Fixed field

Let $E$ be a field , and let $G$ be a set (typically a subgroup) of field automorphisms of $E$.

Definition (fixed field). The fixed field of $G$ is

Then $E^G$ is a subfield of $E$. Moreover, when $G$ contains the identity (as it does for any subgroup), $E^G\subseteq E$ is an intermediate field for the extension $E/E^G$.

Fixed fields are central to Galois theory: if $E/F$ is a Galois extension , then the Galois group $\mathrm{Gal}(E/F)$ has fixed field exactly $F$, and subgroups $H\le \mathrm{Gal}(E/F)$ correspond to intermediate fields via the Galois correspondence . For finite groups of automorphisms, Artin’s theorem on fixed fields describes $[E:E^G]$ and identifies $\mathrm{Gal}(E/E^G)$ with $G$.

Examples.

1. Let $E=\mathbb{Q}(\sqrt2)$ and $G=\mathrm{Gal}(E/\mathbb{Q})=\{1,\sigma\}$ with $\sigma(\sqrt2)=-\sqrt2$. Then $E^G=\mathbb{Q}$. At the other extreme, if $G=\{1\}$ then $E^G=E$.
2. Let $E=\mathbb{C}$ and let $G=\{1,\text{conj}\}$, where $\text{conj}(a+bi)=a-bi$. Then $E^G=\mathbb{R}$.
3. Let $E=\mathbb{F}_{p^n}$ and let $\mathrm{Frob}(x)=x^p$. If $G=\langle \mathrm{Frob}^d\rangle$ (so $d\mid n$), then the fixed field is $E^G=\mathbb{F}_{p^d}$.