Polynomials in Integers is Unique Factorization Domain
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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 $21$