An integral domain in which a Division Algorithm holds. Specifically, an integral domain R with a function d:R\\{0}→ℕ such that (i) d(a) ≤ d(ab) for all a,b ∈ R\\{0}; (ii) for any a,b ∈ R\\{0} there exist q,r ∈ R such that a = qb + r and d(r)<d(b) or r = 0. Examples include ℤ with d(n) = |n|, the Gaussian integers with d(a+ib) = a² + b², and polynomials over a field with d(f) being the degree of f. Euclidean domains are principal ideal domains (see quadratic field).