Definition:Galois Extension

From ProofWiki
Jump to navigation Jump to search

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

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


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.


Also see