# fixed field

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 $F\subset K^{H}\subset K$.

Title fixed field FixedField 2013-03-22 12:08:24 2013-03-22 12:08:24 djao (24) djao (24) 6 djao (24) Definition msc 12F10