{
  "abstract": "Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\\mathcal{O}_S^\\times$ for its unit group. Let $π: X \\rightarrow \\mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\\operatorname{Red}(π)$ the set of $α\\in \\mathbb{P}^1(K)$ such that the fibre $π^{-1}(α)$ is reducible (i.e. the Galois action on the fibre is intransitive). Hilbert's Irreducibility Theorem asserts that $\\operatorname{Red}(π)$ is contained in a thin subset of $\\mathbb{P}^1(K)$. In this paper we give an explicit description of $\\mathcal{O}_S^\\times \\cap \\operatorname{Red}(π)$. As an application we prove the following result inspired by a classical theorem of Pólya and Siegel. Let $p_1,\\dotsc,p_s$ be rational primes. Let $f \\in \\mathbb{Q}[x]$.Then the following are equivalent: - There are infinitely many tuples $(e_1,\\dotsc,e_s) \\in \\mathbb{N}^s$ such that the polynomial $f(x)-p_1^{e_1} \\cdots p_s^{e_s}$ is reducible. - $f=p_1^{a_1} \\cdots p_s^{a_s} g^\\ell$ (with $\\ell$ prime) or $f=-4 p_1^{a_1} \\cdots p_s^{a_s} g^4$ for some $g \\in \\mathbb{Q}[x]$ and some integers $a_1,\\dotsc,a_s$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04551",
  "author_total": 2,
  "authors": [
    {
      "name": "Michael Stoll"
    },
    {
      "name": "Samir Siksek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fdaqrevwhb122.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04551",
  "fulltext_url": null,
  "id": "fdaqrevwhb122",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04551",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7211872276",
  "pdf_url": "https://arxiv.org/pdf/2609.04551",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fdaqrevwhb122.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04551",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04551"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Hilbert's Irreducibility for $\\mathbb{G}_m$",
  "title_text": "Hilbert's Irreducibility for 𝔾_m",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/fdaqrevwhb122",
  "venue": "arXiv",
  "versions": []
}