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arXiv · 2512.00337

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

Sunil Kumar Pasupulati
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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.

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Sunil Kumar Pasupulati. 2025-11-29. Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective. https://arxiv.org/abs/2512.00337

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