The procedure is compatible with product: . The construct can be used to show the statement: A polynomial ring over a UFD is a UFD. Indeed, by induction, it is enough to show is a UFD when is a UFD. Let be a non-zero polynomial. Now, is a unique factorization domain (since it is a principal ideal domain) and so, as a polynomial in , can be factorized as: where are irreducible polynomials of . Now, we write for the gcd of the coefficients of (and is the primitive part) and then: Now, is a product of prime elements of (since is a UFD) and a prime element of is a prime element of , as is an integral domain. Hence, admits a prime factorization (or a unique factorization into irreducibles). Next, observe that is a unique factorization into irreducible elements of , as (1) each is irreducible by the irreducibility statement and (2) it is unique since the factorization of can also be viewed as a factorization in and factorization there is unique. Since and are uniquely determined by up to unit elements, the above factorization of is a unique factorization into irreducible elements. The condition that "R is a unique factorization domain" is not superfluous because it implies that every irreducible element of this ring is also a prime element, which in turn implies that every non-zero element of R has at most one factorization into a product of irreducible elements and a unit up to order and associate relationship. In a ring where factorization is not unique, say with p and q irreducible elements that do not divide any of the factors on the other side, the product shows the failure of the primitivity statement. For a concrete example one can take , , , , . In this example the polynomial (obtained by dividing the right hand side by ) provides an example of the failure of the irreducibility statement (it is irreducible over R, but reducible over its field of fractions ). Another well-known example is the polynomial , whose roots are the golden ratio and its conjugate showing that it is reducible over the field , although it is irreducible over the non-UFD which has as field of fractions. In the latter example the ring can be made into an UFD by taking its integral closure in (the ring of Dirichlet integers), over which