Fixed Fields and Galois Extensions

Field Extension Automorphisms

Galois Group

Suppose is a finite field extension. Then the Galois group is the group of all -automorphisms of , where an -automorphism is a field extension isomorphism .

Galois Extension

A finite extension is a Galois extension if the order of its Galois group is equal to the degree of the extension .

e.g. The field of complex numbers is a Galois extension of the field of real numbers , with Galois group being a cyclic group of order two, generated by the complex conjugation.

Lemma

Let be a finite field extension. If a polynomial is irreducible, then acts transitively on its roots.

Fixed Fields

Fixed Field Theorem

Let be a finite group of automorphisms of a field , and let be its fixed field. Then is a finite extension of , and its degree , which is the order of .

e.g.

Lüroth's theorem

Let be a subfield of the field of rational functions that contains and is not itself. Then is isomorphic to a field of rational functions.

Galois Extensions

Intermediate Field

If is a field extension of , an intermediate field is a field such that . An intermediate field is proper if and .

Characterization of Galois Extensions

Let be a finite extension and let be its Galois group. The following are equivalent:

• is a Galois extension;
• The fixed field is equal to ;
• is the splitting field over .

Corollary

One can deduce the following properties from the above theorem:

• Every finite extension is contained in a Galois extension.
• If is a Galois extension with Galois group , and if is an intermediate field, then is also a Galois extension of , and the Galois group of over is a subgroup of .

Proof The first statement holds because every finite field extension is contained in a splitting field, which is a Galois extension. For the second statement, if is a Galois extension with Galois group , then is a splitting field over and . Suppose is an intermediate field, then must also be a splitting field over , thus is also Galois. Let its Galois group be , then . So , which means every automorphism in must fix every element in , thus , and furthermore, .

The Main Theorem

Main Theorem of Galois Theory

Let be a Galois extension. Let be its Galois group. There is a one-to-one correspondence between the subgroups of and the intermediate fields such that . The correspondence is given by where is the fixed field of , or conversely , sending an intermediate field to its Galois group.

Theorem

Let be a Galois extension with Galois group , and let be the fixed field of a subgroup . The extension is a Galois extension if and only if is a normal subgroup of . In this case, the Galois group is isomorphic to the quotient group .