Infinite Galois theory

Abstract

[en] For a finite Galois extension $K | k$, the fundamental theorem of classical Galois theory establishes a one-to-one correspondence between the intermediate fields $E | k$ and the subgroups of Gal $( K | k )$ , the Galois group of the extension. With this correspondence, we can examine the finite field extension by using group theory, which is, in some sense, better understood.

A natural question may arise: does this correspondence still hold for infinite Galois extensions? It is very tempting to assume the correspondence still exists. Unfortunately, this correspondence between the intermediate fields of $K | k$ and the subgroups of Gal $( K | k )$ does not necessarily hold when $K | k$ is an infinite Galois extension.

A naive approach to why this correspondence fails is to observe that Gal $( K | k )$ has "too many" subgroups, so there is no subfield E of K containing k that can correspond to most of its subgroups. Therefore, it is necessary to find a way to only look at the "relevant subgroups" of the infinite Galois group. This is where topology comes to the rescue, letting us introduce a topology on an arbitrary group and study its subgroups with a different perspective.

This new study of groups with a topological perspective will lead to our main goal for this work, the discovery that the fundamental theorem of classical Galois theory holds for infinite Galois extensions $K | k$, whenever we associate a particular topology to the Galois group Gal $( K | k )$.

After this theorem is proved, we are going give some examples, two of them with more details than the others. We are going to first characterize the absolute Galois group, that is, the Galois group of the extension $\bar{k} | k$, where $\bar{k}$ is the algebraic closure of $k$. This will be achieved by the means of the Artin-Shreier theorem. Then, we are going to explore the field of $p$-adic numbers, $\mathbb{Q}_p$ . We will briefly discuss the structures of the Galois extensions of this field.

In this dissertation we assume some previous knowledge. This previous knowledge corresponds to the subjects taught at the University of Barcelona: Algebraic Structures, Algebraic Equations, Toplogy and Mathematical Analysis.