Irreducibility by degree-preserving reduction modulo a prime

Let $f\in \mathbb{Z}[X]$ be primitive polynomial and let $p$ be a prime number not dividing its leading coefficient. If its coefficientwise reduction is irreducible in ${\mathbb{F}}_p[X]$, then $f$ is irreducible in $\mathbb{Q}[X]$. Indeed, a rational factorization would give an integer factorization by Gauss's lemma. The leading coefficients of both factors remain nonzero modulo $p$, so reduction preserves their positive degrees, contradicting irreducibility of the reduction. For monic polynomials the leading-coefficient condition is automatic.