Gauss’s Lemma is not a horrible combinatorial calculation

I’m not really sure why I’ve never seen this argument before, but it’s much cleaner and easier to digest than the arguments I’m used to that make Gauss’s Lemma look like it’s not part of algebra at all.

Definition. A polynomial with integer coefficients is primitive if its coefficients are relatively prime.

Lemma. (Gauss) The product of primitive polynomials is primitive.

Proof. Notice first that a polynomial is primitive iff no prime divides all its coefficients, iff it is not the zero polynomial (mod p) for any prime p.

Suppose two polynomials f and g are primitive. Then for each prime p neither f nor g is the zero element in . Since this ring is an integral domain, it follows that f g is not the zero element either. As this holds for all primes p, f g is primitive.