infinite Galois theory
Let be a Galois extension, not necessarily finite dimensional.
1 Topology on the Galois group
Recall that the Galois group of is the group of all field automorphisms that restrict to the identity map on , under the group operation of composition. In the case where the extension is infinite dimensional, the group comes equipped with a natural topology, which plays a key role in the statement of the Galois correspondence.
We define a subset of to be open if, for each , there exists an intermediate field such that
-
•
The degree is finite,
-
•
If is another element of , and the restrictions and are equal, then .
The resulting collection of open sets forms a topology on , called the Krull topology, and is a topological group under the Krull topology. Another way to define the topology is to state that the subgroups for finite extensions form a neighborhood basis for at the identity.
2 Inverse limit structure
In this section we exhibit the group as a projective limit of an inverse system of finite groups. This construction shows that the Galois group is actually a profinite group.
Let denote the set of finite normal extensions of which are contained in . The set is a partially ordered set under the inclusion relation. Form the inverse limit
consisting, as usual, of the set of all such that for all with . We make into a topological space by putting the discrete topology on each finite set and giving the subspace topology induced by the product topology on . The group is a closed subset of the compact group , and is therefore compact.
Let
be the group homomorphism which sends an element to the element of whose –th coordinate is the automorphism . Then the function has image equal to and in fact is a homeomorphism between and . Since is profinite, it follows that is profinite as well.
3 The Galois correspondence
Theorem 1 (Galois correspondence for infinite extensions).
Let , , be as before. For every closed subgroup of , let denote the fixed field of . The correspondence
defined for all intermediate field extensions , is an inclusion reversing bijection between the set of all intermediate extensions and the set of all closed subgroups of . Its inverse is the correspondence
defined for all closed subgroups of . The extension is normal if and only if is a normal subgroup of , and in this case the restriction map
has kernel .
Theorem 2 (Galois correspondence for finite subextensions).
Let , , be as before.
-
•
Every open subgroup is closed and has finite index in .
-
•
If is an open subgroup, then the field extension is finite.
-
•
For every intermediate field with finite, the Galois group is an open subgroup of .
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 |