On the existence of a non-principal Euclidean ideal class in biquadratic fields with class number two

TL;DR

This paper constructs a family of biquadratic fields with class number two that possess a Euclidean ideal class, extending previous results and providing new examples beyond principal ideal classes.

Contribution

It introduces new biquadratic fields with Euclidean ideal classes when the class number is two, expanding known families beyond principal ideal classes.

Findings

Constructed biquadratic fields with Euclidean ideal classes and class number two

Extended previous families of such fields by Graves, Hsu, Chattopadhyay, and Muthukrishnan

Demonstrated existence of non-principal Euclidean ideal classes in these fields

Abstract

Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a family of biquadratic fields with a Euclidean ideal whenever the class number is 2. This extends the families given by Graves, Hsu, Chattopadhyay, and Muthukrishnan.