Ex. 3, ~ 185, also ~ 104.