= 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.
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