Transitive subgroup and Galois group

Question

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~

Answer 1

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 $n$.