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.