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 relation to groups. It was proved by Évariste Galois in his development of Galois theory.
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. ("Intermediate fields" are fields "K" satisfying "F" ⊆ "K" ⊆ "E"; they are also called "subextensions" of "E"/"F".)
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 (and only if) "E"/"F" is a Galois extension.
For example, the topmost field "E" corresponds to the trivial subgroup of Gal("E"/"F"), and the base field "F" corresponds to the whole group Gal("E"/"F").
The notation Gal("E"/"F") is only used for Galois extensions. If "E"/"F" is Galois, then Gal("E"/"F") = Aut("E"/"F"). If "E"/"F" is not Galois, then the "correspondence" gives only an injective (but not surjective) map from formula_1 to formula_2, and a surjective (but not injective) 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("E"/"F").
Properties of the correspondence.
The correspondence has the following useful properties.
Example 1.
Consider the field
Since is constructed from the base field formula_4 by adjoining , then , each element of can be written as:
Its Galois group formula_6 comprises the automorphisms of which fix . Such automorphisms must send to or , and send to or , since they permute the roots of any irreducible polynomial. Suppose that exchanges and , so
and exchanges and , so
These are clearly automorphisms of , respecting its addition and multiplication. There is also the identity automorphism which fixes each element, and the composition of and which changes the signs on "both" radicals:
Since the order of the Galois group is equal to the degree of the field extension, formula_10, there can be no further automorphisms:
which is isomorphic to the Klein four-group. Its five subgroups correspond to the fields intermediate between the base formula_4 and the extension . 
Example 2.
The following is the simplest case where the Galois group is not abelian.
Consider the splitting field "K" of the irreducible polynomial formula_17 over formula_18; that is, formula_19 where "θ" is a cube root of 2, and "ω" is a cube root of 1 (but not 1 itself). If we consider "K" inside the complex numbers, we may take formula_20, the real cube root of 2, and formula_21 Since "ω" has minimal polynomial formula_22"," the extension formula_23 has degree:formula_24, with formula_18-basis formula_26 as in the previous example. Therefore the Galois group formula_27 has six elements, determined by permutations of the three roots of formula_17:formula_29Since there are only 3! = 6 such permutations, "G" must be isomorphic to the symmetric group of all permutations of three objects. The group can be generated by two automorphisms "f" and "g" defined by:
and formula_32, obeying the relations formula_33. Their effect as permutations of formula_34 is (in cycle notation): formula_35. Also, "g" can considered as the complex conjugation mapping.
The subgroups of "G" and corresponding subfields are as follows:
Example 3.
Let formula_49 be the field of rational functions in the indeterminate "λ," and consider the group of automorphisms:
here we denote an automorphism formula_51 by its value formula_52, so that formula_53. This group is isomorphic to formula_54 (see: six cross-ratios).
Let formula_55 be the fixed field of formula_56, so that formula_57.
If formula_58 is a subgroup of formula_56, then the coefficients of the polynomial
generate the fixed field of formula_58. The Galois correspondence implies that every subfield of formula_62 can be constructed this way. For example, for formula_63, the fixed field is formula_64 and if formula_65 then the fixed field is formula_66. The fixed field of formula_56 is the base field formula_68 where is the -invariant written in terms of the modular lambda function:formula_69Similar examples can be constructed for each of the symmetry groups of the platonic solids as these also have faithful actions on the projective line formula_70 and hence on formula_71.
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 (see Abel–Ruffini theorem). One first determines the Galois groups of radical extensions (extensions of the form "F"(α) where α is an "n"-th root of some element of "F"), 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 formula_72 be a Galois extension (possible infinite) and let formula_73 be the Galois group of the extension. Let formula_74be the set of the Galois groups of all finite intermediate Galois extension. Note that for all formula_75 we can define the maps formula_76 by formula_77. We then define the "Krull Topology" on formula_56 to be weakest topology such that for all formula_75 the maps formula_76 are continuous, where we endow each formula_81 with the discrete topology. Stated differently formula_82 as an inverse limit of topological groups (where again each formula_81 is endowed with the discrete topology). This makes formula_56 a profinite group (in fact every profinite group can be realised as the Galois group of a Galois extension, see for example ). Note that when formula_62 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 formula_86 denote the set of all finite intermediate field extension of formula_62 and let formula_88 denote the set of all closed subgroups of formula_73 endowed with the Krull topology. Then there exists a bijection between formula_86 and formula_88 given by the mapformula_92defined by formula_93 and the map formula_94defined by formula_95. One important thing one need to check is that formula_96 is a well-defined map, that is that formula_97 is a closed subgroup of formula_56 for all intermediate. For a proof see for example 