## Characterization of Galois Extensions

For a finite extension , each of the following statements is equivalent to the statement that is Galois:

1) is a normal extension and a separable extension.

2) Every irreducible polynomial in with at least one root in splits over and is separable.

3) is a splitting field of a separable polynomial with coefficients in .

4) , that is, the number of automorphisms equals the degree of the extension.

5) is the fixed field of .

## Fundamental Theorem of Galois Theory

Given a field extension that is finite and Galois, there is a one-to-one correspondence between its intermediate fields and subgroups of its Galois group.

where and is the corresponding fixed field (the set of those elements in which are fixed by every automorphism in ).

where is an intermediate field of and is the set of those automorphisms in which fix every element of .

This correspondence is a one-to-one correspondence if and only if is a Galois extension.

Examples

1) , the trivial subgroup of .

2) .