For a primitive polynomial over a UFD, irreducibility is unchanged on passing to the fraction field.Suggest a slogan
Let \(R\) be a unique factorization domain with field of fractions \(F\), and let \(p\in R[x]\) be primitive. Then \(p\) is irreducible in \(R[x]\) if and only if \(p\) is irreducible in \(F[x]\).
More precisely, if \(p=AB\) with \(A,B\in F[x]\) nonconstant, then there exist \(r,s\in F^\times\) with \(rs=1\) such that \(rA,sB\in R[x]\), so \(p=(rA)(sB)\) is a factorization of \(p\) in \(R[x]\) into nonconstant polynomials.
Every monic polynomial in \(R[x]\) is primitive, since its leading coefficient \(1\) is a unit; so a monic \(p\in\ZZ[x]\) is irreducible in \(\ZZ[x]\) if and only if it is irreducible in \(\QQ[x]\).