{
  "abstract": "We calculate the Galois group of $f = x^n + ax^k + b \\in \\mathbb{Z}\\left[ x \\right]$ over $\\mathbb{Q}$ for all $1 \\leq \\left| ab \\right| \\leq 3$, $n$ sufficiently large and $k$ coprime to $n$. In particular, we prove that the Galois group of $f$ is the maximal one that its factorization type allows. We give a criterion for the Galois group of a factor of a trinomial to be symmetric, with an eye towards future applications. To the best of our knowledge, the only previously known explicit example of polynomials of degree $n$ with bounded coefficients and Galois group $S_n$ is $x^n - x - 1$, while recent results show that the probability that a random polynomial of degree $n$ with reasonably bounded coefficients has Galois group $S_n$ tends to $1$ as $n$ tends to infinity. Using our calculation, one can construct many new such explicit examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22764",
  "author_total": 1,
  "authors": [
    {
      "name": "Hilel Garmi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2q6c7h0mrgkre.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22764",
  "fulltext_url": null,
  "id": "2q6c7h0mrgkre",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22764",
  "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",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7222944857",
  "pdf_url": "https://arxiv.org/pdf/2610.22764",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2q6c7h0mrgkre.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.22764",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22764"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Galois Groups of Trinomials with Small Coefficients",
  "title_text": "Galois Groups of Trinomials with Small Coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/2q6c7h0mrgkre",
  "venue": "arXiv",
  "versions": []
}