Artin's theorem on fixed fields

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

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

1. $L/K$ is a finite Galois extension ;
2. the restriction map identifies $G$ with the full Galois group : $\mathrm{Gal}(L/K)=G;$
3. in particular, $[L:K]=|G|.$ This is a foundational input to the fundamental theorem of Galois theory and explains why fixed fields and automorphism groups match so tightly.

Examples

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

2. $L=\mathbb{Q}(\sqrt2,\sqrt3)$, 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 $[L:\mathbb{Q}]=4$.

3. $L=\mathbb{F}_{p^n}$ 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 $n$.