Let such that . Prove that the product of any coefficient of with any coefficient of is an integer.
Note that is in and factors in . By Gauss’ Lemma, there exist such that and . Since is an integral domain, in fact . Let and denote the coefficients of and , respectively; we have and , so that for all and .
Note that the proof still works if we replace by an arbitrary unique factorization domain and by its field of fractions.