This lecture is part of an online graduate course on Galois theory.
We show how to extend Galois theory to infinite Galois extensions. The main difference is that the Galois group has a topology, and intermediate field extensions now correspond to closed subgroups of the Galois group. We give some examples, such as the absolute Galois group of a finite field, and the Galois group of the cylotomic extension of the rationals.
We also show that the Gaussian integers cannot be extended to a Galois extension with Galois group Z/4Z, which put some restrictions on the absolute Galois group of the rationals.