# 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 $G:=\operatorname{Gal}(L/F)$ of $L/F$ is the group of all field automorphisms $\sigma:L\longrightarrow L$ 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 $\operatorname{Gal}(L/K)$ for finite extensions $K/F$ form a neighborhood basis for $\operatorname{Gal}(L/F)$ 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

 $\Gamma:=\,\underset{\longleftarrow}{\lim}\,\operatorname{Gal}(K/F)\subset\prod% _{K\in\mathcal{A}}\operatorname{Gal}(K/F)$

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

Let

 $\phi:G\longrightarrow\prod_{K\in\mathcal{A}}\operatorname{Gal}(K/F)$

be the group homomorphism which sends an element $\sigma\in G$ to the element $(\sigma_{K})$ of $\prod_{K}\operatorname{Gal}(K/F)$ whose $K$–th coordinate is the automorphism $\sigma|_{K}\in\operatorname{Gal}(K/F)$. Then the function $\phi$ has image equal to $\Gamma$ and in fact is a homeomorphism between $G$ and $\Gamma$. Since $\Gamma$ 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

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

defined for all intermediate field extensions $F\subset K\subset L$, 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 $K/F$ is normal if and only if $\operatorname{Gal}(L/K)$ is a normal subgroup of $G$, and in this case the restriction map

 $G\longrightarrow\operatorname{Gal}(K/F)$

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

###### 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 $\operatorname{Gal}(L/K)$ is an open subgroup of $G$.

Title infinite Galois theory InfiniteGaloisTheory 2013-03-22 12:39:06 2013-03-22 12:39:06 djao (24) djao (24) 7 djao (24) Topic msc 12F10 msc 13B05 FundamentalTheoremOfGaloisTheory GaloisGroup InverseLimit Krull topology