Gauss lemma (content multiplicativity)

Gauss content lemma: Let $R$ be a UFD . For $f,g\in R[x]$ in the polynomial ring , let $\operatorname{cont}(f)$ denote the content of $f$ (a gcd of its coefficients, defined up to units). Then

where $\sim$ denotes equality up to associates . Equivalently, the product of two primitive polynomials is primitive.

This lemma is the technical engine behind Gauss-type transfer results between $R[x]$ and $\mathrm{Frac}(R)[x]$.