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

Description
Summary:	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.

Characterization of number fields by their integral trace form

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