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.

## Comments

Sorry if this is a silly question, but could you explain why Q[x] being an integral domain forces rs = 1?

Look at the leading coefficient of . (Perhaps a better justification would be ‘since is an integral domain’.)