Biquadratic fields having a non-principal euclidean ideal class

TL;DR

This paper constructs new families of biquadratic number fields with non-principal Euclidean ideal classes, extending known examples and advancing the understanding of Euclidean ideal classes in algebraic number theory.

Contribution

It introduces novel families of biquadratic fields with non-principal Euclidean ideal classes, expanding the classification beyond previously known cases.

Findings

Constructed new biquadratic fields with non-principal Euclidean ideal classes.

Extended the classification of Euclidean ideal classes in number fields.

Provided explicit examples surpassing earlier known families.

Abstract

H. W. Lenstra \cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in \cite{hester1} and \cite{cathy}). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves \cite{hester1} and C. Hsu \cite{cathy}.