Suppose that is an extension of the field (written as and read "E over F "). 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 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.
Suppose are Galois extensions of with Galois groups The field with Galois group has an injection which is an isomorphism whenever .
As a corollary, this can be inducted finitely many times. Given Galois extensions where then there is an isomorphism of the corresponding Galois groups:
In the following examples is a field, and 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.
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,
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
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 ,
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
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.
Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If q is a prime power, and if and denote the Galois fields of order and respectively, then is cyclic of order n 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 .
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 .
Comparing Galois groups of field extensions of global fields
Given a global field extension (such as ) the and an equivalence class of valuations on (such as the -adic valuation), and on such that their completions give a Galois field extension
of local fields. Then, 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. This means if then there is an induced isomorphic of local fields
Since we have taken the hypothesis that lies over (i.e. there is a Galois field extension ), the field morphism is in fact an isomorphism of -algebras. If we take the isotropy subgroup of for the valuation class
then there's a surjection of the global Galois group to the local Galois group such that there's and isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means
where the vertical arrows are isomorphisms. 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 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