or unique factorization domain, n. an integral domain in which every non-zero non-unit is uniquely representable as a finite product of irreducible elements, up to permutation. Since the Gaussian integers are a Gaussian domain, and 5 = (1+ 2i)(1- 2i), 5 is reducible in the Gaussian integers; 3 is irreducible therein, and hence prime, because it is in a Euclidean domain. The domain Q[x, y] is a Gaussian domain, but not a principal ideal domain; for a quadratic number field, a Gaussian domain is a principal ideal domain.