Field automorphism

A field automorphism is a bijective ring homomorphism from a field to itself. The automorphisms fixing a specified subfield form a Galois group when the extension is Galois.

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 $K$.

The field automorphism group fixing $K$ consists of all field automorphisms of $L$ fixing $K$ pointwise, with composition. It is defined without assuming that $L/K$ is Galois. For $L=K(\alpha )$ with algebraic degree $d$, the action on the roots of the minimal polynomial inside $L$ is faithful, so this group embeds in $S_d$ and is finite.