Result 201, Algebra

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.

Disproof or counterexample

The bigger picture

Why it matters

Satisfying a polynomial equation can make an individual element look finite, but it need not make a whole algebra small. The manuscript reports a counterexample showing this gap even when just two elements generate everything.

What changes?

A division ring is a number system with division by every nonzero element, but multiplication need not commute. Its center consists of elements commuting with everything. The reported example is countable and has characteristic zero, so no positive multiple of one is zero. Every element satisfies a nonzero polynomial equation with coefficients in the center. Yet two elements generate the ring as an algebra over that center, and its vector-space dimension over the center is infinite.

What does that help mathematicians do?

Kurosh's problem asks whether being algebraic over the center forces local finiteness: every subalgebra generated by finitely many elements over the center would have finite dimension. Here, the two generators already produce an infinite-dimensional algebra. The claimed example therefore rules out that implication, showing that polynomial relations for individual elements do not ensure finite-dimensional control when those elements are combined.

Are there practical applications?

Its immediate value is foundational: it marks a boundary for reducing questions about division rings to finite-dimensional algebra. Researchers seeking such reductions would need assumptions beyond algebraicity over the center. Neither restricting to characteristic zero nor requiring just two algebra generators suffices, according to the reported construction.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Counterexample to Kurosh’s Division-Ring Problem

September 23, 2026 45 pages

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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Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.