In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
forms a group with the operation of function composition.
factors as a product of linear polynomials over the field
One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory.
then there is an isomorphism of the corresponding Galois groups: In the following examples
are the fields of complex, real, and rational numbers, respectively.
The notation F(a) indicates the field extension obtained by adjoining an element a to the field F. One of the basic propositions required for completely determining the Galois group[3] of a finite field extension is the following: Given a polynomial
Then the order of the Galois group is equal to the degree of the field extension; that is, A useful tool for determining the Galois group of a polynomial comes from Eisenstein's criterion.
is the trivial group that has a single element, namely the identity automorphism.
must preserve the ordering of the real numbers and hence must be the identity.
This example generalizes for a prime number
Using the lattice structure of Galois groups, for non-equal prime numbers
is Another useful class of examples comes from the splitting fields of cyclotomic polynomials.
, then a corollary of this is In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.
is cyclic of order n and generated by the Frobenius homomorphism.
, the Klein four-group, they determine the entire Galois group.
is isomorphic to S3, the dihedral group of order 6, and L is in fact the splitting field of
For example, the field extension has the prescribed Galois group.
with rational coefficients and exactly two non-real roots, then the Galois group of
with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is
such that their completions give a Galois field extension
of local fields, there is an induced action of the Galois group
on the set of equivalence classes of valuations such that the completions of the fields are compatible.
[8] This gives a technique for constructing Galois groups of local fields using global Galois groups.
A basic example of a field extension with an infinite group of automorphisms is
One of the most studied classes of infinite Galois group is the absolute Galois group, which is an infinite, profinite group defined as the inverse limit of all finite Galois extensions
and Another readily computable example comes from the field extension
The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.