Trace and norm of indecomposable integers in cubic orders

We study the structure of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal traces of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surpr...

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Bibliographic Details
Published in:The Ramanujan journal 2023-08, Vol.61 (4), p.1121-1144
Main Author: Tinková, Magdaléna
Format: Article
Language:English
Subjects:
Codification
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Integers
Mathematics
Mathematics and Statistics
Norms
Number Theory
Upper bounds
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Summary:We study the structure of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal traces of indecomposable integers multiplied by totally positive elements of the codifferent can be arbitrarily large. This is very surprising, as in the so-far studied examples of quadratic and simplest cubic fields, this minimum is 1 or 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-022-00669-y