{
  "abstract": "In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power $q=p^e\\equiv1\\pmod3$ and every $n\\ge5$ with $\\gcd(n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $φ(n)$ isotopy classes if $p\\equiv1\\pmod3$, and $φ(n)/2$ if $p\\equiv2\\pmod3$, where $φ$ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension $n\\ge5$, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32651",
  "author_total": 2,
  "authors": [
    {
      "name": "Gábor P. Nagy"
    },
    {
      "name": "Yue Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1yn6fvjqq8ms1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32651",
  "fulltext_url": null,
  "id": "1yn6fvjqq8ms1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32651",
  "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.CO"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7214791339",
  "pdf_url": "https://arxiv.org/pdf/2609.32651",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1yn6fvjqq8ms1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32651",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32651"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Semifields in prime dimensions and counterexamples to Kaplansky's conjecture",
  "title_text": "Semifields in prime dimensions and counterexamples to Kaplansky's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/1yn6fvjqq8ms1",
  "venue": "arXiv",
  "versions": []
}