{
  "schema_version": 1,
  "problem_number": "KOU-21.76",
  "title": "Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An elementary net (carpet) of order n over a field K is closed if its elementary net group contains no new elementary transvections, and it is completable if its diagonal can be supplemented to a full net. Completable nets are closed. Koibaev gave closed nets that are not completable over fields of characteristic 0 and 2 and asked for such nets in odd characteristic (Kourovka Notebook, Problem 21.76). Nuzhin (2026) has answered this question, with examples in every characteristic, using closures of finitely generated subgroups over a rational function field in two variables. We give explicit examples with short proofs. An elementary lifting lemma turns a closed pair of additive subgroups of a quotient ring R/J into a closed elementary net of every order n ≥ 3. With F[x,y] → F[x] and a degree argument it gives, for every field F, the net with Fx + yF[x,y] in positions (1,2) and (2,1) and yF[x,y] elsewhere. Lifting the two exceptional pairs of Levchuk’s refinement of Dickson’s theorem along F₃[t] → F₉ and F₂[t] → F₄ gives closed nets over F₃(t) and F₂(t) that are not completable. These one-variable nets also satisfy the hypotheses of Kourovka Problem 19.48. Together with a theorem of Koibaev and Nuzhin on algebraic extensions, it follows that in characteristics 2 and 3 a field carries such nets if and only if it is not algebraic over its prime field. For p ≥ 5 we do not know whether examples exist over F_p(t); by Levchuk’s theorem, lifting from a finite field cannot produce them. This is an unrefereed note.",
  "result_type": "COMPLETE_AFFIRMATIVE_ANSWER",
  "categories": [
    "math.GR",
    "math.RA"
  ],
  "keywords": [
    "elementary nets",
    "carpets",
    "closed nets",
    "elementary transvections",
    "Kourovka Notebook",
    "KOU-21.76",
    "open mathematics",
    "mathematical proof",
    "math.GR",
    "math.RA"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-21-76/",
  "pdf_url": "https://eulersolve.org/papers/kou-21-76/paper.pdf?v=dc31e2c1bace",
  "doi": "10.5281/zenodo.23004034",
  "zenodo_record_url": "https://zenodo.org/records/23004034",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "To our knowledge Problem 21.76 was first answered by Ya. N. Nuzhin (Sib. Math. J. 67 (2026) 840–845, Corollary 1). This note gives explicit examples with short proofs, including one-variable examples in characteristic 3, and a characterisation in characteristics 2 and 3; no priority is claimed for the answer to Problem 21.76.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}