The Euclidean Algorithm in Cubic Number Fields

TL;DR

This paper develops algorithms to compute Euclidean minima in cubic number fields and successfully identifies all norm-Euclidean cubic fields within a specific discriminant range, advancing understanding of Euclidean properties in algebraic number theory.

Contribution

It introduces new algorithms for calculating Euclidean minima and classifies all norm-Euclidean cubic fields within a large discriminant interval.

Findings

Identified all norm-Euclidean cubic fields with -999 < d < 10000.

Developed algorithms for Euclidean minima computation in cubic fields.

Enhanced classification of Euclidean properties in algebraic number theory.

Abstract

In this note we present algorithms for computing Euclidean minima of cubic number fields; in particular, we were able to find all norm-Euclidean cubic number fields with discriminants -999 < d < 10000.