I wanna show, that if $f(x)$ is a primitive polynomial we have the following is equivalent: $f(x)$ is irreducible over $\Bbb Z$ if and only if it is irreducible over $\Bbb Q$.
The first direction we can show by contraposition. We suppose $f(x)$ is a reducible polynomial over $\Bbb Q[x]$. So if we write $f=g \cdot h$ we have that $g$ or $h$ are $\in \Bbb Q^*$. And because $\Bbb Q^* \subset \Bbb Z^*$, we are done. Now for the other side I have some trouble to show it. Can someone please help me?