Ring of integer-valued polynomials over rational integers is not a UFD

Statement

The ring of integer-valued polynomials over rational integers is not a unique factorization domain.

Related facts

• Every binomial polynomial is irreducible but not prime in the ring of integer-valued polynomials over rational integers

Proof

For the proof, we observe that $x,x-1,{\binom {x}{2}},2$ are all irreducible in this ring, with no two of them being associates, but:

$x(x-1)=2{\binom {x}{2}}$.

Thus, $x(x-1)$ does not have a unique factorization into irreducibles, and we obtain that the ring of integer-valued polynomials over rational integers is not a unique factorization domain.