Category:Finite Galois Extensions

This category contains results about Finite Galois Extensions.
Definitions specific to this category can be found in Definitions/Finite Galois Extensions.

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

Definition 1

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

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

Definition 2

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

Definition 3

$L / K$ is a (finite) Galois extension if and only if the order of the automorphism group $\Aut {L / K}$ equals the degree $\index L K$:

$\order {\Aut {L / K} } = \index L K$