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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.00337 |
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| _version_ | 1866915684791877632 |
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| author | Pasupulati, Sunil Kumar |
| author_facet | Pasupulati, Sunil Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective Pasupulati, Sunil Kumar Number Theory 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. |
| title | Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.00337 |