{
  "abstract": "For an integer $b\\neq0,1$ let $θ$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\\mathbb{Q}(θ)$. We exhibit an explicit set of $b$ of positive density for which $η_b=-1/(θ-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $η_b$ is the square of a unit: it is parametrized by the Pell equation $D^2-3E^2=1$, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all $b$ for which $η_b$ is not the fundamental unit. That condition fails for only four $b$ with $|b|\\le3000$, and at one of them, $b=3$, the conclusion itself fails. In the order $\\mathbb{Z}[η_b]$ generated by $η_b$, by contrast, $η_b$ is the fundamental unit for every $b$, so these exceptions are a phenomenon of the maximal order. As an application we construct infinitely many biquadratic fields whose $3$-class field tower has length greater than $1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37389",
  "author_total": 2,
  "authors": [
    {
      "name": "Iwao Kimura"
    },
    {
      "name": "Hikaru Umemoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aj6sxq6cw9npk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37389",
  "fulltext_url": null,
  "id": "aj6sxq6cw9npk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37389",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7214960033",
  "pdf_url": "https://arxiv.org/pdf/2609.37389",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aj6sxq6cw9npk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.37389",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37389"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On an infinite family of cubic fields with explicit fundamental units",
  "title_text": "On an infinite family of cubic fields with explicit fundamental units",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/aj6sxq6cw9npk",
  "venue": "arXiv",
  "versions": []
}