{
  "abstract": "We prove the degree-seven case of the Hermite--Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with $\\text{tr}_{E/F}(a)=\\text{tr}_{E/F}(a^{3})=0$, that is, with minimal polynomial of the form $λ^{7}+c_{2}λ^{5}+c_{4}λ^{3}+c_{5}λ^{2}+c_{6}λ+c_{7}$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $θ$ and divided by the derivative of its minimal polynomial evaluated at $θ$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_{1}(a)=c_{3}(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_{1}(a)=c_{3}(a)=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15786",
  "author_total": 4,
  "authors": [
    {
      "name": "Sunil Chebolu"
    },
    {
      "name": "Ján Mináč"
    },
    {
      "name": "Behzad Nikzad"
    },
    {
      "name": "Charlotte Ure"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c43g35rg774nj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15786",
  "fulltext_url": null,
  "id": "c43g35rg774nj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15786",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213327991",
  "pdf_url": "https://arxiv.org/pdf/2609.15786",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c43g35rg774nj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15786",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15786"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A septic covariant and the Hermite--Joubert problem in degree seven",
  "title_text": "A septic covariant and the Hermite--Joubert problem in degree seven",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/c43g35rg774nj",
  "venue": "arXiv",
  "versions": []
}