PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: No information on entry rating
infinite Galois theory (Topic)

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

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'$ is another element of $ G$, and the restrictions $ \sigma\vert _K$ and $ \sigma'\vert _K$ are equal, then $ \sigma' \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}(K/F)$ for finite extensions $ K/F$ form a neighborhood basis for $ \operatorname{Gal}(L/F)$ at the identity.

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

$\displaystyle \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'}\vert _K = \sigma_K$ for all $ K,K' \in \mathcal{A}$ with $ K \subset K'$. 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

$\displaystyle \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\vert _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.

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
$\displaystyle 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
$\displaystyle 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
$\displaystyle 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$.



"infinite Galois theory" is owned by djao.
(view preamble)

View style:

See Also: fundamental theorem of Galois theory, Galois group, inverse limit

Also defines:  Krull topology
Log in to rate this entry.
(view current ratings)

Cross-references: index, open subgroup, kernel, normal subgroup, inverse, bijection, field extensions, fixed field, closed, homeomorphism, image, function, coordinate, group homomorphism, compact, closed subset, product topology, induced, subspace topology, finite set, discrete topology, inverse limit, relation, inclusion, partially ordered set, contained, normal extensions, profinite group, finite groups, inverse system, projective limit, section, identity, basis, neighborhood, finite extensions, subgroups, topological group, collection, restrictions, degree, open, subset, Galois correspondence, topology, infinite dimensional, extension, composition, group operation, identity map, automorphisms, field, group, Galois group, finite dimensional, Galois extension
There are 5 references to this entry.

This is version 3 of infinite Galois theory, born on 2002-05-18, modified 2007-09-20.
Object id is 2918, canonical name is InfiniteGaloisTheory.
Accessed 7470 times total.

Classification:
AMS MSC: 12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
 13B05 (Commutative rings and algebras :: Ring extensions and related topics :: Galois theory)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
References by fourmanifold on 2008-02-24 04:32:19
Ca you please add some references?
[ reply | up ]
Interact
post | correct | update request | add example | add (any)