Gauss's lemma

Statement

Gauss's lemma states that, in a gcd domain (in particular, in a Bezout domain or a unique factorization domain), we have the following:

• A product of primitive polynomials is primitive.
• The content of a product of polynomials is the product of their contents (upto associates).

Related facts

Applications

• Unique factorization is polynomial-closed: The polynomial ring over a unique factorization domain is again a unique factorization domain. The proof of this is a direct consequence of Gauss's lemma.