Division algorithms for norm-Euclidean real quadratic fields -- part I
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917254455623680 |
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| author | Morain, François |
| author_facet | Morain, François |
| contents | We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each case, we cover the square $[-1/2, 1/2] \times [-1/2, 1/2]$ with hyperbolas and give a list of these, together with regions covered. We mechanize the proofs as much as we can, using exact computations, in order to be able to reproduce them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06538 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Division algorithms for norm-Euclidean real quadratic fields -- part I Morain, François Number Theory We give a Euclidean division algorithm for the real quadratic fields $\mathbb{Q}(\sqrt{m})$ for $m \in \{2, 3, 6, 7, 11, 19\}$, with the property that the norm of the remainder depends on the first Euclidean minimum of the field. In each case, we cover the square $[-1/2, 1/2] \times [-1/2, 1/2]$ with hyperbolas and give a list of these, together with regions covered. We mechanize the proofs as much as we can, using exact computations, in order to be able to reproduce them. |
| title | Division algorithms for norm-Euclidean real quadratic fields -- part I |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.06538 |