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In a UFD, the content of a product equals the product of contents up to associates.
Gauss content lemma: Let be a UFD. For in the polynomial ring, choose generators of their coefficient content ideals. Then
where denotes equality up to associates. Equivalently, the product of two primitive polynomials is primitive.
This lemma is the technical engine behind Gauss-type transfer results between and .
A unique factorization domain (UFD) is an integral domain such that:
Let be a commutative ring with . The polynomial ring consists of finite sums with coefficients , with addition termwise and multiplication determined by distributivity and .
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 an integral domain. Elements are associates if there exists a unit 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).