{
  "abstract": "Following Johnson, Queen, and Sevilla, we call an order $\\mathcal{O}$ in a number field $K$ \\textsf{generalized Euclidean} if for all $α,β\\in \\mathcal{O}$ with $β\\ne 0$ for which $α\\mathcal{O}+β\\mathcal{O}$ is a principal ideal of $\\mathcal{O}$, there are $γ, ρ\\in \\mathcal{O}$ with $α=βγ+ ρ$ and $|N_{K/\\mathbb{Q}}(ρ)| < |N_{K/\\mathbb{Q}}(β)|$. We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed $d > 1$ there are only finitely many integers $m$ for which $T^d-m$ is irreducible over $\\mathbb{Q}$ and $\\mathbb{Z}[m^{1/d}] = \\mathbb{Z}[T]/(T^d-m)$ is generalized Euclidean. When $d$ is even, we can remove the assumption of GRH.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25793",
  "author_total": 4,
  "authors": [
    {
      "name": "Carlo Francisco Adajar"
    },
    {
      "name": "Kevin J. McGown"
    },
    {
      "name": "Paul Pollack"
    },
    {
      "name": "Enrique Treviño"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5x8cf4tb5fmrt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25793",
  "fulltext_url": null,
  "id": "5x8cf4tb5fmrt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25793",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.25793",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5x8cf4tb5fmrt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25793",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25793"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Orders in the Euclidean Court",
  "title_text": "Orders in the Euclidean Court",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/5x8cf4tb5fmrt",
  "venue": "arXiv",
  "versions": []
}