{
  "abstract": "Let $F$ be a global field and let $p$ be an odd prime with $p\\neq\\operatorname{char}F$. Assuming first that $μ_p\\subset F$, we record in a uniform global-field form the length-four common-slot chain for equal degree-$p$ symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from \\[(a,b)_p=(c,d)_p\\in\\textrm{Br}(F)[p]\\] one obtains $x,y\\in F^\\times$ such that \\[(a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p.\\] For number fields, the underlying chain is the length-four chain lemma of Gille--Szamuely, based on Tate's simultaneous local--global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product $H_{p^3}*H_{p^3}$ over a $C_p^4$-Kummer extension, the central-embedding obstruction is $(a,b)_p-(c,d)_p$, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain $μ_p$, we show by a restriction--corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension $F(μ_p)$. Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00244",
  "author_total": 1,
  "authors": [
    {
      "name": "Marina Palaisti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f51fq58t4gbgh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00244",
  "fulltext_url": null,
  "id": "f51fq58t4gbgh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00244",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7206164295",
  "pdf_url": "https://arxiv.org/pdf/2609.00244",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f51fq58t4gbgh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00244",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00244"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "From Common-Slot Chains to Heisenberg Central Products over Global Fields",
  "title_text": "From Common-Slot Chains to Heisenberg Central Products over Global Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/f51fq58t4gbgh",
  "venue": "arXiv",
  "versions": []
}