Definition:Galois Extension/Finite

Definition

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 $K$:

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

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 $\Aut {L / K}$ equals the degree $\index L K$:

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

Also see

• Equivalence of Definitions of Finite Galois Extension