You are here
Homeinfinite Galois theory
Primary tabs
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

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 $G$.
Mathematics Subject Classification
12F10 no label found13B05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new problem: Geometry by parag
Aug 24
new question: Scheduling Algorithm by ncovella
new question: Scheduling Algorithm by ncovella
Comments
References
Ca you please add some references?
An open invitation
Riemann Hypothesis:
My gut feeling with regard to RH is that it is true.I may be wrong but to prove it it may need an OR approach i.e. only a team consisting of a)analytical number theorists b)algebraists c)topologists d) geometers e) programmers and, perhaps a physicist
jointly attacking the problem can crack it.
Perhaps PM can take the initiative in forming such a team.
A.K.Devaraj