Automorphism group of a field extension (reviewed)

If $K/F$ is an extension of fields, its automorphism group is $$ \textrm{Aut}(K/F) = \{\sigma:K\to K\mid \forall a\in F, \sigma(a)=a, \text{ and } \sigma \text{ is an isomorphism}\}.$$ Note, a finite extension is Galois if and only if $|\textrm{Aut}(K/F)| = [K:F]$.