Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective
Sunil Kumar Pasupulati
TL;DR
This work establishes an unconditional route to Euclidean ideals for real biquadratic fields under the hypothesis that the Hilbert class field is abelian and the class group is cyclic, by leveraging explicit genus-field descriptions and a Galois-theoretic criterion for Euclidean ideals. It airbrushes genus theory with Hilbert class-field structure to show that K contains a Euclidean ideal whenever Cl_K is cyclic and H(K)/Q is abelian, and then uses a refined counting framework (via the Rome–Rom18 parametrization and Sathe–Selberg-type estimates) to prove that the subset of biquadratic fields admitting a Euclidean ideal has density zero in the natural family. The analysis highlights how genus numbers constrain which multiquadratic fields can support Euclidean ideals and demonstrates that such favorable fields are statistically scarce. Together, these results extend unconditional existence results for Euclidean ideals to a broad class of biquadratic fields and clarify their distribution within the genus landscape.
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.