Low degree extensions with Cyclic class group

TL;DR

This paper proves the existence of Euclidean ideal classes in certain abelian quartic, biquadratic, cubic, and quadratic fields, extending the understanding of Euclidean structures in number fields with cyclic class groups.

Contribution

It establishes the existence of Euclidean ideal classes in abelian quartic fields and specific biquadratic, cubic, and quadratic extensions, broadening the class of fields known to have Euclidean ideal classes.

Findings

Euclidean ideal classes exist in abelian quartic fields

Certain biquadratic fields with class number two have Euclidean ideal classes

Investigation of Euclidean ideal classes in specific cubic and quadratic fields

Abstract

Lenstra introduced the notion of the Euclidean ideal class, a generalization of the Euclidean domain that captures cyclic class groups. In this article, we establish the existence of Euclidean ideal classes in abelian quartic fields. As a corollary, we demonstrate that certain biquadratic fields with class number two possess a Euclidean ideal class. Additionally, we investigate the presence of Euclidean ideal classes in specific cubic and quadratic extensions.