Field automorphism

Let $L$ be a field . A field automorphism of $L$ is a bijective field homomorphism $\sigma:L\to L$. The set $\mathrm{Aut}(L)$ of all field automorphisms is a group under composition (an instance of automorphism group ).

If $L/K$ is a field extension , a $K$-automorphism of $L$ is a field automorphism $\sigma\in\mathrm{Aut}(L)$ such that $\sigma|_K=\mathrm{id}_K$. The group of all $K$-automorphisms is denoted $\mathrm{Aut}_K(L)$, and when $L/K$ is Galois it is the Galois group $\mathrm{Gal}(L/K)$.

Automorphisms are the “symmetries” that permute conjugates, and they control invariants such as fixed fields , trace , and norm .

Examples

1. Complex conjugation. The map $\sigma:\mathbb{C}\to\mathbb{C}$, $\sigma(a+bi)=a-bi$, is a field automorphism. It is a $\mathbb{R}$-automorphism of $\mathbb{C}/\mathbb{R}$.

2. Quadratic extension. For $L=\mathbb{Q}(\sqrt{d})$, the map $\sigma(a+b\sqrt{d})=a-b\sqrt{d}$ is a nontrivial $\mathbb{Q}$-automorphism. Thus $\mathrm{Aut}_\mathbb{Q}(L)\cong C_2$.

3. Finite fields and Frobenius. If $L=\mathbb{F}_{p^n}$ is a finite field , then the Frobenius map $x\mapsto x^p$ is an automorphism of $L$, and its powers generate $\mathrm{Gal}(L/\mathbb{F}_p)$ (see finite-field Galois groups are cyclic ).