Gauss's Lemma on Primitive Rational Polynomials

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

Let $\Q \left[{X}\right]$ be the ring of polynomials over $\Q$ in one indeterminate $X$.

Let $f \left({X}\right), g \left({X}\right) \in \Q \left[{X}\right]$ be primitive polynomials.

Then their product $fg$ is also a primitive polynomial.

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

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

Then $fg$ is primitive.

Stronger results

 * Content of Polynomials is Multiplicative