Artin's theorem on fixed fields - Imported Knowlpedia Content

Let $L$ be a field and let $G$ be a finite subgroup of the group of field automorphisms of $L$ . Define the fixed field

L^G=\{x\in L:\sigma(x)=x\ \text{for all}\ \sigma\in G\}.

Theorem (Artin). If $G\le \mathrm{Aut}(L)$ is finite and $K=L^G$ , then:

$L/K$ is a finite Galois extension;
the restriction map identifies $G$ with the full Galois group: $\mathrm{Gal}(L/K)=G;$
in particular, $[L:K]=|G|.$

Remarks

This is a foundational input to the fundamental theorem of Galois theory and explains why fixed fields and automorphism groups match so tightly.

Examples

$L=\mathbb{C}$ , $G=\{1,\mathrm{conj}\}$ where $\mathrm{conj}(z)=\overline{z}$ . Then $L^G=\mathbb{R}$ , so $\mathbb{C}/\mathbb{R}$ is Galois with $[\mathbb{C}:\mathbb{R}]=2=|G|$ .

$L=\mathbb{Q}(\sqrt2,\sqrt3)$ , and let $G$ be the subgroup of $\mathrm{Aut}(L)$ generated by $\sqrt2\mapsto-\sqrt2$ and $\sqrt3\mapsto-\sqrt3$ . Then $G$ has order $4$ , its fixed field is $\mathbb{Q}$ , and Artin’s theorem gives $[L:\mathbb{Q}]=4$ .

$L=\mathbb{F}_{p^n}$ and let $G=\langle \varphi\rangle$ where $\varphi(x)=x^p$ is Frobenius (see Frobenius endomorphism). Then $|G|=n$ , $L^G=\mathbb{F}_p$ , and Artin’s theorem recovers that $\mathbb{F}_{p^n}/\mathbb{F}_p$ is Galois of degree $n$ .