infinite Galois theory

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

1 Topology on the Galois group

Recall that the Galois groupMathworldPlanetmath G:=Gal(L/F) of L/F is the group of all field automorphisms σ:LL that restrict to the identity map on F, under the group operationMathworldPlanetmath of compositionMathworldPlanetmathPlanetmath. In the case where the extensionPlanetmathPlanetmathPlanetmathPlanetmath 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 correspondencePlanetmathPlanetmath.

We define a subset U of G to be open if, for each σU, there exists an intermediate field KL such that

  • The degree [K:F] is finite,

  • If σ is another element of G, and the restrictionsPlanetmathPlanetmathPlanetmath σ|K and σ|K are equal, then σU.

The resulting collectionMathworldPlanetmath of open sets forms a topologyMathworldPlanetmath on G, called the Krull topology, and G is a topological groupMathworldPlanetmath under the Krull topology. Another way to define the topology is to state that the subgroupsMathworldPlanetmathPlanetmath Gal(L/K) for finite extensionsMathworldPlanetmath K/F form a neighborhoodMathworldPlanetmathPlanetmath basis for Gal(L/F) at the identityPlanetmathPlanetmathPlanetmathPlanetmath.

2 Inverse limit structure

In this sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we exhibit the group G as a projective limitPlanetmathPlanetmath of an inverse systemMathworldPlanetmath of finite groupsMathworldPlanetmath. This construction shows that the Galois group G is actually a profinite group.

Let 𝒜 denote the set of finite normal extensionsMathworldPlanetmath K of F which are contained in L. The set 𝒜 is a partially ordered setMathworldPlanetmath under the inclusion relation. Form the inverse limit


consisting, as usual, of the set of all (σK)KGal(K/F) such that σK|K=σK for all K,K𝒜 with KK. We make Γ into a topological space by putting the discrete topology on each finite setMathworldPlanetmath Gal(K/F) and giving Γ the subspace topology induced by the product topology on KGal(K/F). The group Γ is a closed subset of the compact group KGal(K/F), and is therefore compactPlanetmathPlanetmath.



be the group homomorphismMathworldPlanetmath which sends an element σG to the element (σK) of KGal(K/F) whose K–th coordinate is the automorphismPlanetmathPlanetmathPlanetmathPlanetmath σ|KGal(K/F). Then the function ϕ has image equal to Γ and in fact is a homeomorphismPlanetmathPlanetmath between G and Γ. Since Γ 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 LH denote the fixed field of H. The correspondence


defined for all intermediate field extensions FKL, is an inclusion reversing bijection between the set of all intermediate extensions K and the set of all closed subgroups of G. Its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath is the correspondence


defined for all closed subgroups H of G. The extension K/F is normal if and only if Gal(L/K) is a normal subgroupMathworldPlanetmath of G, and in this case the restriction map


has kernel Gal(L/K).

Theorem 2 (Galois correspondence for finite subextensions).

Let G, L, F be as before.

  • Every open subgroup HG is closed and has finite index in G.

  • If HG is an open subgroup, then the field extension LH/F is finite.

  • For every intermediate field K with [K:F] finite, the Galois group Gal(L/K) is an open subgroup of G.

Title infinite Galois theory
Canonical name InfiniteGaloisTheory
Date of creation 2013-03-22 12:39:06
Last modified on 2013-03-22 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