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Over a UFD, content(fg) is associate to content(f)content(g) for polynomials.
Content formula: Let be a UFD and let . Then the content satisfies
i.e., is associate to . In particular, the product of primitive polynomials is primitive.
Here content is computed in the polynomial ring over a UFD. The formula implies that the product of two primitive polynomials is primitive and is a standard ingredient in Gauss's content lemma.
Let be a commutative ring with , and let . The content of is the ideal
i.e. the ideal generated by the coefficients of .
Let be a commutative ring with . The polynomial ring consists of finite sums with coefficients , with addition termwise and multiplication determined by distributivity and .
A unique factorization domain (UFD) is an integral domain such that:
Let be an integral domain and let be nonzero. The polynomial is primitive if its content is the whole ring, i.e. (equivalently, the coefficients generate the unit ideal).
Gauss content lemma: Let be a UFD. For in the polynomial ring, let denote the content of (a gcd of its coefficients, defined up to units). Then
where denotes equality up to associates. Equivalently, the product of two primitive polynomials is primitive.