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.
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.
For finite extensions, the correspondence can be described explicitly as follows.
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 not Galois, then the "correspondence" gives only an injective (but not surjective) map from
, 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).
by adjoining √2, then √3, each element of K can be written as: Its Galois group
There is also the identity automorphism e which fixes each element, and the composition of f and g which changes the signs on both radicals: Since the order of the Galois group is equal to the degree of the field extension,
, 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
and the extension K. The following is the simplest case where the Galois group is not abelian.
Consider the splitting field K of the irreducible polynomial
has six elements, determined by all permutations of the three roots of
be the field of rational functions in the indeterminate λ, and consider the group of automorphisms: here we denote an automorphism
The Galois correspondence implies that every subfield of
where j is the j-invariant written in terms of the modular lambda function:
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
is a finite extension, but not a splitting field over
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.
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.
Therefore we amend this by introducing a topology on the Galois group.
be a Galois extension (possibly infinite) and let
{\displaystyle {\text{Int}}_{\text{F}}(E/F)=\{G_{i}={\text{Gal}}(L_{i}/F)~|~L_{i}/F{\text{ is a finite Galois extension and }}L_{i}\subseteq E\}}
Now that we have defined a topology on the Galois group we can restate the fundamental theorem for infinite Galois extensions.
denote the set of all intermediate field extensions of
