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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.
This video is part of a lecture course by Richard Borcherds from 2020-21.