Prove: Let $Gal(f)$ acts transitively on $Z(f)$ if and only if $f$ is irreducible in $F[x]$

Question

Can someone provide a proof for this, please? Particularly for the backward direction.

Let $F$ be a field. Let $f(x)$ be a separable polynomial in $F[x]$. Let $K/F$ be the splitting field of $f(x)$. Let $Z(f)$ be the set of roots of $f(x)$ in K. Then $G$ acts transitively on $S$ if and only if $f(x)$ is irreducible in $F[x]$.

Thanks

Answer 1

See theorem 4.10(b) here. The link is to an expository paper by Keith Conrad. He has many others here, including some very nice ones on Galois theory, which you may find useful.