Galois group


In mathematics, more specifically 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.

Definition

Suppose that is an extension of the field . An automorphism of is defined to be an automorphism of that fixes pointwise. In other words, an automorphism of is an isomorphism such that for each. The set of all automorphisms of forms a group with the operation of function composition. This group is sometimes denoted by
If is a Galois extension, then is called the Galois group of, and is usually denoted by.
If is not a Galois extension, then the Galois group of is sometimes defined as, where is the Galois closure of.

Galois group of a polynomial

Another definition of the Galois group comes from the Galois group of a polynomial. If there is a field such that factors as a product of linear polynomials
over the field, then the Galois group of the polynomial is defined as the Galois group of where is minimal among all such fields.

Structure of Galois groups

Fundamental theorem of Galois theory

One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension, there is a bijection between the set of subfields and the subgroups Then, is given by the set of invariants of under the action of, so
Moreover, if is a normal subgroup then. And conversely, if is a normal field extension, then the associated subgroup in is a normal group.

Lattice structure

Suppose are Galois extensions of with Galois groups The field with Galois group has an injection which is an isomorphism whenever.

Inducting

As a corollary, this can be inducted finitely many times. Given Galois extensions where then there is an isomorphism of the corresponding Galois groups:

Examples

In the following examples is a field, and are the fields of complex, real, and rational numbers, respectively. The notation indicates the field extension obtained by adjoining an element to the field.

Computational tools

Cardinality of the Galois group and the degree of the field extension

One of the basic propositions required for completely determining the Galois groups of a finite field extension is the following: Given a polynomial, let be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,

Eisenstein's criterion

A useful tool for determining the Galois group of a polynomial comes from Eisenstein's criterion. If a polynomial factors into irreducible polynomials the Galois group of can be determined using the Galois groups of each since the Galois group of contains each of the Galois groups of the

Trivial group

is the trivial group that has a single element, namely the identity automorphism.
Another example of a Galois group which is trivial is Indeed, it can be shown that any automorphism of must preserve the ordering of the real numbers and hence must be the identity.
Consider the field The group contains only the identity automorphism. This is because is not a normal extension, since the other two cube roots of, and are missing from the extension—in other words is not a splitting field.

Finite Abelian groups

The Galois group has two elements, the identity automorphism and the complex conjugation automorphism.

Quadratic extensions

The degree two field extension has the Galois group with two elements, the identity automorphism and the automorphism which exchanges and −. This example generalizes for a prime number

Product of quadratic extensions

Using the lattice structure of Galois groups, for non-equal prime numbers the Galois group of is

Cyclotomic extensions

Another useful class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials defined as
whose degree is, Euler's totient function at. Then, the splitting field over is and has automorphisms sending for relatively prime to. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group. If then
If is a prime, 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.

Finite Fields

Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If is a prime power, and if and denote the Galois fields of order and respectively, then is cyclic of order and generated by the Frobenius homomorphism.

Degree 4 examples

The field extension is an example of a degree field extension. This has two automorphisms where and Since these two generators define a group of order, the Klein four-group, they determine the entire Galois group.
Another example is given from the splitting field of the polynomial
Note because the roots of are There are automorphisms
generating a group of order. Since generates this group, the Galois group is isomorphic to.

Finite Non-Abelian groups

Consider now where is a primitive cube root of unity. The group is isomorphic to, the dihedral group of order 6, and is in fact the splitting field of over

Quaternion Group

The Quaternion group can be found as the Galois group of a field extension of. For example, the field extension
has the prescribed Galois group.

Symmetric group of prime order

If is an irreducible polynomial of prime degree with rational coefficients and exactly two non-real roots, then the Galois group of is the full symmetric group
For example, is irreducible from Eisenstein's criterion. Plotting the graph of with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is.

Infinite Groups

A basic example of a field extension with an infinite group of automorphisms, is since it contains every algebraic field extension. For example, the field extensions for a square-free element each have a unique degree automorphism, inducing an automorphism in
One of the most studied classes of examples of infinite Galois groups come from the Absolute Galois group, which are profinite groups. These are infinite groups defined as the inverse limit of Galois groups all finite Galois extensions for a fixed field. The inverse limit is denoted
where is the separable closure of a field. Note this group is a Topological group. Some basic examples include and
Another readily computable example comes from the field extension containing the square root of every positive prime. It has Galois group
which can be deduced from the profinite limit
and using the computation of the Galois groups.

Properties

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed subgroups of the Galois group correspond to the intermediate fields of the field extension.
If is a Galois extension, then can be given a topology, called the Krull topology, that makes it into a profinite group.