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The shape of cyclic number fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  06 September 2022

Wilmar Bolaños  and
Guillermo Mantilla-Soler Open the ORCID record for Guillermo Mantilla-Soler [Opens in a new window]
Wilmar Bolaños*
Affiliation:
Department of Mathematics, Fundación Universitaria Konrad Lorenz, Bogotá, Colombia
Guillermo Mantilla-Soler
Affiliation:
Department of Mathematics, Universidad Nacional de Colombia sede Medellín, Medellín, Colombia e-mail: gmantelia@gmail.com
*
e-mail: wilmarr.bolanosc@konradlorenz.edu.co
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Abstract

Let $m>1$ and $\mathfrak {d} \neq 0$ be integers such that $v_{p}(\mathfrak {d}) \neq m$ for any prime p. We construct a matrix $A(\mathfrak {d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak {d}$ with the following property: For any tame $ \mathbb {Z}/m \mathbb {Z}$ -number field K of discriminant $\mathfrak {d}$ , the matrix $A(\mathfrak {d})$ represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.

Keywords

Shapescyclic number fieldsreal number fields

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R20: Other abelian and metabelian extensions 11R29: Class numbers, class groups, discriminants
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 599 - 609
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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