Gauss's Lemma on Primitive Rational Polynomials

Rational Polynomial
Let $\Q$ be the field of rational numbers.

Let $\Q \sqbrk X$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $\map f X, \map g X \in \Q \sqbrk X$ be primitive polynomials.

Then their product $f g$ is also a primitive polynomial.

General Ring
Let $R$ be a commutative ring with unity.

Let $f, g \in R \sqbrk X$ be primitive polynomials.

Then $f g$ is primitive.

Stronger results

 * Content of Polynomial is Multiplicative