{
  "abstract": "Ozaki proved that every finite $p$-group occurs as the Galois group of a maximal unramified $p$-extension of a number field. Hajir, Maire and Ramakrishna made this theorem effective, obtaining a base-field degree of order $|G|$. In this article, we reduce that degree by taking the Frattini structure of $G$ into account. More precisely, for an odd prime $p$ and a finite $p$-group $G$ of order $p^n$, we prove \\[ τ_p(G) \\leq p^{\\ell_Φ(G)+ \\left\\lceil\\log_p\\left(\\binom{n+2}{2}+1\\right)\\right\\rceil}. \\] Here $τ_p(G)$ is the least degree of a number field realizing $G$ as its $p$-class tower group, and $\\ell_Φ(G)$ is the iterated Frattini length of $G$. Thus, for groups of bounded Frattini length, the order-scale bound $p^n$ is replaced by the quadratic bound $O_p(n^2)$. For $E_m=(\\Z/p\\Z)^m$, we further prove the sharp estimate $τ_p(E_m)\\asymp_p m^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22766",
  "author_total": 1,
  "authors": [
    {
      "name": "Kwang-Seob Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cf7a2mr41r6c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22766",
  "fulltext_url": null,
  "id": "1cf7a2mr41r6c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22766",
  "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": "W7214002287",
  "pdf_url": "https://arxiv.org/pdf/2609.22766",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cf7a2mr41r6c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22766",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22766"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Maximal Unramified $p$-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem",
  "title_text": "Maximal Unramified p-Extensions with Prescribed Galois Groups: A Quantitative Refinement of Ozaki's Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/1cf7a2mr41r6c",
  "venue": "arXiv",
  "versions": []
}