Kurosh's problem for division rings: are algebraic division rings locally finite?
A division ring with center is algebraic over if every element satisfies a nonzero polynomial over , and locally finite if every finite subset generates a division subring of finite dimension over . In 1941, in connection with Burnside's problem, Kurosh asked whether a division ring that is algebraic and finitely generated as an algebra over a central subfield must be finite-dimensional over it. Golod's nil algebras answer the analogous question for general algebras but contain nilpotents, so they do not touch division rings; Jacobson and Kaplansky settled the bounded-degree and polynomial-identity cases. Is every division ring that is algebraic over its center locally finite?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Noncommutative ring theory; division algebras
- Posed by
- A. G. Kurosh, Ringtheoretische Probleme, die mit dem Burnsideschen Problem uber periodische Gruppen in Zusammenhang stehen (1941), Problem (K)
- Year posed
- 1941
- Years open
- 85y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 48 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Theorem 1.1: there is a countable division ring of characteristic zero with center and such that , , and every element of is algebraic over (with no uniform degree bound). So algebraicity over the center does not imply local finiteness. It does NOT give an example in positive characteristic or over a prescribed center, and polynomial degrees grow without bound, as they must by Jacobson's theorem.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. The construction is a countable inductive scheme using symbol and cyclic algebras, formal series in two noncommuting variables and a lifting theorem that makes each scheduled element algebraic.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against Kurosh's Problem (K): a countable division ring of characteristic zero, algebraic over its center , with for two elements and . Since is central, this is a negative answer to the problem as Kurosh posed it (over a central subfield). No Lean formalisation exists for this family. The release README warns that some unformalised results could have issues.