Gauss's lemma

Gauss’s lemma: Let $R$ be a UFD with fraction field $K$. If $f\in R[x]$ is primitive , then any factorization $f=gh$ in the polynomial ring $K[x]$ can be written (after multiplying $g,h$ by nonzero scalars in $K$) as a factorization $f=g'h'$ with $g',h'\in R[x]$ primitive. In particular, a primitive $f\in R[x]$ is irreducible in $R[x]$ if and only if it is irreducible in $K[x]$.

This rests on the content multiplicativity lemma and is the key bridge between factorization over $R$ and over its field of fractions.

Proof sketch: Clear denominators to write $f=c\cdot \tilde g\tilde h$ with $\tilde g,\tilde h\in R[x]$ and $c\in R\setminus\{0\}$. Taking contents and using multiplicativity forces $c$ to be a unit when $f$ is primitive, so $f$ factors in $R[x]$. Irreducibility equivalence follows by contrapositive.