Show that if \(K/F\) is the splitting field of a separable polynomial, then \(K/F\) is Galois.
Solution
Let \(K\) be the splitting field over \(F\) of a separable polynomial \(f\in F[x]\).
1. The extension \(K/F\) is normal.
Every splitting field over \(F\) is normal.
2. The extension \(K/F\) is separable.
The field \(K\) is generated over \(F\) by the roots of \(f\). Since \(f\) is separable, each of these roots is separable over \(F\), and an algebraic extension generated by separable elements is separable.
Therefore \(K/F\) is normal and separable, hence Galois.