or valuation, n. a mapping g from an integral domain E\{0} into the non-negative integers, such that g(ab) ≥ g(a) for all a and b in E\{0}, and such that for b in E and a in E\{0} there exist q and r in E for which b = qa + r and either r = 0 or g(r) < g(a). For example, the degree of polynomials over a field is a gauge. See also Euclidean domain.