Gauss's Lemma on Unique Factorization Domains

Theorem
Let $R$ be a unique factorization domain.

Then the Ring of Polynomial Functions $R \left[{X}\right]$ is a unique factorization domain.

Proof
Since a UFD is Noetherian, and a Noetherian Domain is UFD if every irreducible element is prime, it is sufficient to prove that every irreducible element of $R[X]$ is prime.

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