fixed field

Let $K/F$ be a field extension with Galois group $G=\mathrm{Gal}(K/F)$, and let $H$ be a subgroup of $G$. The fixed field of $H$ in $K$ is the set

$$K^H:=\{x\in K\mid \sigma (x)=x\text{ for all }\sigma \in H\}.$$

The set $K^H$ is always a field, and $F\subset K^H\subset K$.