Field is Galois over Fixed Field of Automorphism Group

Theorem
Let $E/F$ be a finite field extension.

Let $K = \operatorname{Fix}_E(\operatorname{Aut}(E/F))$ be the fixed field of the automorphism group of $E/F$.

Then $E/K$ is Galois.

Proof
Follows from Closed Fields in Galois Connection for Field Extension (and does not use Fundamental Theorem of Galois Theory).