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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),

 
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 Z_n^×.

 
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 = 2^k + 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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