Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective

TL;DR

This paper proves that certain real biquadratic fields with cyclic class groups and abelian Hilbert class fields contain Euclidean ideal classes, and shows that such fields are rare among all biquadratic fields.

Contribution

It provides unconditional existence results for Euclidean ideal classes in specific biquadratic fields using genus theory, removing reliance on GRH assumptions.

Findings

If the class group is cyclic and the Hilbert class field is abelian over Q, then the field contains a Euclidean ideal class.

The set of biquadratic fields with Euclidean ideal classes has density zero among all such fields.

The distribution of genus numbers in biquadratic fields is analyzed to support these results.

Abstract

We study Euclidean ideal classes in real biquadratic fields and obtain unconditional existence results via genus theory. Lenstra showed (assuming the Generalized Riemann Hypothesis) that a number field with unit rank at least one admits a Euclidean ideal precisely when its class group is cyclic; subsequent work has aimed to remove the GRH hypothesis in special families. Focusing on real biquadratic fields $K=\mathbb{Q}\left(\sqrt{d_1},\sqrt{d_2}\right)$ with $2\nmid d_1d_2$, we prove that if the class group $\mathrm{Cl}_K$ is cyclic and the Hilbert class field $H(K)$ is abelian over $\mathbb{Q}$, then $K$ contains a Euclidean ideal class (unconditionally). We also analyse the distribution of genus numbers in a natural family of biquadratic fields and, using these statistics, show that the set of biquadratic fields admitting a Euclidean ideal has density zero.