The determination of norm-Euclidean cyclic cubic fields

2025/07/08 by Gustav Kjærbye Bagger, Andrew R. Booker, Bagger, Gustav Kjærbye +9
Mathematics · #11A05 #11R04 #11Y40 #Advanced Harmonic Analysis Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.05862

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• An Investigation Into Several Explicit Versions of Burgess' Bound
• Explicit upper bounds on the least primitive root
• Explicit bounds for small prime nonresidues
• The Euclidean algorithm in quintic and septic cyclic fields
• On Congruences with Products of Variables from Short Intervals and Applications
• Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
• Approximate formulas for some functions of prime numbers

Related

• The Euclidean algorithm in quintic and septic cyclic fields
• A note on Euclidean cyclic cubic fields
• Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis
• Families of Cyclic Cubic Fields
• Group theory of cyclic cubic number fields