# Gauss's Lemma on Unique Factorization Domains

## Theorem

Let $R$ be a unique factorization domain.

Then the ring of polynomials $R \sqbrk X$ is also 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 $R \sqbrk X$ is prime.

etc,

## Source of Name

This entry was named for Carl Friedrich Gauss.