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 .
2 Inverse limit structure
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.
3 The Galois correspondence
Theorem 1 (Galois correspondence for infinite extensions).
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|
|Date of creation||2013-03-22 12:39:06|
|Last modified on||2013-03-22 12:39:06|
|Last modified by||djao (24)|