Author
Oscar A. Cámpoli
Bio: Oscar A. Cámpoli is an academic researcher from Valparaiso University. The author has contributed to research in topics: Euclidean distance & Principal ideal domain. The author has an hindex of 1, co-authored 1 publications receiving 16 citations.
Papers
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TL;DR: A Principal Ideal Domain that is not a Euclidean Domain this article is a principal ideal domain that does not have an ideal domain and is not an ideal space-time domain.
Abstract: (1988). A Principal Ideal Domain That Is Not a Euclidean Domain. The American Mathematical Monthly: Vol. 95, No. 9, pp. 868-871.
18 citations
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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
TL;DR: An elementary proof of this strong failure of elementary generation for SL2 over imaginary quadratic rings is given.
Abstract: Abstract It is almost always the case that the elementary matrices generate the special linear group SLn over a ring of integers in a number field. The only exceptions to this rule occur for SL2 over rings of integers in imaginary quadratic fields. The surprise is compounded by the fact that, in these cases when elementary generation fails, it actually fails rather badly: the group E2 generated by the elementary 2-by-2 matrices turns out to be an infinite-index, non-normal subgroup of SL2. We give an elementary proof of this strong failure of elementary generation for SL2 over imaginary quadratic rings.
20 citations
01 Jan 2013
TL;DR: The ring Z[ √−19 2 ] is a principal ideal domain (PID) that is not a Euclidean domain this article, and it is known to be a PID.
Abstract: The ring Z[ √−19 2 ] is usually given as a first example of a principal ideal domain (PID) that is not a Euclidean domain. This paper gives an elementary and more direct proof that Z[ √−19 2 ] is indeed a PID. Mathematics Subject Classification: 11R04, 13F07, 13F10
6 citations
TL;DR: The Principal Ideal Domains are Almost Euclidean as discussed by the authors, which is a generalization of the notion of principal ideal domains in the classical ideal domain model, has been proposed.
Abstract: (1997). Principal Ideal Domains are Almost Euclidean. The American Mathematical Monthly: Vol. 104, No. 2, pp. 154-156.
4 citations
01 Jan 2008
TL;DR: In this article, it was shown that every Euclidean ring is a principal ideal ring and that the converse is not valid under the rings R of integral algebraic numbers in quadratic complex fields.
Abstract: It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the rings R of integral algebraic numbers in quadratic complex flelds Q £p iD ⁄ , for D = 19;43;67, and 163. In conection with these counterexamples several results were published in an efiort to make them somewhat more accessible. The aim of this note is to present and complete these results.
2 citations