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.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $21$