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arXiv · 2507.05862

The determination of norm-Euclidean cyclic cubic fields

Gustav Kjærbye Bagger·Andrew R. Booker·Bryce Kerr·Kevin J. McGown·Valeriia Starichkova·Tim Trudgian
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Abstract

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.

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Gustav Kjærbye Bagger, Andrew R. Booker, Bryce Kerr, Kevin J. McGown, Valeriia Starichkova, Tim Trudgian. 2025-07-08. The determination of norm-Euclidean cyclic cubic fields. https://arxiv.org/abs/2507.05862

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