I have a question about the following result:
If $f ∈ \mathbb{Z}[X]$ is primitive and there is a prime $p$ not dividing the leading coefficient of $f$ such that $f$ is irreducible in $(\mathbb{Z}/p \mathbb{Z})[X]$ then $f(x)$ is irreducible in $\mathbb{Z}[X]$.
My question: Does this result still hold if $p$ is composite? I have looked at a few proofs of this statement and I’m not sure how the fact that $p$ is prime is being used. Any insight would be appreciated.