The Euclidean algorithm

Abstract

In this note a constructive criterion for the existence of a Euclidean algorithm within a given integral domain is derived, and from among the different possible Euclidean algorithms in an integral domain one is singled out. The same is done for "transfinite" Euclidean algorithms. The criterion obtained is applied to some special rings, in particular rings of quadratic integers. By an example it is shown that there exist principal ideal rings with no Euclidean algorithm. Finally, different sets of axioms for the Euclidean algorithm and related notions are compared, and the possible implications for the classification of principal ideal rings, and other integral domains, indicated.The question of the relationship between different Euclidean algorithms in the same integral domain was raised (orally) by O. Zariski.1. The derived sets. Let Q be an integral domain. A subset P of Q -0 (Q except zero) shall be called a product ideal if P(Q -0)QP.For any subset S of Q, the set B of all b in Q for which there exists an a in Q such that a+bQQS is called the total derived set of S, and the intersection Bl^iS is called the derived set S'. With S also S' is a product ideal. If SiQS, then S{ QS'.A Euclidean algorithm (or process) is given by a norm |a| defined in Q -0, with positive integral (or zero) values and such that \a\ è|&| for b dividing a and that for any b in Q -0 and any a not divisible by b there exist q and r in Q satisfying a = qb+r t \r\ <\b\.Let P it i = 0, 1, 2, • • • , be the set of all b in Q with \b\ ^i. Obviously Pi is a product ideal. For any b in P{, let a be an element with a+bQQPi, whence a -bq^O and (for any r~a -bq with |H<M) \ r \=i> |&|è*+l; we see that P!QP i+1 . Conversely, given a sequence Q -0 = Po2-Pi2 • • • of product ideals with empty intersection CiPi such that Pi QPi+i, the norm defined by \b\ -i for every b in Pi -P i+ i will fulfil the conditions for a Euclidean algorithm. Hence there is a one-one correspondence between sequences of this kind and Euclidean algorithms.If for another Euclidean algorithm, with the sequence P t -, always PiQ7i, we say that the first algorithm is the faster one (under cer-

Euclidean Totally Definite Quaternion Fields Over the Rational Field and Over Quadratic Number Fields

“…The transfinite construction. Now, let us follow the approach of Motzkin [9] and Samuel [11] for the commutative case 1 . Suppose that Λ is right-Euclidean for some stathm Φ : Λ −→ W .…”

Euclidean function fields.

“…Denoting the canonical class by W and by /*(9l) the dimension of the class WC~l dual to C, the Statement of the Riemann-Roch Theorem is (7) From (6) and (7), we have (8) The following Proposition gives, in the language of adeles, a sufficient condition for a function field FjK to be euclidean. For a divisor 9l of Ff K, let δ (9l) denote its degree and let /(9l) be the dimension of the divisor class C to which 9l" 1 belongs.…”

Number Systems Pertaining to Euclidean Rings of Imaginary Quadratic Integers

“…Furthermore, only the first five examples of those are Euclidean domains, whose Euclidean functions are induced by the norms; whereas, the other four have no Euclidean functions whatsoever. A brilliant proof for the latter claim was presented by Motzkin [12] around 1949, who came up with a criterion for an integral domain to be Euclidean. But the proof seems too terse for laymen.…”

“…For applications in data processing Khmeinik [9] Such a number system is said to be unitary. Indeed, bases of unitary number systems are referred to as the universal side divisors by Motzkin [12]. However, it seems rather a unfamiliar expression (c.f.…”