Fundamental theorem of Galois theory


In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.
In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.

Explicit description of the correspondence

For finite extensions, the correspondence can be described explicitly as follows.
The fundamental theorem says that this correspondence is a one-to-one correspondence if E/F is a Galois extension.
For example, the topmost field E corresponds to the trivial subgroup of Gal, and the base field F corresponds to the whole group Gal.
The notation Gal is only used for Galois extensions. If E/F is Galois, then Gal = Aut. If E/F is not Galois, then the "correspondence" gives only an injective map from to, and a surjective map in the reverse direction. In particular, if E/F is not Galois, then F is not the fixed field of any subgroup of Aut.

Properties of the correspondence

The correspondence has the following useful properties.
Consider the field
Since is first determined by adjoining, then, each element of can be written as:
Its Galois group can be determined by examining the automorphisms of which fix. Each such automorphism must send to either or, and must send to either or since the permutations in a Galois group can only permute the roots of an irreducible polynomial. Suppose that exchanges and, so
and exchanges and, so
These are clearly automorphisms of. There is also the identity automorphism which does not change anything, and the composition of and which changes the signs on both radicals:
Therefore,
and is isomorphic to the Klein four-group. It has five subgroups, each of which correspond via the theorem to a subfield of.
The following is the simplest case where the Galois group is not abelian.
Consider the splitting field K of the polynomial over that is, where θ is a cube root of 2, and ω is a cube root of 1. For example, if we imagine K to be inside the field of complex numbers, we may take θ to be the real cube root of 2, and ω to be
It can be shown that the Galois group has six elements, and is isomorphic to the group of permutations of three objects. It is generated by two automorphisms, say f and g, which are determined by their effect on θ and ω,
and then
The subgroups of G and corresponding subfields are as follows:
Let be the field of rational functions in and let
which is a group under composition, isomorphic to .
Let be the fixed field of, then.
If is a subgroup of then the coefficients of the following polynomial
generate the fixed field of. Galois correspondence means that every subfield of can be constructed this way. For example, if then the fixed field is and if then the fixed field is. Likewise, one can write, the fixed field of, as where is the -invariant.
Similar examples can be constructed for each of the symmetry groups of the platonic solids as these also have faithful actions on the projective line and hence on.

Applications

The theorem classifies the intermediate fields of E/F in terms of group theory. This translation between intermediate fields and subgroups is key
to showing that the general quintic equation is not solvable by radicals. One first determines the Galois groups of radical extensions, and then uses the fundamental theorem to show that solvable extensions correspond to solvable groups.
Theories such as Kummer theory and class field theory are predicated on the fundamental theorem.

Infinite case

Given an infinite algebraic extension we can still define it to be Galois if it is normal and separable. The problem that one encounters in the infinite case is that the bijection in the fundamental theorem does not hold as we get too many subgroups generally. More precisely if we just take every subgroup we can in general find two different subgroups that fix the same intermediate field. Therefore we amend this by introducing a topology on the Galois group.
Let be a Galois extension and let be the Galois group of the extension. Let be the set of the Galois groups of all finite intermediate Galois extension. Note that for all we can define the maps by. We then define the Krull Topology on to be weakest topology such that for all the maps are continuous, where we endow each with the discrete topology. Stated differently as an inverse limit of topological groups. This makes a profinite group. Note that when is finite, the Krull topology is the discrete topology.
Now that we have defined a topology on the Galois group we can restate the fundamental theorem for infinite Galois extension.
Let denote the set of all finite intermediate field extension of and let denote the set of all closed subgroups of endowed with the Krull topology. Then there exists a bijection between and given by the mapdefined by and the map defined by. One important thing one need to check is that is a well-defined map, that is that is a closed subgroup of for all intermediate. For a proof see for example