{
  "abstract": "We investigate the stability of the $2$-rank in the cyclotomic $\\mathbb{Z}_2$-extension of real biquadratic number fields. We determine several families of real biquadratic fields $F$ for which the ranks of the $2$-class group $A(F)$ and the associated $2$-Iwasawa module $A_\\infty(F)$ are both equal to $3$. More precisely, we provide a complete classification of all fields $F=\\mathbb{Q}(\\sqrt{q},\\sqrt{d}),$ where $q\\equiv 3\\pmod{4}$ is a prime number and $d$ is an odd positive square-free integer, and all fields $F=\\mathbb{Q}(\\sqrt{2q},\\sqrt{d}),$ where $q\\equiv 3\\pmod{4}$ is a prime number and $d\\equiv 1\\pmod{4}$ is a positive square-free integer, for which $ \\operatorname{rank}(A(F))=\\operatorname{rank}(A_\\infty(F))=3.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34400",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed Mahmoud Chems-Eddin"
    },
    {
      "name": "Ikrame Daqaq"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aa1refzrsd5fk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34400",
  "fulltext_url": null,
  "id": "aa1refzrsd5fk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34400",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7223875928",
  "pdf_url": "https://arxiv.org/pdf/2610.34400",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aa1refzrsd5fk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34400",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34400"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$",
  "title_text": "On real biquadratic number fields for which the $2$-Iwasawa module has rank $3$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/aa1refzrsd5fk",
  "venue": "arXiv",
  "versions": []
}