Algebra·OpenAI Math·Result 201
A counterexample to Kurosh’s division-ring problem
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebra: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a counterexample to Kurosh’s division-ring problem (algebra) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Kurosh's Division Ring Problem: OpenAI Claims a Counterexample, is on YouTube.
What OpenAI claims
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.
Paper
A Counterexample to Kurosh’s Division-Ring Problem
OpenAI's 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.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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.
- Where is the proof?
- In the paper, A Counterexample to Kurosh’s Division-Ring Problem (PDF).
- Is there a video?
- Yes, on YouTube (0:21).
Also in
- Chapter 39 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 13:09 · watch from there on YouTube →
The video
Kurosh's Division Ring Problem: OpenAI Claims a Counterexample · Short · 0:21 · description, sources and transcript →
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