Let be a UFD and write for its fraction field; write for the canonical inclusion of rings. My goal is to show that any polynomials that are relatively prime in continue to be relatively prime polynomials as elements of .
Take two non-zero, non-unit polynomials and in . Because we are working in a UFD, they both admit a unique prime factorization: where each and are irreducible polynomials. Suppose that and are not relatively prime in ; we will show that in this case and are also not relatively prime in .
Notice that if all ’s were constants, then would be invertible, contrary to our hypothesis that and are not relatively prime; the same argument shows that at least one of the ’s is not a constant. Since for our purposes it suffices to exhibit a common irreducible factor, we can, without loss of generality, suppose that none of the ’s and ’s are constant polynomials.
By Gauss’ Lemma on polynomials, all of the ’s and ’s are irreducible polynomials in . Let be an irreducible factor of and . We must have and for some indices and . Because is an injective function, this yields . Hence and share an irreducible factor, so they are not relatively prime.
EDIT In fact, this does not yield , but only that divides . This is still sufficient for the proof to conclude.