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A Counterexample to Kurosh’s Division-Ring Problem
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Abstract
We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- AI Proved What?, Every element tame, yet infinite? Kurosh’s division-ring problem (Episode 201, status checked 19 October 2026). Reports an uncatalogued Lean development in the release that derives the counterexample assuming the lifting theorem, with that theorem left unproved; the exposition states that its authors did not run the Lean check. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Bui Xuan Hai, Trinh Thanh Deo, Mai Hoang Bien, On subgroups in division rings of type 2 (arXiv:1007.0791, 2010; v2 2019). It defines division rings of "type 2" ($F(x,y)$ finite-dimensional for all $x,y$), says none is known that is not locally finite, and lists the cases where Kurosh's problem has a positive answer ($F$ uncountable, finite, or with only finite algebraic extensions); these bear on Theorem 1.1's two-generated example with a countable center.
- Jason P. Bell, Lance W. Small, Agata Smoktunowicz, Primitive algebraic algebras of polynomially bounded growth (arXiv:1011.4133, 18 November 2010). Over a countable field it constructs finitely generated (including two-generated), infinite-dimensional, primitive algebraic algebras, a comparison for Theorem 1.1's stronger requirement that the constructed algebra be a division ring.
Digests and expositions
- Gradient Damage, Kurosh’s Division Ring Problem: OpenAI Claims a Counterexample (video and accompanying description, 16 October 2026). Illustrates the claimed failure of local finiteness by comparing quaternion word spans with an expanding tree, explicitly describing the animation as an illustration rather than the paper’s construction or proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 201
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026,
author = {{OpenAI}},
title = {{A Counterexample to Kurosh's Division-Ring Problem}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026}},
year = {2026}
}Family 201: A counterexample to Kurosh’s division-ring problem
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.
This is the only paper in this family.