Pisot unit generators in number fields

TL;DR

This paper develops algorithms to find the smallest Pisot units generating various number fields, including real and complex cases, expanding understanding of algebraic integers with special conjugate properties.

Contribution

It introduces algorithms for identifying minimal Pisot units in all real and certain complex number fields, including those without CM.

Findings

Algorithms successfully find minimal Pisot units in real fields.

Extended methods to complex Pisot numbers in non-CM fields.

Provides a comprehensive approach to generating specific algebraic integers.

Abstract

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates have all modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular all fields without CM, but not only those.