I would like to know whether a polynomial in $\mathbb Z[x]$ is a prime element if and only if it is irreducible.
Since $\mathbb Z[x]$ is an integral domain, a prime element in $\mathbb Z[x]$ is always irreducible. It remains to prove that each irreducible element is a prime. Let $p$ be an irreducible in $\mathbb Z[x]$ and assume $p|ab$. It is clear that $p$ is nonzero and non-unit. Then we have $ab=pc$ for some $c \in \mathbb Z[x]$.
How can I proceed?