infinite Galois theory

Let $L/F$ be a Galois extension, not necessarily finite dimensional.

1 Topology on the Galois group

Recall that the Galois group $G:=\mathrm{Gal}(L/F)$ of $L/F$ is the group of all field automorphisms $\sigma :L⟶L$ that restrict to the identity map on $F$, under the group operation of composition. In the case where the extension $L/F$ is infinite dimensional, the group $G$ comes equipped with a natural topology, which plays a key role in the statement of the Galois correspondence.

We define a subset $U$ of $G$ to be open if, for each $\sigma \in U$, there exists an intermediate field $K\subset L$ such that

• The degree $[K:F]$ is finite,

• If ${\sigma }^{\prime }$ is another element of $G$, and the restrictions ${\sigma |}_K$ and ${{\sigma }^{\prime }|}_K$ are equal, then ${\sigma }^{\prime }\in U$.

The resulting collection of open sets forms a topology on $G$, called the Krull topology, and $G$ is a topological group under the Krull topology. Another way to define the topology is to state that the subgroups $\mathrm{Gal}(L/K)$ for finite extensions $K/F$ form a neighborhood basis for $\mathrm{Gal}(L/F)$ at the identity.

2 Inverse limit structure

In this section we exhibit the group $G$ as a projective limit of an inverse system of finite groups. This construction shows that the Galois group $G$ is actually a profinite group.

Let $𝒜$ denote the set of finite normal extensions $K$ of $F$ which are contained in $L$. The set $𝒜$ is a partially ordered set under the inclusion relation. Form the inverse limit

$$\mathrm{\Gamma }:=\underset{⟵}{lim}\mathrm{Gal}(K/F)\subset \prod _{K\in 𝒜}\mathrm{Gal}(K/F)$$

consisting, as usual, of the set of all $({\sigma }_K)\in {\prod }_K\mathrm{Gal}(K/F)$ such that ${{\sigma }_{K^{\prime }}|}_K={\sigma }_K$ for all $K,K^{\prime }\in 𝒜$ with $K\subset K^{\prime }$. We make $\mathrm{\Gamma }$ into a topological space by putting the discrete topology on each finite set $\mathrm{Gal}(K/F)$ and giving $\mathrm{\Gamma }$ the subspace topology induced by the product topology on ${\prod }_K\mathrm{Gal}(K/F)$. The group $\mathrm{\Gamma }$ is a closed subset of the compact group ${\prod }_K\mathrm{Gal}(K/F)$, and is therefore compact.

Let

$$\varphi :G⟶\prod _{K\in 𝒜}\mathrm{Gal}(K/F)$$

be the group homomorphism which sends an element $\sigma \in G$ to the element $({\sigma }_K)$ of ${\prod }_K\mathrm{Gal}(K/F)$ whose $K$–th coordinate is the automorphism ${\sigma |}_K\in \mathrm{Gal}(K/F)$. Then the function $\varphi$ has image equal to $\mathrm{\Gamma }$ and in fact is a homeomorphism between $G$ and $\mathrm{\Gamma }$. Since $\mathrm{\Gamma }$ is profinite, it follows that $G$ is profinite as well.

3 The Galois correspondence

Theorem 1 (Galois correspondence for infinite extensions).

Let $G$, $L$, $F$ be as before. For every closed subgroup $H$ of $G$, let $L^H$ denote the fixed field of $H$. The correspondence

$$K\mapsto \mathrm{Gal}(L/K),$$

defined for all intermediate field extensions $F\mathrm{\subset }K\mathrm{\subset }L$, is an inclusion reversing bijection between the set of all intermediate extensions $K$ and the set of all closed subgroups of $G$. Its inverse is the correspondence

$$H\mapsto L^H,$$

defined for all closed subgroups $H$ of $G$. The extension $K\mathrm{/}F$ is normal if and only if $\mathrm{Gal}\mathit{}\mathrm{(}L\mathrm{/}K\mathrm{)}$ is a normal subgroup of $G$, and in this case the restriction map

$$G⟶\mathrm{Gal}(K/F)$$

has kernel $\mathrm{Gal}\mathit{}\mathrm{(}L\mathrm{/}K\mathrm{)}$.

Theorem 2 (Galois correspondence for finite subextensions).

Let $G$, $L$, $F$ be as before.

• Every open subgroup $H\subset G$ is closed and has finite index in $G$.

• If $H\subset G$ is an open subgroup, then the field extension $L^H/F$ is finite.

• For every intermediate field $K$ with $[K:F]$ finite, the Galois group $\mathrm{Gal}(L/K)$ is an open subgroup of $G$.

Title	infinite Galois theory
Canonical name	InfiniteGaloisTheory
Date of creation	2013-03-22 12:39:06
Last modified on	2013-03-22 12:39:06
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Topic
Classification	msc 12F10
Classification	msc 13B05
Related topic	FundamentalTheoremOfGaloisTheory
Related topic	GaloisGroup
Related topic	InverseLimit
Defines	Krull topology