Characterization of number fields by their integral trace form

We prove that the integral trace form is a complete invariant for totally real number fields of fundamental discriminant. We also study the relations of this invariant with the trace-zero form and the shape of a number field, and give analog results for such invariants. As a consequence, we settle a...

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Autores:
Rivera Guaca, Carlos Andrés
Tipo de recurso:
Fecha de publicación:
2018
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
spa
OAI Identifier:
oai:repositorio.unal.edu.co:unal/76670
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/76670
http://bdigital.unal.edu.co/73309/
Palabra clave:
Trace form
Totally real number fields
Shapes of number fields
Casimir invariant
Higher composition laws
Rights
openAccess
License
Atribución-NoComercial 4.0 Internacional
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oai_identifier_str oai:repositorio.unal.edu.co:unal/76670
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
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_abf2Mantilla Soler, GuillermoRodriguez Vega, John JaimeRivera Guaca, Carlos Andrésdd5dff07-c166-429d-b4db-ab13c48f22113002020-03-30T06:24:29Z2020-03-30T06:24:29Z2018-11https://repositorio.unal.edu.co/handle/unal/76670http://bdigital.unal.edu.co/73309/We prove that the integral trace form is a complete invariant for totally real number fields of fundamental discriminant. We also study the relations of this invariant with the trace-zero form and the shape of a number field, and give analog results for such invariants. As a consequence, we settle a conjecture from 2012 by Mantilla-Soler about tamely ramified quartic fields of fundamental discriminant. Our method of proof is based on what we call Casimir elements and Casimir pairings, new tools we introduce in this work, which are related to (and generalize) the Casimir elements from the representation theory of Lie algebras. Additionally, we give an alternative proof of this conjecture via Bhargava's parametrization of quartic rings.Resumen: Probamos que la forma traza entera es un invariante completo para cuerpos de números totalmente reales de discriminante fundamental, también estudiamos la relación de este invariante con la forma traza-cero y la forma geométrica de un cuerpo de números, y damos resultados análogos para estos invariantes. Como consecuencia, probamos una conjetura del 2012 propuesta por Mantilla-soler sobre cuerpos cuarticos moderadamente ramificados de discriminante fundamental. Nuestro método de prueba se basa en lo que llamamos elementos de Casimir y emparejamientos de Casimir, herramientas nuevas introducidas en este trabajo, las cuales están relacionadas con (y generalizan) los elementos de Casimir de la teoría de representación de álgebras de Lie. Adicionalmente, damos una prueba alternativa de esta conjetura via la parametrización de anillos cuárticos de Bhargava.Maestríaapplication/pdfspaUniversidad Nacional de Colombia Sede Bogotá Facultad de Ciencias Departamento de MatemáticasDepartamento de Matemáticas51 Matemáticas / MathematicsRivera Guaca, Carlos Andrés (2018) Characterization of number fields by their integral trace form. Maestría thesis, Universidad Nacional de Colombia - Sede Bogotá.Characterization of number fields by their integral trace formTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMTrace formTotally real number fieldsShapes of number fieldsCasimir invariantHigher composition lawsORIGINALThesis.ms.pdfapplication/pdf722277https://repositorio.unal.edu.co/bitstream/unal/76670/1/Thesis.ms.pdf144e906240476cb1bbe050a0a17080e4MD51THUMBNAILThesis.ms.pdf.jpgThesis.ms.pdf.jpgGenerated Thumbnailimage/jpeg4347https://repositorio.unal.edu.co/bitstream/unal/76670/2/Thesis.ms.pdf.jpg293e386011be8f6b18135bde988e616fMD52unal/76670oai:repositorio.unal.edu.co:unal/766702024-07-14 01:05:19.139Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.co
dc.title.spa.fl_str_mv Characterization of number fields by their integral trace form
title Characterization of number fields by their integral trace form
spellingShingle Characterization of number fields by their integral trace form
Trace form
Totally real number fields
Shapes of number fields
Casimir invariant
Higher composition laws
title_short Characterization of number fields by their integral trace form
title_full Characterization of number fields by their integral trace form
title_fullStr Characterization of number fields by their integral trace form
title_full_unstemmed Characterization of number fields by their integral trace form
title_sort Characterization of number fields by their integral trace form
dc.creator.fl_str_mv Rivera Guaca, Carlos Andrés
dc.contributor.author.spa.fl_str_mv Rivera Guaca, Carlos Andrés
dc.contributor.spa.fl_str_mv Mantilla Soler, Guillermo
Rodriguez Vega, John Jaime
dc.subject.proposal.spa.fl_str_mv Trace form
Totally real number fields
Shapes of number fields
Casimir invariant
Higher composition laws
topic Trace form
Totally real number fields
Shapes of number fields
Casimir invariant
Higher composition laws
description We prove that the integral trace form is a complete invariant for totally real number fields of fundamental discriminant. We also study the relations of this invariant with the trace-zero form and the shape of a number field, and give analog results for such invariants. As a consequence, we settle a conjecture from 2012 by Mantilla-Soler about tamely ramified quartic fields of fundamental discriminant. Our method of proof is based on what we call Casimir elements and Casimir pairings, new tools we introduce in this work, which are related to (and generalize) the Casimir elements from the representation theory of Lie algebras. Additionally, we give an alternative proof of this conjecture via Bhargava's parametrization of quartic rings.
publishDate 2018
dc.date.issued.spa.fl_str_mv 2018-11
dc.date.accessioned.spa.fl_str_mv 2020-03-30T06:24:29Z
dc.date.available.spa.fl_str_mv 2020-03-30T06:24:29Z
dc.type.spa.fl_str_mv Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/masterThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.content.spa.fl_str_mv Text
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/TM
status_str acceptedVersion
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/76670
dc.identifier.eprints.spa.fl_str_mv http://bdigital.unal.edu.co/73309/
url https://repositorio.unal.edu.co/handle/unal/76670
http://bdigital.unal.edu.co/73309/
dc.language.iso.spa.fl_str_mv spa
language spa
dc.relation.ispartof.spa.fl_str_mv Universidad Nacional de Colombia Sede Bogotá Facultad de Ciencias Departamento de Matemáticas
Departamento de Matemáticas
dc.relation.haspart.spa.fl_str_mv 51 Matemáticas / Mathematics
dc.relation.references.spa.fl_str_mv Rivera Guaca, Carlos Andrés (2018) Characterization of number fields by their integral trace form. Maestría thesis, Universidad Nacional de Colombia - Sede Bogotá.
dc.rights.spa.fl_str_mv Derechos reservados - Universidad Nacional de Colombia
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.license.spa.fl_str_mv Atribución-NoComercial 4.0 Internacional
dc.rights.uri.spa.fl_str_mv http://creativecommons.org/licenses/by-nc/4.0/
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
dc.format.mimetype.spa.fl_str_mv application/pdf
institution Universidad Nacional de Colombia
bitstream.url.fl_str_mv https://repositorio.unal.edu.co/bitstream/unal/76670/1/Thesis.ms.pdf
https://repositorio.unal.edu.co/bitstream/unal/76670/2/Thesis.ms.pdf.jpg
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293e386011be8f6b18135bde988e616f
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MD5
repository.name.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
repository.mail.fl_str_mv repositorio_nal@unal.edu.co
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