{
  "abstract": "Let $K^h$ be a Henselization of a valued field $(K,v),$ $w$ a valuation-transcendental extension of $v$ to $K[x],$ and $ϕ$ a key polynomial for $w$. In this paper, we prove that if $ϕ$ is irreducible over $K^h,$ then the simple extension $L|K,$ generated by a root of any key polynomial for $w,$ is unibranched and its defect is independent of the choice of key polynomial. As a consequence, we generalize some well-known results for Henselian valued fields to arbitrary valued fields. Moreover, we provide some criteria for $L|K$ to be unibranched and defectless in terms of Mac Lane-Vaquié chains, complete sequences of abstract key polynomials and saturated distinguished chains. As an application, we generalize a classical result of Ore about the existence of $p$-regular generators for number fields.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15450",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikita Dwivedi"
    },
    {
      "name": "Anuj Bishnoi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ynh8q1zz9dwx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15450",
  "fulltext_url": null,
  "id": "4ynh8q1zz9dwx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15450",
  "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.AC"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213367619",
  "pdf_url": "https://arxiv.org/pdf/2609.15450",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ynh8q1zz9dwx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15450",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15450"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On simple extensions generated by key polynomials",
  "title_text": "On simple extensions generated by key polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4ynh8q1zz9dwx",
  "venue": "arXiv",
  "versions": []
}