Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 8.5 Cyclotomic Polynomials

Definition 8.5.1. Let n be a positive integer, and let be the complex number = cos + i sin , where = 2 / n. The polynomial

n (x) = k (x - k),

where k belongs to the set of positive integers less than n and relatively prime to n, is called the nth cyclotomic polynomial.

Proposition 8.5.2. Let n be a positive integer, and let n(x) be the nth cyclotomic polynomial. The following conditions hold:

(a) deg ( n (x)) = (n);

(b) xn - 1 = d | n d (x);

(c) n (x) is monic, with integer coefficients.

Theorem 8.5.3. The nth cyclotomic polynomial n(x) is irreducible over Q, for every positive integer n.

Theorem 8.5.4. For every positive integer n, the Galois group of the nth cyclotomic polynomial n(x) over Q is isomorphic to Zn×.

Example. 8.5.2. It can be shown that a regular n-gon is constructible if and only if (n) is a power of 2. If p is an odd prime, and (p) is a power of 2, then p must have the form p = 2k + 1, where k is a power of 2. Such primes are called Fermat primes. The only known examples are 3, 5, 17, 257, and 65537. This implies, for example, that a regular 17-gon is constructible.

A set that satisfies all the axioms of a field except for commutativity of multiplication is called a division ring or skew field.

Theorem 8.5.6. [Wedderburn] Any finite division ring is a field.


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