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Determining norm-Euclidean complex cubic fields
Abstract
We prove that Cavallar and Lemmermeyer's list of $58$ norm-Euclidean complex cubic fields is complete. The proof combines a classical discriminant bound with exact certificates of non-Euclideanity. The unit action reduces their verification to finite lattice enumeration with rigorous interval arithmetic. An independent covering computation confirms norm-Euclideanity for the $58$ fields.
Related
- Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
- Optimal Gradient-Norm Minimization in Non-Euclidean Hölder-Smooth Convex Optimization
- Primitive norm extraction and a signed descent for the cubic Fermat equation
- Norm-Euclideanity in cyclic fields
- All totally real abelian number fields of class number one are Euclidean
BibTeX
@misc{bagger2026determining,
title = {{Determining norm-Euclidean complex cubic fields}},
author = {Gustav Kjærbye Bagger and Andrew R. Booker and Tim Trudgian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38073},
eprint = {2610.38073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38073},
}
RIS
TY - UNPB
TI - Determining norm-Euclidean complex cubic fields
AU - Gustav Kjærbye Bagger
AU - Andrew R. Booker
AU - Tim Trudgian
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.38073
UR - https://arxiv.org/abs/2610.38073
AB - We prove that Cavallar and Lemmermeyer's list of $58$ norm-Euclidean complex cubic fields is complete. The proof combines a classical discriminant bound with exact certificates of non-Euclideanity. The unit action reduces their verification to finite lattice enumeration with rigorous interval arithmetic. An independent covering computation confirms norm-Euclideanity for the $58$ fields.
ER -