Let $K/F$ be a field extension with Galois group $G = \operatorname{Gal}(K/F)$ 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 $F \subset K^H \subset K$