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

324 CONSTRUCTIONS BY RULER AND COMPASSES [CH. XVII

Hence if r can be expressed in terms of i and real square roots,
cos 2r/n can be expressed in terms of real square roots. The
converse is seen to be true by an inspection of (2), since the
sine can be found from the cosine by a real square root. Hence
a regular n-gon can be constructed by ruler and compasses if
and only if the nth root (2) of unity can be found by the extraction of square roots, all except the last one of which is real.
If n is an odd prime p, r is a root of an equation of degree
p-1 irreducible in the domain R of all rational numbers and
having as its group for R a regular cyclic group C of order
p-1 (~ 161-3). The adjunction of any root reduces C to
the identity. If a regular p-gon can be constructed, the adjunction of the root r is equivalent to that of several square
roots, the adjunction of each of which causes either no reduction
in the group or a reduction to a subgroup of index 2. Hence
a regular p-gon can be constructed by ruler and compasses
if and only if p-1 is a power 2h of 2. But if h=fq, where f
is odd, then 2h+1 has the factor 2q +1. Hence a prime of the
form 2h+1 is of the form
(3) 22t+1.
For t=O, 1, 2, 3, 4, the corresponding numbers are 3, 5, 17,
257, 65537, and are all primes. But for t=5, 6, 7, 8, 9, 11, 12,
etc., the number is known to be not prime.
Next, let n=ab, where a and b are relatively prime integers
> 1. If a regular a-gon and a regular b-gon can be constructed
by ruler and compasses, the same is true of a regular n-gon.
For, multiples of the angles 2r/a and 2r/b can then be constructed and hence also the sum of these multiples. Since
there exist integers c and d such that ca+db=l, the angle
27r 27 2r 27
d*-+c -=-(db+ca) =
a b ab ab'
and therefore also the ab-gon, can be constructed. Conversely,
from the latter we obtain a regular a-gon by using the 1st,
(b+l)th, (2b+l)th,..., [(a-1)b+l]th vertices. Hence if
n=pq..., where p, q,... are distinct primes, a regular