Polynomials in Integers is Unique Factorization Domain

Theorem
Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$.

Then $\Z \sqbrk X$ is a unique factorization domain.

Proof
We have that Integers form Unique Factorization Domain.

The result follows from Gauss's Lemma on Unique Factorization Domains.