Fixed field

For a group G of automorphisms of L, the subfield L^G fixed pointwise by every element of G.

Fixed field

Definition. Let $L$ be a field and let $G\le \mathrm{Aut}(L)$ be a subgroup of its field automorphism group . The fixed field of $G$ is

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

It is a subfield of $L$ . In Galois settings, fixed fields of subgroups are exactly the intermediate fields (see Galois correspondence and Artin's theorem on fixed fields ).

See also. group action (automorphisms act on field elements).

Examples.

If $G=\langle\text{conjugation}\rangle\le \mathrm{Aut}(\mathbb{C})$ , then $\mathbb{C}^G=\mathbb{R}$ .
For $L=\mathbb{F}_{p^n}$ and $G=\mathrm{Gal}(L/\mathbb{F}_p)$ , one has $L^G=\mathbb{F}_p$ .
In $L=\mathbb{Q}(\sqrt2)$ , the full Galois group $G\cong C_2$ fixes exactly $\mathbb{Q}$ : $L^G=\mathbb{Q}$ .