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Journal ArticleDOI

The Euclidean algorithm

01 Dec 1949-Bulletin of the American Mathematical Society (American Mathematical Society)-Vol. 55, Iss: 12, pp 1142-1146
TL;DR: In this paper, a constructive criterion for the existence of Euclidean algorithms within a given integral domain is derived, and from among the different possible algorithms in an integral domain one is singled out, and the same criterion is applied to some special rings, in particular rings of quadratic integers.
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.

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Citations
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Journal ArticleDOI
Pieter Moree1
01 Dec 2012-Integers
TL;DR: A survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927) and the contributions in the survey on `elliptic Artin' are due to Alina Cojocaru.
Abstract: One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean

139 citations


Cites background or methods from "The Euclidean algorithm"

  • ...Motzkin [363] proved that the condition (21) is also necessary if R is to be Euclidean....

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  • ...Motzkin [363] constructed the so-called minimal Euclidean algorithm....

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Book
01 Jan 2002
TL;DR: Jungnickel et al. as discussed by the authors presented a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings.
Abstract: Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory. Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

115 citations


Cites background from "The Euclidean algorithm"

  • ...’s which are not Euclidean domains ([58]); expecially in Number Theory (see, for example, [48], [59] or [70]), one can find quadratic fields that are not Euclidean domains....

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Journal ArticleDOI
TL;DR: In this paper, the authors give the following definition of Euclidean rings: all rings are commutative with unit, all modules are unitary, and given a ring A, its multiplicative group of units (i.e. invertible elements) is denoted by A*.

110 citations

01 Jan 2004
TL;DR: The authors survey what is known about Euclidean number fields from a number theoretical (and number geometrical) point of view and put some emphasis on the open problems in this field.
Abstract: This article, which is an update of a version published 1995 in Expo. Math., intends to survey what is known about Euclidean number fields; we will do this from a number theoretical (and number geometrical) point of view. We have also tried to put some emphasis on the open problems in this field.

79 citations


Cites background or result from "The Euclidean algorithm"

  • ...Motzkin Sets....

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  • ...The following observation is due to Motzkin [147]:...

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  • ...Motzkin Sets 4 2.4. k-stage Euclidean Rings 5 2.5....

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  • ...The following observation is due to Motzkin [147]: Proposition 2.2....

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  • ...For any integral domain R, define the Motzkin sets Ek, k ≥ 0, by E0 = {0}, E1 = {0} ∪R∗, the unit group of R and, generally, Ek = {0} ∪ {α ∈ R : each residue class mod α contains a β ∈ Ek−1}, E∞ = ⋃ k≥0 Ek The Motzkin sets of R = Z are easily computed: E0 = {0}, E1 = {0,±1}, E2 = {0,±1,±2,±3}, . . . , Ek = {0,±1 . . . ,±(2k − 1)}....

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Journal ArticleDOI
TL;DR: It is argued that no such equivalence relation exists between algorithms and programs with respect to a suitable equivalences relation.
Abstract: People usually regard algorithms as more abstract than the programs that implement them. The natural way to formalize this idea is that algorithms are equivalence classes of programs with respect to a suitable equivalence relation. We argue that no such equivalence relation exists.

59 citations

References
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Book
01 Jan 1863
TL;DR: In this article, the authors present an approach to bestimmung der Anzahl der Classen Supplemente in the quadratischen formen, den quadrarischen resten and den quadratical formen.
Abstract: Vorwort 1. Von der Theilbarkeit der Zahlen 2. Von der Congruens der Zahlen 3. Von den quadratischen Resten 4. Von den quadratischen Formen 5. Bestimmung der Anzahl der Classen Supplemente.

60 citations

Book
01 Jan 1927

39 citations


"The Euclidean algorithm" refers background or methods in this paper

  • ...Further, such a tea determines, and is determined by, a transfinite sequence Sx, O ^ X ^ J U , of subsets of Q-0 with (1) Sx 'CSx+i, (2) S\QS\-i, but S\~C\Si, i<\, if X —1 does not exist, (3) empty S^....

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  • ...It is easily seen that in every principal ideal ring the before mentioned norm 7* fulfils (1,1,2) (see, for example, [4, pp....

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  • ...Let a tea (transfinite Euclidean algorithm) be an algorithm as before but where (1) we allow \b\ to take any ordinal numbers as values; (2) we do not require \a\ è |&| for 6 dividing a....

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  • ...On the other hand, even the weakest condition with Z = l, which is (3, 3, 1), is not always fulfilled in principal ideal rings, as we have shown; and (3, 3, 1) is equivalent to (3, 2,1), (2 ,3, 1) is equivalent to (2, 2,1), (1, 3, l ) , and (1,2,1), and finally (2, 1, 1) to (1, 1, 1), while it remains open whether these three sets of conditions are really of different strength....

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Journal ArticleDOI
TL;DR: For m positive, the Euclidean algorithm has been shown to exist for the following values of m : for m = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 11.2 as mentioned in this paper.
Abstract: 2. Previous results. In order that a field be Euclidean the class number must be 1. However, this condition is not sufficient for, as Dedekind pointed out [l j 1 , the field R( —19) has class number 1 but is not Euclidean. L. E. Dickson [2] showed that for m negative the Euclidean algorithm exists only if m = — 1, —2, —3, —7, and —11. For m positive, the algorithm has been shown to exist for the following values of m :

8 citations


"The Euclidean algorithm" refers background or methods in this paper

  • ...Further, such a tea determines, and is determined by, a transfinite sequence Sx, O ^ X ^ J U , of subsets of Q-0 with (1) Sx 'CSx+i, (2) S\QS\-i, but S\~C\Si, i<\, if X —1 does not exist, (3) empty S^....

    [...]

  • ...It is easily seen that in every principal ideal ring the before mentioned norm 7* fulfils (1,1,2) (see, for example, [4, pp....

    [...]

  • ...Let a tea (transfinite Euclidean algorithm) be an algorithm as before but where (1) we allow \b\ to take any ordinal numbers as values; (2) we do not require \a\ è |&| for 6 dividing a....

    [...]

  • ...On the other hand, even the weakest condition with Z = l, which is (3, 3, 1), is not always fulfilled in principal ideal rings, as we have shown; and (3, 3, 1) is equivalent to (3, 2,1), (2 ,3, 1) is equivalent to (2, 2,1), (1, 3, l ) , and (1,2,1), and finally (2, 1, 1) to (1, 1, 1), while it remains open whether these three sets of conditions are really of different strength....

    [...]