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

326 CONSTRUCTIONS BY RULER AND COMPASSES [CH. XVII

that the group of (4) for R is a regular cyclic group of order k.
If s> 1, k is not a power of 2, and by the usual argument the
regular p8-gon cannot be constructed by ruler and compasses.
Combining our results, we have the
THEOREM. A regular polygon of n sides can be constructed
by ruler and compasses if and only if n=2 pl1p2..., where
pi, p2,... are distinct primes of the form (3).
Since therefore a regular 9-gon cannot be constructed, we
have a new proof that angle 120~ cannot be trisected by ruler
and compasses.
Gauss * was the first to prove that a regular p-gon can
be constructed if p is a prime of the form (3); he stated,t but
apparently did not publish a proof of, the remaining part of
the above theorem. For the elegant method invented by
Gauss for finding the series of quadratic equations leading
to a 17th root of unity and the actual geometrical construction
of a regular 17-gon, as well as for a longer proof of the above
theorem without the aid of group theory, the reader may consult the monograph by Dickson,: where references to other
books are given.
* Disquisitiones Arithmeticc, 1801, Art. 335-366 [=Werke, 1]; German
translation by Maser, 1889, pp. 397-448, 630-652.
t Gauss-Maser, p. 447.
t Monographs on Modern Mathematics, edited by J. W. A. Young, New York,
1911. A brief, but more elementary, treatment is given in Dickson's Elementary
Theory of Equations, 1914, pp. 84-92. A still more elementary discussion is that
by Dickson, Amer. Math. Monthly, vol. 21 (1914), 259-262.