Fundamental theorem of Galois theory

Let \(K/F\) be a finite Galois extension with Galois group \(G=\Gal(K/F)\). The maps \[ \begin{aligned} \theset{E\suchthat F\subseteq E\subseteq K\text{ a field}}&\longleftrightarrow\theset{H\suchthat H\le G}\\ E&\mapsto\Gal(K/E)=\theset{\tau\in G\suchthat\tau|_E=\id_E}\\ K^H&\mapsfrom H \end{aligned} \] are mutually inverse bijections, where \(K^H\) is the fixed field of \(H\). If \(E\) corresponds to \(H\), and \(E_1,E_2\) correspond to \(H_1,H_2\), then:

1. \(E_1\subseteq E_2\) if and only if \(H_2\le H_1\).

2. \(\abs H=[K:E]\) and \([E:F]=[G:H]\).

3. \(K/E\) is Galois with \(\Gal(K/E)=H\).

4. \(E/F\) is Galois if and only if \(H\) is normal in \(G\), in which case \(\Gal(E/F)\cong G/H\).

5. The intersection \(E_1\cap E_2\) corresponds to the subgroup \(\gens{H_1,H_2}\) generated by \(H_1\) and \(H_2\), and the compositum \(E_1E_2\) corresponds to \(H_1\cap H_2\).