Cyclotomic Extensions

(Section 55) Cyclotomic Extensions

(subsection) Galois group of a Cyclotomic Extension

(Def 55.1) The splitting field of x^{n} -1 over F is the nth cyclotomic extension of F.

(Def 55.2) The polynomial \Phi_{n} (x) = \prod_{i = 1}^{\phi (n)} (x- \alpha_{i} ) where the \alpha_{i} are the primitive nth roots of unity in \bar{F}, is the nth cyclotomic polynomial over F.

(prop) Over \mathbb{Q}, \Phi_{n}(x) is irreducible.

(Thm 55.4) The Galois group of the nth cyclotomic extension of \mathbb{Q} has \phi (n) elements and is isomorphic to the group consisting of the positive integers less than n and relatively prime to n under multiplication modulo n.

(Cor 55.6) The Galois group of the pth cyclotomic extension of \mathbb{Q} for a prime p is cyclic of order p-1.

(subsection) Constructible Polygons

(Def) Fermat prime p = 2^{2^{k}} + 1 for k \in \mathbb{N} which is prime.

(Thm 55.8) The regular n-gon is constructible with a compass and a straightedge iff all the odd primes dividing n are Fermat primes whose squares do not divide n.