Trace and norm of indecomposable integers in cubic orders

TL;DR

This paper investigates the structure of indecomposable integers in totally real cubic fields, revealing that their minimal trace can be arbitrarily large and providing bounds on their norms, contrasting with quadratic cases.

Contribution

It introduces new bounds and properties of indecomposable integers in cubic orders, highlighting differences from quadratic and simpler cubic fields.

Findings

Minimal trace of indecomposables can be arbitrarily large

Sharp upper bounds on norms of indecomposable integers

Contrasts with quadratic and simpler cubic field cases

Abstract

We study the structure of the codifferent and of additively indecomposable integers in families of totally real cubic fields. We prove that for cubic orders in these fields, the minimal trace 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 and 2. We further give sharp upper bounds on the norms of indecomposable integers in our families.