# Transitive subgroup and Galois group

Given H is a transitive subgroup of $$S_n$$, I know that $$|H|$$ is a multiple of $$n$$. I then want to deduce that if $$f(x) ∈ F[x]$$ is irreducible and separable of degree $$n$$ and $$L$$ is a splitting field of $$f(x)$$, then $$n$$ divides $$|Gal(L|F)|$$. I know of a result that says we have the subgroup of $$S_n$$ corresponding to $$Gal (L/F)$$ is transitive iff f is irreducible over F. But I am not sure how to continue~

## 1 Answer

By definition, since $$f$$ is separable its has $$n$$ distinct roots, $$Gal(L:F)$$ acts transtively on the $$n$$-distinct roots of $$f$$ and you know that its order is divisible by $$n$$.