Past exam of the galois theory course of the University of Oxford / 2023 / 2 / a / Solution

The Galois group acts transitively on the $n$ roots because $f$ is irreducible and separable. The orbit-stabilizer theorem givesso $n$ divides $|G|$. Equivalently, the tower law gives $[K:F]=[K:F({\alpha }_1)]n$, while the finite Galois extension has group order $[K:F]$.