Automorphism of a field extension ($\sigma \in {\mathrm{Aut}}_F(K)$)

An automorphism of a field extension $K/F$ is a field automorphism $\sigma :K\to K$ fixing every element of $F$. Such maps form the field automorphism group ${\mathrm{Aut}}_F(K)$ under composition. For a Finite Galois extension this is the Galois group $\mathrm{Gal}(K/F)$. Specifying the fixed base field distinguishes these maps from arbitrary field automorphisms of $K$.