 (n)
For a positive integer n, the number of positive integers
less than or equal to n and relatively prime to n.
Primitive n-th roots of unity
Let n be a positive integer and let
  cos(2 / n)  i sin(2 / n) .
The quantities  k , where 1 k  n and gcd(n,k)=1.
n-th cyclotomic polynomial over Q
For positive integer n, let 1,2 ,..., (n) be the primitive
roots of unity.
The polynomial n ( x)  ( x  1)( x  2 )...( x   (n) ) .
Theorem 33.1
For every positive integer n, xn 1   d |n d ( x) , where
the product runs over all positive divisors d of n.
Theorem 33.2
For every positive integer n, n ( x) has integer
coefficients.
Theorem 33.3 (Gauss)
The cyclotomic polynomials n ( x) are irreducible over
Z.
Theorem 33.4
Let  be a primitive n-th root of unity. Then
Gal(Q()/ Q) U (n) .
Lemma, p. 569
Let n be a positive integer and let
  cos(2 / n)  i sin(2 / n) . Then
Q(cos(2 / n))  Q() .
Theorem 33.5 (Gauss)
It is possible to construct the regular n-gon with a
straightedge and compass if and only if n has the form
2k p1 p2... pt , where k  0 and the pi are distinct primes
of the form 2m 1. [Note: if 2m 1 is prime, with m  1 ,
it can be shown that m is a power of 2.]