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

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