The determination of norm-Euclidean cyclic cubic fields
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911045339054080 |
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| author | Bagger, Gustav Kjærbye Booker, Andrew R. Kerr, Bryce McGown, Kevin Starichkova, Valeriia Trudgian, Tim |
| author_facet | Bagger, Gustav Kjærbye Booker, Andrew R. Kerr, Bryce McGown, Kevin Starichkova, Valeriia Trudgian, Tim |
| contents | It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The determination of norm-Euclidean cyclic cubic fields Bagger, Gustav Kjærbye Booker, Andrew R. Kerr, Bryce McGown, Kevin Starichkova, Valeriia Trudgian, Tim Number Theory 11A05, 11R04, 11Y40 It is known on the Generalised Riemann Hypothesis that there are precisely $13$ cyclic cubic fields that are norm-Euclidean. Unconditionally, there is a gap between analytic estimates which hold for all sufficiently large conductors and computational techniques. In this paper, we establish new results concerning explicit bounds for cubic non-residues and refine previous computational techniques, enabling us to completely characterise all norm-Euclidean cyclic cubic fields. |
| title | The determination of norm-Euclidean cyclic cubic fields |
| topic | Number Theory 11A05, 11R04, 11Y40 |
| url | https://arxiv.org/abs/2507.05862 |