Let $K$ be a field, $f\in K[t]$ separable over $K$, $L$ the splitting field of $f$ over $K$, and $G=\Gamma(L:K)$. I need to prove that $f$ is irreducible if and only if $G$ acts transitively on the roots of $f$.
I was able to show one direction, that $f$ irreducible implies $G$ acts transitively on the roots. I'm stuck on the other direction of implication; supposing $G$ acts transitively on the roots of $f$. Any guidance or a point in the right direction would be much appreciated!