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

~ 172]

REGULAR POLYGONS

325

n-gon can be constructed if and only if a regular p-gon, q'-gon,. can be constructed. A 2S-gon can be constructed by
repeated bisections of 180~.
It therefore remains only to discuss the regular p'-gon,
where p is an odd prime. By De Moivre's theorem,
27r 27r
p = cos- +i sin -
is a root of xp =1, but not of xp ='1, and hence is a root of
(4) x - Xt(P-1) +X t(p-2)+... + _=0 (t= p-1).
Since pP, p2p,.. ptp give the t roots of xt=1, the remaining
tp-t powers of p, with positive exponents less than tp and not
divisible by p, are roots of (4) and give all of the roots of (4).
They are called the primitive path roots of unity.
For pS= 9, the six primitive ninth roots of unity are p, p2, p4, p5, p7, p8
and are the roots of x6 +x+1=0.
The proof that (4) is irreducible in the domain R of all
rational numbers differs from that in ~ 163 for the special
case s=l1 only in the detail of having, instead of e,,...
e~-1 in the former case, the roots p, pa, pb,..., p of (4),
where 1, a, b,..., I denote the positive integers less than p*
and not divisible by p, and p is an arbitrary primitive psth
root of unity.
As shown in the theory of numbers, there exists a primitive
root g of p8, where p is an odd prime, i.e., an integer g such that
1, 9a g22., -1 ( = ps_ s-l),
when divided by ps, give as remainders in some order the positive integers less than p* and not divisible by p. Thus the
roots of (4) are
P, Pg pg2... T pg-1
In the former example pS = 9, we may take g=2. Then the preceding
roots are p, p2, p4, p8, p7, p5, respectively.
Since each root of (4) can therefore be expressed as the
gth power of the preceding root, we readily find as in ~ 161