Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set of notes, in addition to elementary number theory, group and ring theory, and linear algebra: Fact 0.2. If F is a field, then F [x] is a PID, so all nonzero prime ideals are maximal and are generated by a single irreducible polynomial. This irreducible polynomial is the polynomial of lowest positive degree in the ideal and is unique modulo units.