{
  "abstract": "Let $\\mathbf{E}$ be the ring of values taken at algebraic points by $E$-functions. The exponential function and the Bessel function $J_0$ are classical examples of $E$-functions. We prove that $\\mathbf{E}$ is not a field. More precisely, we prove that it is neither noetherian nor factorial, and construct non-units of $\\mathbf{E}$ explicity, including $e+1$, $J_0(1)$, and ${}_1F_1(a;b;1)$ for rational parameters $a$ and $b$ such that $a\\in (0,1)$ and $b-a\\in (0,1)$.",
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  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16603",
  "author_total": 2,
  "authors": [
    {
      "name": "Stéphane Fischler"
    },
    {
      "name": "Tanguy Rivoal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tqkenvc23nq0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16603",
  "fulltext_url": null,
  "id": "0tqkenvc23nq0",
  "issns": [],
  "landing_url": "https://hal.science/hal-05763493v2",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
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      "arxiv",
      "math.NT"
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      "hal",
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      "openalex",
      "2608|Algebraic Geometry and Number Theory"
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  "openalex": "W7214568397",
  "pdf_url": "https://hal.science/hal-05763493/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tqkenvc23nq0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:54Z",
      "native_id": "hal-05763493",
      "source": "hal",
      "url": "https://hal.science/hal-05763493v2"
    },
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16603",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16603"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The ring of values of $E$-functions is not a field",
  "title_text": "The ring of values of E-functions is not a field",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0tqkenvc23nq0",
  "venue": "HAL",
  "versions": []
}