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

~ 164] SUFFICIENT CONDITION FOR SOLVABILITY

311

This will follow if proved for the case of a regular cyclic
group of prime order. Assuming the theorem for this case,
let f(x) =0 be an equation whose group for R has the prime
factors of composition v, p,... As in the proof of Theorem
4 of ~ 159, there is a series of equations n(Q) =0, r(x) =0,.
of prime degrees v, p,..., the solution of which is equivalent
to the solution of f(x)=0. The group for R of n(4)=0 is a
regular cyclic group of prime order v so that this auxiliary
equation is solvable by radicals relatively to R. The coefficients of r(x) =0 are in the domain R'= (R, 4) and its group
for R' is a regular cyclic group of prime order p; hence it is
solvable by radicals relatively to R'. In view of the earlier
result, this second auxiliary equation is solvable by radicals
relatively to R. A repetition of this argument shows that
f(x) =0 is solvable by radicals relatively to R.
It remains only to prove that an equation C(x)=0 having
a regular cyclic group G of prime order p for a domain R is
solvable by radicals relatively to R. This is true for p=2.
To proceed by induction, suppose that every equation having
a regular cyclic group of prime order <p for any domain D
is solvable by radicals relatively to D. As in the proof above,
this implies that the equation for the imaginary pth roots of
unity is solvable by radicals (i.e., relatively to the domain of
rational numbers). In fact, its group for that domain is a
regular cyclic group of order p-1 (~ 161), each of whose factors
of composition is a prime <p.
Adjoin to R an imaginary pth root e of unity. The group
of C(x)=0 for (R, e) is either the initial cyclic group G or the
identity group. In the latter case, the roots are in (R, e)
and can be found from the quantities in R by rational operations and root extractions, since e was shown to be derivable
from the rational number by those operations. In the former
case, C(x)=0 is solvable (~ 160) by radicals relatively to (R, e)
and hence, as before, relatively to R. Hence the induction is
complete.
COROLLARY. If p is an odd prime, the equation for the
p-1 imaginary pth roots of unity is solvable by radicals.