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 thegroup of all field automorphisms $\sigma :L⟶L$ that restrict tothe identity map on $F$, under the group operation of composition. Inthe case where the extension $L/F$ is infinite dimensional, the group$G$ comes equipped with a natural topology, which plays a key role inthe 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 $G$, calledthe Krull topology, and $G$ is a topological group under theKrull topology. Another way to define the topology is to state thatthe subgroups $\mathrm{Gal}(L/K)$ for finite extensions $K/F$ form a neighborhoodbasis 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 aninverse system of finite groups. This construction shows that theGalois group $G$ is actually a profinite group.

Let $𝒜$ denote the set of finite normal extensions $K$ of $F$ whichare contained in $L$. The set $𝒜$ is a partially ordered set underthe 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 byputting the discrete topology on each finite set $\mathrm{Gal}(K/F)$ andgiving $\mathrm{\Gamma }$ the subspace topology induced by the product topologyon ${\prod }_K\mathrm{Gal}(K/F)$. The group $\mathrm{\Gamma }$ is a closed subset of thecompact 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 theelement $({\sigma }_K)$ of ${\prod }_K\mathrm{Gal}(K/F)$ whose $K$–th coordinateis the automorphism ${\sigma |}_K\in \mathrm{Gal}(K/F)$. Then the function$\varphi$ has image equal to $\mathrm{\Gamma }$ and in fact is a homeomorphismbetween $G$ 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 allintermediate 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 $K\mathrm{/}F$ isnormal if and only if $\mathrm{Gal}\mathit{}\mathrm{(}L\mathrm{/}K\mathrm{)}$ is a normal subgroup of $G$, and inthis 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 indexin $G$.

• 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 Galoisgroup $\mathrm{Gal}(L/K)$ is an open subgroup of $G$.