# A counterexample to Kurosh’s division-ring problem > A division ring made from two elements, each element algebraic, can still be infinite-dimensional over its center. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-201/ Last updated: 2026-10-14 **A division ring made from two elements, each element algebraic, can still be infinite-dimensional over its center.** **Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Algebra · **Entry:** 201 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A division ring is a number system where you can divide by anything nonzero, but multiplication need not commute. Kurosh asked in 1941 about such rings generated by finitely many elements, where every element solves a polynomial equation. Must they be finite in size over their center? This paper claims not. It builds a two-generated ring where every element solves some polynomial, yet the ring is infinite-dimensional over its center. ## At a glance - **First step:** 1941 (Alexander Kurosh, “Ringtheoretische Probleme, die mit dem Burnsideschen Problem über periodische Gruppen in Zusammenhang stehen”) - **Significance:** Major. If the proof holds, it answers a question Kurosh posed in 1941, which positive results had settled only under extra hypotheses. - **Trust:** Claimed, not checked - **Field:** Algebra (1 paper) ## The explanation Start with the quaternions, numbers like q = 1 + 2i + 3j. Here i² = j² = −1, and ij = −ji, so order matters. Square q. The cross terms 6ij and 6ji cancel, and q² = −12 + 4i + 6j. Then q² − 2q = −14. So q solves the equation q² − 2q + 14 = 0, whose coefficients are ordinary real numbers. Every quaternion solves such an equation. The 2 is twice the real part, and the 14 is 1 + 4 + 9. The real numbers are the center of the quaternions: the numbers that commute with everything. A ring is algebraic over its center when every element solves a nonzero polynomial with central coefficients. The quaternions are also small: every quaternion is built from 1, i, j and ij, so they have dimension 4 over the reals. A division ring is a number system where every nonzero element can be divided by, though multiplication need not commute. The Kurosh problem asks whether algebraic over the center forces local finiteness[1]. Locally finite means anything generated by finitely many elements has finite dimension over the center. Alexander Kurosh asked this in 1941, inspired by Burnside's problem about groups[2]. Positive results came under extra conditions. If one degree bound works for every element, the ring satisfies a polynomial identity, and theorems of Jacobson and Kaplansky then give local finiteness[3]. Bell, Drensky and Sharifi sharpened this to dimension at most d² over the center[4]. Golod built infinite-dimensional algebras of a related kind in 1964. But they contain nonzero elements whose powers vanish, so they are not division rings[5]. This paper gives a negative answer[6]. It builds a countable division ring of characteristic zero, generated over its center by two elements x and y[7]. Every element is algebraic over the center, yet the ring has infinite dimension over it[7]. The trick is that each element may need its own polynomial, of growing degree[8]. That is exactly what the uniform-degree results forbid. The construction is a long iteration. Countability matters here: it lets the paper list every polynomial expression in x and y and treat them one at a time[9]. At each stage a lifting theorem makes the next listed expression algebraic over a larger central field[10]. The same step enlarges the division algebra and keeps every earlier equation true[11]. A limiting construction, called an ultraproduct, then keeps all these equations at once[12]. The one obstacle is keeping the process alive. Each step must leave two independent directions free and keep x and y generating everything[10]. Otherwise the next step would have nothing to work with. Most of the paper proves that these properties survive every step, using matrix calculations and formal power series. A last argument shows the dimension is infinite over the true center, which the construction does not fix in advance[13]. How sure is this? No Lean formalization is listed for this paper. The claim rests on the paper's own long argument. No specialist review of the paper has been published. **What it doesn't show.** This does not give a ring where one polynomial degree works for every element; that case was already known to force finite size. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major if the proof holds. No Lean check is listed and no specialist review is published. ## Words used - **Division ring:** A number system where every nonzero element can be divided by; multiplication need not commute. - **Center:** The elements of a ring that commute with every other element. - **Algebraic over the center:** Every element solves some nonzero polynomial equation with coefficients from the center. - **Locally finite:** Anything generated by finitely many elements has finite dimension over the center. - **Characteristic zero:** Adding 1 to itself never gives 0. - **Polynomial identity:** A fixed polynomial equation that holds for all choices of elements in the ring. ## The original paper - [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf) - September 23, 2026 ## Quoted lines from the paper (13) [1] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > The Kurosh problem for division rings asks whether algebraicity over the center implies local finiteness. [2] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > Kurosh formulated his question in 1941 in connection with Burnside's finiteness problem for groups. [3] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > Jacobson's bounded-degree theorem gives a polynomial identity, and Kaplansky's theorem makes algebraic algebras satisfying a polynomial identity locally finite; see. [4] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > Bell, Drensky, and Sharifi later proved that a division ring whose elements satisfy nonzero polynomial relations of uniformly bounded degree $d$, with coefficients on the left in a possibly noncentral subfield, has dimension at most $d^2$ over its center. [5] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > Their unitizations are algebraic algebras with nonzero nilpotents, so they do not answer the division-ring question. [6] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > We give a negative answer. [7] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > There is a countable division ring $D$ of characteristic zero, with center $F=Z(D)$, and elements $x,y\in D$ such that \[ D=F[x,y]=F(x,y),\qquad [D:F]=\infty, \] while every element of $D$ is algebraic over $F$. [8] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > The annihilating polynomial is allowed to depend on the element. [9] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > Countability plays a different role here: it allows us to schedule all polynomial expressions in the construction. [10] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > The main ingredient is Theorem …. [11] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > At the same time it increases the degree of the division algebra, preserves all previous polynomial identities, and restores both generation and the two independent directions. [12] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > A division-ring ultraproduct retains the resulting relations. [13] From [A Counterexample to Kurosh’s Division-Ring Problem](https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf): > The construction does not prescribe the center. ## Other sources - [A. Kurosch, “Ringtheoretische Probleme, die mit dem Burnsideschen Problem über periodische Gruppen in Zusammenhang stehen”, Izv. Akad. Nauk SSSR Ser. Mat. 5 (1941), 233–240](https://mathscinet.ams.org/mathscinet/relay-station?mr=5723) - [N. Jacobson, “Le problème de Kurosch”, Séminaire Bourbaki, Vol. 2 (1951–1954), exposé 62](https://www.numdam.org/item/SB_1951-1954__2__145_0/) ## Related - [A counterexample to Kaplansky’s zero-divisor conjecture](https://closedform.nishachay.tech/e/openai-math-2026-196/) - Algebra - [Eisenbud–Green–Harris and lex-plus-powers](https://closedform.nishachay.tech/e/openai-math-2026-200/) - Algebra - [The blockwise Alperin weight conjecture](https://closedform.nishachay.tech/e/openai-math-2026-202/) - Algebra - [Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures](https://closedform.nishachay.tech/e/openai-math-2026-199/) - Algebra - [Donovan's conjecture over fields and complete mixed-characteristic DVRs](https://closedform.nishachay.tech/e/openai-math-2026-203/) - Algebra - [Algebra](https://closedform.nishachay.tech/science/mathematics/algebra/) - Field · 18 results - [Open since the 1940s](https://closedform.nishachay.tech/open-since/1940s/) - Era · 12 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20201%5D%20)