= Solution
The <Fundamental theorem of Galois theory> applies to a finite <Galois extension> $L/K$ with group $G$. It gives inclusion-reversing inverse bijections
$$
\boxed{H\leq G\ \longleftrightarrow\ L^H,\qquad K\subseteq M\subseteq L\ \longleftrightarrow\operatorname{Gal}(L/M).}
$$
The <fixed field> $L^H$ consists of elements fixed by every member of $H$. Moreover $[L:L^H]=|H|$ and $[L^H:K]=[G:H]$. An intermediate extension $M/K$ is Galois iff $H=\operatorname{Gal}(L/M)$ is normal, in which case restriction gives $\operatorname{Gal}(M/K)\cong G/H$. All intermediate $L/M$ are themselves Galois. Finiteness and normality/separability of $L/K$ are essential for this stated form.
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