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

THE ALGEBRAIC SOLUTION OF EQUATIONS 223

the same order, are identical. Hence the Galois group is the
cyclic group, s, s2,..., s"', and the normal equation is a cyclic
equation, ~ 170.
195. Metacyclic Equations. An equation is called metacyclic
or solvable, when its solution can be reduced to the solution of
a series of cyclic equations. Abelian equations are a special
class of metacyclic equations. The latter embrace all equations
that are solvable by radicals, and no others.
In ~ 191 it was shown that any equation which can be solved
by radicals can be solved by the aid of cyclic equations of
prime index. In ~ 193 it was shown that if the adjunction of
a root of a cyclic equation of prime degree reduces the group
G, there exists a normal sub-group whose index is a prime
number; while in ~ 194 it was shown that, if G has a normal
sub-group, the reduction can always be effected by the adjunction of such a root.
196. Criterion of Solvability. That a given calgebraic equation
be metacyclic it is necessary and sufficient that there exist a series
of gro'ps G 1, G,, '", GC =1,
the first of which is the Galois grou?.p of the equatiol in Q, the last
of iwhich is the identical grotup, each grolupe being a normal subgroup of the preceding and of a prime index.
The group G of a metacyclic equation must have a normal
sub-group of an index j that is a prime number. Call this subgroup G1. If GC consists of the identical substitution only
(whose order is 1), then j =- That is, the order of G itself
is prime, and G has no sub-groups, except 1. This can happen
only when G itself is a cyclic group, and the given metacyclic
equation is itself only a cyclic equation.
If GC is not 1, then, since the equation is, by hypothesis,
solvable by radicals, C], must again have a normal sub-group G2,