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Euclidean domainsEuclidean domains|
∀a,b (a,b ≠ 0, b ≠ ⇒ d(a) ≤ d(ab)) and |
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∀a,b (a ≠ 0 and b ≠ 0 ⇒ d(a) ≤ d(ab)) and |
A Euclidean domain is an integral domain (D, + ,−,0,·,1) together with a function d : D \ {0} → N such that
∀a,b (a ≠ 0 and b ≠ 0 ⇒ d(a) ≤ d(ab)) and
∀a,b ∃q,r (a = b·q + r and (r = 0 or d(r) < d(b))).
(Z, + ,−,0,·,1,d), the ring of integers with addition, subtraction, zero, and multiplication is a Euclidean domain with d(a) = |a|.