PreprintCC BY 4.0arXiv
All totally real abelian number fields of class number one are Euclidean
Abstract
It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions K/ℚ, whose ring of integers 𝒪K is a unique factorization domain (UFD), but not Euclidean. As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis. Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz's result, and show, more generally, that the ring of integers 𝒪K of any totally real abelian number field K is a UFD, if and only if it is Euclidean.
Related
- Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields
- An adelic approach to the Kudla-Millson lifts over totally real number fields
- Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
- Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
- Shapes of tame abelian number fields
BibTeX
@misc{grobner2026totally,
title = {{All totally real abelian number fields of class number one are Euclidean}},
author = {Harald Grobner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24900},
eprint = {2610.24900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24900},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - All totally real abelian number fields of class number one are Euclidean
AU - Harald Grobner
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24900
UR - https://arxiv.org/abs/2610.24900
AB - It is a famous result of Narkiewicz that there are at most two real quadratic and at most one totally real cubic Galois extensions K/ℚ, whose ring of integers 𝒪_K is a unique factorization domain (UFD), but not Euclidean. As predicted by another famous result due to Weinberger, these exceptions should in fact not exist, as their existence would contradict a generalized version of the Riemann hypothesis. Building on recent spectacular progress on the Quasi-Riemann hypothesis for Hecke characters, in this short note we close the above final gap provided by Narkiewicz's result, and show, more generally, that the ring of integers 𝒪_K of any totally real abelian number field K is a UFD, if and only if it is Euclidean.
ER -