Field automorphism

Definition. A field automorphism of a field $L$ is a bijective homomorphism $\sigma:L\to L$ (preserving $1$). The set $\mathrm{Aut}(L)$ of all field automorphisms is a group under composition (compare automorphism groups ).

Given a subfield $K\subseteq L$, the automorphisms that fix $K$ pointwise form the Galois group $\mathrm{Gal}(L/K)$.

See also. bijective functions , field embeddings .

Examples.

1. Complex conjugation $a+bi\mapsto a-bi$ is a nontrivial automorphism of $\mathbb{C}$ fixing $\mathbb{R}$.
2. In $\mathbb{Q}(\sqrt2)$, the map $\sqrt2\mapsto -\sqrt2$ extends uniquely to a field automorphism fixing $\mathbb{Q}$.
3. On $\mathbb{F}_{p^n}$, the Frobenius map $x\mapsto x^p$ is an automorphism.