Cyclotomic Extensions

For any field K, a field K(ζn) where ζn is a root of unity (of order n) is called a cyclotomic extension of K. The term cyclotomic means circle-dividing, and comes from the fact that the nth roots of unity divide a circle into arcs of equal length. We will see that the extensions K(ζn)/K have abelian Galois groups and we will look in particular at cyclotomic extensions of Q and finite fields. There are not many general methods known for constructing Galois extensions with abelian Galois groups; cyclotomic extensions are essentially the only construction that works for all base fields. (Other constructions of Galois extensions with abelian Galois groups are Kummer extensions, Artin-Schreier-Witt extensions, and Carlitz extensions, but these all require special conditions on the base field.) We will look at nth roots of unity over K when Xn−1 is separable in K[X], so Xn−1 has n different roots in a splitting field over K. The polynomial Xn − 1 is separable when it is relatively prime to its derivative nXn−1 in K[X], so separability is equivalent to n 6= 0 in K: eitherK has characteristic 0 and n is arbitrary orK has characteristic p and n is not divisible by p. To emphasize that, we repeat: assume throughout that Xn − 1 is separable in K[X]: char(K) = 0, or char(K) = p and (p, n) = 1. When Xn − 1 is separable, its n roots are a multiplicative group of size n. In C we can write down the nth roots of unity analytically as e2πik/n for 0 ≤ k ≤ n − 1 and see they form a cyclic group with generator e2πi/n. It turns out to be a general phenomenon in all fields (including characteristic p) that the nth roots of unity are a cyclic group.