- Show that if \(K/F\) is the splitting field of a separable polynomial then it is Galois.
Solution
Let \(K/F\) be the splitting field of a separable polynomial \(f\in F[x]\).
1. The extension \(K/F\) is finite and normal.
A splitting field of a polynomial of finite degree is finite over the base. By definition, \(K\) is generated by all roots of \(f\), and every \(F\)-embedding of \(K\) into an algebraic closure permutes those roots. Hence every such embedding sends \(K\) onto itself, so \(K/F\) is normal.
2. The extension \(K/F\) is separable.
Because \(f\) is separable, each of its roots is separable over \(F\). The field \(K\) is generated over \(F\) by those separable elements, so \(K/F\) is separable.
3. Therefore \(K/F\) is Galois.
A finite extension is Galois iff it is both normal and separable. Apply 1 and 2.