Finite fields via Galois theory
The Galois theory of finite fields is completely understood and we present it in this video. In fact, we will use Galois theory to recover the classical classification of finite fields. The key is the Frobenius homomorphism which generates the Galois group of finite fields over their prime subfield. We end by showing how the Galois groups fit together in an inverse system by applying the Galois correspondence to the poset of finite subfields of the algebraic closure of Z/p. This foreshadows how Galois theory works in infinite extensions.