Gauss's theorem (UFD ⇒ polynomial ring is UFD)

Gauss’s theorem: If $R$ is a UFD , then the polynomial ring $R[x]$ is a UFD. More generally, $R[x_1,\dots,x_n]$ is a UFD for all $n\ge 1$.

Proof sketch: Let $K$ be the fraction field of $R$. Since $K$ is a field, $K[x]$ is a PID (hence a UFD). Using Gauss's lemma , factorizations and irreducibility questions for primitive polynomials lift between $R[x]$ and $K[x]$, yielding unique factorization in $R[x]$.