# Definition:Galois Extension

## Finite Galois Extension

Let $L/K$ be a finite field extension.

### Definition 1

$L/K$ is a Galois extension if and only if the fixed field of its automorphism group is $K$:

$\operatorname{Fix}_L(\operatorname{Gal}(L/K)) = K$

### Definition 2

$L/K$ is a Galois extension if and only if it is normal and separable.

### Definition 3

$L / K$ is a Galois extension if and only if the order of the automorphism group $\operatorname{Aut} \left({L / K}\right)$ equals the degree $\left[{L : K}\right]$:

$\left\vert{\operatorname{Aut} \left({L / K}\right)}\right\vert = \left[{L : K}\right]$

## Arbitrary Galois Extension

Let $L/K$ be a field extension.

$L/K$ is a Galois extension if and only if it is normal and separable.