Galois theory: Cyclotomic polynomials

This lecture is part of an online graduate course on Galois theory.
We introduce cyclotomic polynomials, and use Frobenius automorphisms to show they are irreducible. We give two applications of them: we prove a special case of Dirichlet's theorem on primes in arithmetic progressions, and we show that any finite abelian group is the Galois group of a finite extension of the rational numbers Q.