Infinite Galois theory
In this video, we look at Galois theory for infinite Galois field extensions. The key idea is to think of such extensions as unions or limits of finite ones. We show that this induces an inverse system of Galois groups which exhibits the Galois group of an infinite Galois extension as a pro-finite group, and hence one that has a topology. We look at the Galois correspondence in this setting which is much the same as in the finite case, with the added subtlety that one must only consider subgroups of the Galois group which are closed in the topology of the pro-finite group. An illustrative example is the absolute Galois group of a finite field with a prime number of elements. We explore this example in some detail. The concepts here are naturally related to the fundamental group in topology and the simply connected cover.