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 .

e.g.

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 followings 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, .

Normal Extension and Separability

Normal Extension

is a normal extension if every element of has its minimal polynomial over split completely in .

In other words, a normal extension never contains “only some” of the conjugates of an algebraic element. If it contains one conjugate, it contains the entire set.

Separable Extension

is separable if the minimal polynomial of any element has no multiple roots.

Proposition

Every algebraic extension of a field of characteristic zero is separable, and every algebraic extension of a finite field is separable.

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 .