On biquadratic fields: when 5 squares are not enough
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911693282476032 |
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| author | Dombek, Daniel |
| author_facet | Dombek, Daniel |
| contents | In this paper we study the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinková towards proving the conjecture by Krásenský, Raška and Sgallová that a biquadratic $K$ satisfies $\mathcal{P}(\mathcal{O}_K)\geq 6$ if and only if it contains neither $\sqrt{2}$ nor $\sqrt{5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\sqrt{6}$ or $\sqrt{7}$ satisfy $\mathcal{P}(\mathcal{O}_K)\geq 6$. Furthermore, we present ideas and computations which further support the conjecture also for $K$ containing $\sqrt{3}$. This enables us to refine the conjecture by explicitly listing the exceptional fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On biquadratic fields: when 5 squares are not enough Dombek, Daniel Number Theory 11E25 In this paper we study the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinková towards proving the conjecture by Krásenský, Raška and Sgallová that a biquadratic $K$ satisfies $\mathcal{P}(\mathcal{O}_K)\geq 6$ if and only if it contains neither $\sqrt{2}$ nor $\sqrt{5}$, with only finitely many exceptions. We fully solve two out of three remaining classes of fields by proving that all but finitely many $K$ containing $\sqrt{6}$ or $\sqrt{7}$ satisfy $\mathcal{P}(\mathcal{O}_K)\geq 6$. Furthermore, we present ideas and computations which further support the conjecture also for $K$ containing $\sqrt{3}$. This enables us to refine the conjecture by explicitly listing the exceptional fields. |
| title | On biquadratic fields: when 5 squares are not enough |
| topic | Number Theory 11E25 |
| url | https://arxiv.org/abs/2506.20820 |