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fundamental theorem of Galois theory
Let $L/F$ be a Galois extension of finite degree, with Galois group $G:=\operatorname{Gal}(L/F)$. There is a bijective, inclusionreversing correspondence between subgroups of $G$ and extensions of $F$ contained in $L$, given by

$K\mapsto\operatorname{Gal}(L/K)$, for any field $K$ with $F\subseteq K\subseteq L$.

$H\mapsto L^{H}$ (the fixed field of $H$ in $L$), for any subgroup $H\leq G$.
The extension $L^{H}/F$ is normal if and only if $H$ is a normal subgroup of $G$, and in this case the homomorphism $G\longrightarrow\operatorname{Gal}(L^{H}/F)$ given by $\sigma\mapsto\sigma_{{L^{H}}}$ induces (via the first isomorphism theorem) a natural identification $\operatorname{Gal}(L^{H}/F)=G/H$ between the Galois group of $L^{H}/F$ and the quotient group $G/H$.
For the case of Galois extensions of infinite degree, see the entry on infinite Galois theory.
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