Kurosh's problem for division rings

Kurosh's problem for division rings asks whether a division ring in which every element is algebraic over the center must be locally finite, that is, finite-dimensional over the center on every finitely generated subring. A manuscript in OpenAI's public mathematics collection, dated 23 September 2026, claims a negative answer: a countable division ring of characteristic zero, algebraic over its center and generated by two elements, of infinite dimension over the center. The collection lists no Lean formalization of this result.

Background

• As recounted in the manuscript's introduction:
  • The question. Kurosh formulated it in 1941 in connection with Burnside's finiteness problem for groups, asking whether a division ring that is algebraic and finitely generated as an algebra over a central subfield must be finite-dimensional over it.
  • Algebraic algebras. Golod's construction, based on the Golod–Shafarevich estimates, gives finitely generated infinite-dimensional nil algebras; their unitizations are algebraic but have nonzero nilpotents, so they do not answer the division-ring question.
  • Positive results. Jacobson's bounded-degree theorem and Kaplansky's theorem make algebraic algebras of bounded degree locally finite, and Bell, Drensky and Sharifi proved that a division ring whose elements satisfy relations of uniformly bounded degree d has dimension at most d² over its center. Amitsur proved that a division algebra finitely generated over an uncountable central field is algebraic, which is a different conclusion from local finiteness.

The original result

• Claim, 23 September 2026. The main theorem of A Counterexample to Kurosh's Division-Ring Problem states that there are a countable division ring D of characteristic zero, with center F, and elements x, y with D = F[x, y] = F(x, y), such that D is infinite-dimensional over F while every element of D is algebraic over F. The same pair witnesses the failure of local finiteness.
• Scope, as stated: the construction does not prescribe the center; the annihilating polynomial depends on the element, with no uniform degree bound. The construction first produces an algebraic division ring over a countable central subfield, and a finite-descent argument then gives infinite dimension over the actual center.
• Method, as described: a lifting theorem makes one prescribed noncommutative polynomial algebraic in a finite-dimensional division algebra while preserving earlier relations, raising the degree and restoring generation by two elements; a countable schedule applies it to every polynomial, and the stages, which need not embed in one another, are glued by an ultraproduct.
• Sources. Commit adc7f12 of the openai/math repository: A Counterexample to Kurosh's Division-Ring Problem.
• Formalization. The repository's contents list gives no Lean scope note or Comparator challenge for this result. Its Lean library contains a folder lean/OAI/RingTheory/Kurosh of 20 files at this commit, not linked from the contents list, whose theorem lifting_implies_main derives the main statement from the lifting theorem taken as a hypothesis; the lifting theorem itself is not formalized there.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-16	J. Jardine, Lattice24, referee review rank_236 in JJardine919/openai-math-audit	Review, AI-assisted audit	Reports no gap. It read all twelve sections and checked in detail the application of the lifting theorem with the ultraproduct and the descent to the center, the generic spectra, the transfer, linkage and global defect steps, the cyclic obstruction, the eleven-block correction, the division realization and the generation step. It names as residual risk several formal-convergence and limit steps stated tersely, which it did not re-derive line by line.	Part of the completed Lattice24 audit; the reviews are published under CC BY 4.0.
2026-10-16	AI Proved What?, episode 201 Every element tame, yet infinite? Kurosh's division-ring problem, remade version of 24 October 2026	Reading, exposition	Video explainer of the 45-page preprint. It recalls Kurosh's 1941 question, Golod's 1964 examples with nilpotent elements and the bounded-degree theorems of Jacobson and Kaplansky, and states that the preprint does not claim anything in positive characteristic, an explicit example or a named center.	Episode status, last checked 30 October 2026: posted as a preprint, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, only the final step, in an uncatalogued folder of about 1,700 lines that assumes the 36-page lifting theorem, which the site did not run; no expert reaction found. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.
2026-10-30	kusasyu36/openai-math-audit, GitHub, commit a98b290, report audits/201	Review, AI-assisted	The report, in Japanese, checked by hand the derivation of the main theorem from the lifting theorem and the descent to a countable field, and checked the eleven-block correction step by exact computation modulo a prime in small cases, including a control experiment in which removing the backward-edge correction makes the system unsolvable. It lists several steps of the lifting theorem whose arguments it finds too brief, and records that an error it first reported about the starting field was withdrawn after a second review found that the source uses the algebraic closure of the rationals.	The repository README states that the audits were written by Anthropic's Claude acting as research agents under the direction of the repository owner, who is not a professional mathematician, and that no domain expert has reviewed the findings, which it asks to be treated as leads rather than verdicts. Apache-2.0. Read on 31 October 2026.

Notes

• The background summarizes the manuscript's own account of earlier work.
• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.