 In fact, in this case, the Euclidean algorithm requires one to compute the reduced form of many fractions, and this makes the Euclidean algorithm less efficient than algorithms which work only with polynomials over the integers (see Polynomial greatest common divisor).The results of the preceding section remain valid if the ring of integers and the field of rationals are respectively replaced by any unique factorization domain and its field of fractions .This is typically used for factoring multivariate polynomials, and for proving that a polynomial ring over a unique factorization domain is also a unique factorization domain.