Let \(L/k\) be a finite Galois extension with Galois group \(G \coloneqq \Gal(L/k)\). For a subgroup \(H\leq G\), let \(L^H\) be its fixed field, and for an intermediate field \(k\subseteq F\subseteq L\), let \[ \Gal(L/F) \coloneqq \theset{ \sigma \in G \suchthat \sigma(\alpha) = \alpha \text{ for all } \alpha \in F}. \] Then the maps \[ \theset{\text{subgroups } H\leq G} \mapstofrom \theset{\text{intermediate fields } k\subseteq F\subseteq L}, \qquad H\mapsto L^H, \qquad \Gal(L/F) \mapsfrom F, \] are mutually inverse bijections, and they reverse inclusions. If \(H\) and \(F\) correspond under these bijections, then:
\([F: k] = [G: H]\) and \([L : F] = \abs{H}\).
\(L/F\) is Galois and \(\Gal(L/F) = H\).
\(F/k\) is Galois if and only if \(H \normal G\), and in that case \(\Gal(F/k) \cong G/H\).
If \(H_1,H_2\) correspond to \(F_1,F_2\), then the compositum \(F_1 F_2\) corresponds to \(H_1 \intersect H_2\), and \(F_1 \intersect F_2\) corresponds to the subgroup generated by \(H_1\) and \(H_2\), \(\gens{H_1, H_2}\).
The group \(\Gal(L/F)\) fixes \(F\) pointwise. For \(\sigma\in G\) one has \(\sigma(L^H)=L^{\sigma H\sigma^{-1}}\), so the setwise stabilizer \(\theset{\sigma\in G \suchthat \sigma(F) = F}\) of \(F=L^H\) is the normalizer \(N_G(H)\), which contains \(H\) and equals \(G\) exactly when \(F/k\) is Galois.
The subgroup \(\gens{H_1,H_2}\) contains the product set \(H_1H_2\), and \(H_1H_2\) is a subgroup if and only if \(H_1H_2=H_2H_1\).