{
  "abstract": "Let $ν$ be a Krull valuation of arbitrary rank on a field with valuation ring $R_ν$, and let $θ$ be a root of the irreducible polynomial $F(x)=(x^k+c)^m-ax^n\\in R_ν[x]$ and $1\\leq n<km$. We establish necessary and sufficient conditions for the integral closedness of $R_ν[θ]$, expressed explicitly in terms of the coefficients $a$, $b$, and the integers $m$, $n$, and $k$. In particular, when $ν$ is the $p$-adic valuation on $\\mathbb{Q}$, our results yield criteria for determining the primes dividing the index $[\\mathbb{Z}_K:\\mathbb{Z}[θ]],$ where $K=\\mathbb{Q}(θ)$ and $\\mathbb{Z}_K$ denotes the ring of integers of $K$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29111",
  "author_total": 1,
  "authors": [
    {
      "name": "V. ADITHYA"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6gdp73nwfg4dw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29111",
  "fulltext_url": null,
  "id": "6gdp73nwfg4dw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29111",
  "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": "W7214329474",
  "pdf_url": "https://arxiv.org/pdf/2609.29111",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6gdp73nwfg4dw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29111",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29111"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the index of generalized trinomials over Valued fields",
  "title_text": "On the index of generalized trinomials over Valued fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/6gdp73nwfg4dw",
  "venue": "arXiv",
  "versions": []
}