A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedeki...

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Autores:: Ramírez V., Victor J.

Tipo de recurso:: Article of journal

Fecha de publicación:: 2016

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: spa

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spelling	Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de Colombiahttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Ramírez V., Victor J.b8375e73-1dee-496d-8594-aa47ae1d475a3002019-07-03T02:08:16Z2019-07-03T02:08:16Z2016-07-01ISSN: 2357-4100https://repositorio.unal.edu.co/handle/unal/66443http://bdigital.unal.edu.co/67471/In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.application/pdfspaUniversidad Nacional de Colombia - Sede Bogotá - Facultad de Ciencias - Departamento de Matemáticas - Sociedad Colombiana de Matemáticashttps://revistas.unal.edu.co/index.php/recolma/article/view/62206Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de MatemáticasRevista Colombiana de MatemáticasRamírez V., Victor J. (2016) A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. Revista Colombiana de Matemáticas, 50 (2). pp. 139-143. ISSN 2357-410051 Matemáticas / MathematicsUnique factorization domainprimeirreducibleA new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163Artículo de revistainfo:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1http://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/ARTORIGINAL62206-316268-1-SM.pdfapplication/pdf346316https://repositorio.unal.edu.co/bitstream/unal/66443/1/62206-316268-1-SM.pdf52aaf986312ef4bb38e2fb3aa49167c2MD51THUMBNAIL62206-316268-1-SM.pdf.jpg62206-316268-1-SM.pdf.jpgGenerated Thumbnailimage/jpeg4755https://repositorio.unal.edu.co/bitstream/unal/66443/2/62206-316268-1-SM.pdf.jpgb48402488af4863f311a9bcf2613cab8MD52unal/66443oai:repositorio.unal.edu.co:unal/664432023-05-25 23:02:43.409Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co
dc.title.spa.fl_str_mv	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
title	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
spellingShingle	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163 51 Matemáticas / Mathematics Unique factorization domain prime irreducible
title_short	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
title_full	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
title_fullStr	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
title_full_unstemmed	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
title_sort	A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163
dc.creator.fl_str_mv	Ramírez V., Victor J.
dc.contributor.author.spa.fl_str_mv	Ramírez V., Victor J.
dc.subject.ddc.spa.fl_str_mv	51 Matemáticas / Mathematics
topic	51 Matemáticas / Mathematics Unique factorization domain prime irreducible
dc.subject.proposal.spa.fl_str_mv	Unique factorization domain prime irreducible
description	In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.
publishDate	2016
dc.date.issued.spa.fl_str_mv	2016-07-01
dc.date.accessioned.spa.fl_str_mv	2019-07-03T02:08:16Z
dc.date.available.spa.fl_str_mv	2019-07-03T02:08:16Z
dc.type.spa.fl_str_mv	Artículo de revista
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dc.identifier.issn.spa.fl_str_mv	ISSN: 2357-4100
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dc.identifier.eprints.spa.fl_str_mv	http://bdigital.unal.edu.co/67471/
identifier_str_mv	ISSN: 2357-4100
url	https://repositorio.unal.edu.co/handle/unal/66443 http://bdigital.unal.edu.co/67471/
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dc.relation.spa.fl_str_mv	https://revistas.unal.edu.co/index.php/recolma/article/view/62206
dc.relation.ispartof.spa.fl_str_mv	Universidad Nacional de Colombia Revistas electrónicas UN Revista Colombiana de Matemáticas Revista Colombiana de Matemáticas
dc.relation.references.spa.fl_str_mv	Ramírez V., Victor J. (2016) A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163. Revista Colombiana de Matemáticas, 50 (2). pp. 139-143. ISSN 2357-4100
dc.rights.spa.fl_str_mv	Derechos reservados - Universidad Nacional de Colombia
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dc.rights.license.spa.fl_str_mv	Atribución-NoComercial 4.0 Internacional
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dc.rights.accessrights.spa.fl_str_mv	info:eu-repo/semantics/openAccess
rights_invalid_str_mv	Atribución-NoComercial 4.0 Internacional Derechos reservados - Universidad Nacional de Colombia http://creativecommons.org/licenses/by-nc/4.0/ http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv	openAccess
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dc.publisher.spa.fl_str_mv	Universidad Nacional de Colombia - Sede Bogotá - Facultad de Ciencias - Departamento de Matemáticas - Sociedad Colombiana de Matemáticas
institution	Universidad Nacional de Colombia
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A new proof of the Unique Factorization of Z 1 + -d 2 for d = 3, 7, 11, 19, 43, 67, 163

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