Euclidean Ideals in Quadratic Imaginary Fields

TL;DR

This paper classifies quadratic imaginary fields with Euclidean ideal classes, identifying seven such fields with class number at most two, where the generator class is norm-Euclidean, advancing understanding of Euclidean properties in algebraic number theory.

Contribution

It provides a complete classification of quadratic imaginary fields with Euclidean ideal classes, highlighting their properties and the norm-Euclidean nature of their class group generators.

Findings

Seven quadratic imaginary fields have Euclidean ideal classes.

All identified fields have class number at most two.

The generator class in each case is norm-Euclidean.

Abstract

We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.