{
  "abstract": "Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\\mathbb{Q}$ during 1984-1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\\mathbb{Q}$. In fact, we produce an explicit degree $23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$. We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Costa, Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\\mathbb{Q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.08538",
  "author_total": 6,
  "authors": [
    {
      "name": "Xiaoyu Huang"
    },
    {
      "name": "Blake Jackson"
    },
    {
      "name": "Kyu-Hwan Lee"
    },
    {
      "name": "Bjorn Poonen"
    },
    {
      "name": "Rachel Pries"
    },
    {
      "name": "Shaowu Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z9r8qe9dtve2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.08538",
  "fulltext_url": null,
  "id": "7z9r8qe9dtve2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.08538",
  "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.AG"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7202211762",
  "pdf_url": "https://arxiv.org/pdf/2608.08538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z9r8qe9dtve2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2608.08538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.08538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Mathieu group $M_{23}$ is a Galois group over $\\mathbb{Q}$",
  "title_text": "The Mathieu group M_23 is a Galois group over ℚ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/7z9r8qe9dtve2",
  "venue": "arXiv",
  "versions": []
}