Artin's Theorem on Fixed Fields

Theorem (Artin).
Let $L$ be a field and $G$ a finite subgroup of field automorphisms of $L$. Let

be the fixed field . Then:

1. $L/K$ is a finite Galois extension ,
2. $\mathrm{Gal}(L/K)=G$, and
3. $[L:K]=|G|$ (compare degree equals group order ).

A standard tool in the proof is the Dedekind independence lemma .

Examples.

1. $L=\mathbb{C}$, $G=\{1,\text{complex conjugation}\}$. Then $L^G=\mathbb{R}$ and $[\mathbb{C}:\mathbb{R}]=2=|G|$.
2. $L=\mathbb{F}_{p^n}$, $G=\langle x\mapsto x^p\rangle$ has order $n$. Then $L^G=\mathbb{F}_p$.
3. Cyclotomic example: $L=\mathbb{Q}(\zeta_m)$ with automorphisms $\zeta_m\mapsto \zeta_m^a$ ($(a,m)=1$); fixed fields correspond to subgroups of $(\mathbb{Z}/m\mathbb{Z})^\times$ via FTGT .

Related. Galois groups , finite-field Galois extensions .