Galois theory: Infinite Galois extensions

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.

[peterclark]

For the Krull topology it's equivalent to take the collection of Galois groups corresponding to finite extensions N_i as a basis for the topology, such that the neighborhood of an element g_i is the translate of N_i by g_i, and the open sets of the topology are precisely the union of open neighborhoods. Then you can show that this is continuous with respect to the group action (g, x) -> gx and the inverse action g -> g^-1

[olivierbegassat]

20:55 why mention the multiplication on the profinite completion of the integers? A priori the only structure that's relevant in the present context is the group structure: is this a misconception? Or does the ring structure have a Galois theoretic interpretation?

[migarsormrapophis]

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