{
  "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 $K/\\mathbb Q$, whose ring of integers $\\mathcal O_K$ 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 $\\mathcal O_K$ of any totally real abelian number field $K$ is a UFD, if and only if it is Euclidean.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24900",
  "author_total": 1,
  "authors": [
    {
      "name": "Harald Grobner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/13wjm9dwrkqy4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24900",
  "fulltext_url": null,
  "id": "13wjm9dwrkqy4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24900",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223013536",
  "pdf_url": "https://arxiv.org/pdf/2610.24900",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/13wjm9dwrkqy4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24900",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24900"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "All totally real abelian number fields of class number one are Euclidean",
  "title_text": "All totally real abelian number fields of class number one are Euclidean",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/13wjm9dwrkqy4",
  "venue": "arXiv",
  "versions": []
}