Definition. automorphism [2101]

Let $F$ be a field.

1. An automorphism of $F$ is an isomorphism from $F$ to itself. We denote by $\mathrm{Aut}(F)$ the set of all automorphisms of $F$.
2. An automorphism $\sigma$ of $F$ is said to fix $\alpha \in F$ if $\sigma(\alpha) = \alpha$, and is said to fix a subset $S \subseteq F$ if $\sigma(\alpha) = \alpha$ for all $\alpha \in S$.