Galois extension definition and fundamental theorem of Galois theory

The finite extension \(K/F\) is Galois if it is both normal and separable. Normality means that every irreducible polynomial in \(F[x]\) with a root in \(K\) splits into linear factors in \(K[x]\). Separability means that the minimal polynomial over \(F\) of every element of \(K\) has distinct roots in an algebraic closure of \(F\). Its Galois group is \[ \operatorname{Gal}(K/F) =\{\sigma:K\to K:\sigma\text{ is a field automorphism and } \sigma(a)=a\text{ for all }a\in F\}, \] with composition as the group operation.

Let \(K/F\) be a finite Galois extension with \(G=\operatorname{Gal}(K/F)\). Then the following statements hold [@DF04].

There is an inclusion-reversing bijection between intermediate fields \(F\subseteq E\subseteq K\) and subgroups \(H\leq G\), given by \[ E\longmapsto\operatorname{Gal}(K/E),\qquad H\longmapsto K^H =\{a\in K:\sigma(a)=a\text{ for every }\sigma\in H\}. \] The maps are inverse: \[ K^{\operatorname{Gal}(K/E)}=E,\qquad \operatorname{Gal}(K/K^H)=H. \] Thus \(F\) corresponds to \(G\), and \(K\) corresponds to the trivial subgroup. For intermediate fields \(E_1,E_2\), \[ E_1\subseteq E_2 \quad\Longleftrightarrow\quad \operatorname{Gal}(K/E_2)\subseteq\operatorname{Gal}(K/E_1). \]

For every intermediate field \(E\), the extension \(K/E\) is Galois. With \(H=\operatorname{Gal}(K/E)\), \[ [K:E]=|H|,\qquad [E:F]=[G:H],\qquad [K:F]=|G|. \]

The extension \(E/F\) is Galois if and only if \(H=\operatorname{Gal}(K/E)\) is normal in \(G\). In that case restriction to \(E\) is a surjective homomorphism \[ G\longrightarrow\operatorname{Gal}(E/F) \] with kernel \(H\), and therefore induces an isomorphism \[ G/H\cong\operatorname{Gal}(E/F). \]