The main theorem of Galois theory gives a inclusion-reversing correspondence between subfields of a finite Galois extension, and subgroups of the group of automorphisms. One might then ask how this situation changes if we consider fields of infinite degree over the ground field. We'll show that the "naive" way of extending the correspondence won't work, but with a suitable restriction on our group, we get a bona fide bijection. We'll then construct some interesting fields with equally interesting Galois groups.
This talk should be accessible to everyone; elementary Galois theory and (very very) elementary topology recommended, but I'll go over all the necessary definitions at the beginning.