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 $\sigma :L\u27f6L$ 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

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The degree $[K:F]$ is finite,

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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 $\mathcal{A}$ denote the set of finite normal extensions^{} $K$ of $F$ which are contained in $L$. The set $\mathcal{A}$ is a partially ordered set^{} under the inclusion relation. Form the inverse limit
$$\mathrm{\Gamma}:=\underset{\u27f5}{lim}\mathrm{Gal}(K/F)\subset \prod _{K\in \mathcal{A}}\mathrm{Gal}(K/F)$$ 
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 $K,{K}^{\prime}\in \mathcal{A}$ 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
$$\varphi :G\u27f6\prod _{K\in \mathcal{A}}\mathrm{Gal}(K/F)$$ 
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
$$G\u27f6\mathrm{Gal}(K/F)$$ 
has kernel $\mathrm{Gal}\mathit{}\mathrm{(}L\mathrm{/}K\mathrm{)}$.
Theorem 2 (Galois correspondence for finite subextensions).
Let $G$, $L$, $F$ be as before.

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

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

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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  20130322 12:39:06 
Last modified on  20130322 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 