{
  "abstract": "We classify the pairs $(q,n)$, where $q$ is a prime power, $n>1$, and $\\gcd(n,q)=1$, for which every irreducible factor of $Φ_n(x)$ over $\\F_q$ has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to $\\{2,3,4\\}$ in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on $(q,n)$ and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00647",
  "author_total": 1,
  "authors": [
    {
      "name": "Li Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1fmt0h7sqr2cy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00647",
  "fulltext_url": null,
  "id": "1fmt0h7sqr2cy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00647",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1fmt0h7sqr2cy.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cyclotomic polynomials with trinomial irreducible factors over finite fields",
  "title_text": "Cyclotomic polynomials with trinomial irreducible factors over finite fields",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/1fmt0h7sqr2cy",
  "venue": "arXiv",
  "versions": []
}