Definition:Galois Extension/Finite

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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 $K$:

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


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 $\left[{L : K}\right]$:

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


Also see