Short··0:21·Algebra
Kurosh's Division Ring Problem: OpenAI Claims a Counterexample
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About this video
In 1941 Kurosh asked whether an algebra that is finitely generated, and in which every element satisfies a polynomial over the centre, must be finite-dimensional. Golod and Shafarevich showed in 1964 that the answer is no for algebras in general, but the question stayed open for division rings, where every nonzero element can be divided by. OpenAI claims a counterexample: a countable division ring of characteristic zero, generated by two elements over its centre, algebraic over that centre, yet infinite-dimensional over it. This is OpenAI's preprint, not yet peer reviewed. The animation first shows the finite case, the quaternions: every word in i and j lands on one of 8 points (drawn as the 16-cell in four dimensions), so the span is only 4-dimensional. The branching tree that never collapses is an illustration of OpenAI's claim, not its construction.
Paper (OpenAI, catalogue result 201):
- A counterexample to Kurosh's division-ring problem: github.com/openai/math/blob/main/preprints/A-Counterexample-t…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/201
About the problem
Algebra·OpenAI Math·Result 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.
Topics
Kurosh problemdivision ringskew fieldalgebraic algebralocally finitenoncommutative algebraring theoryquaternionsquaternion group16-cellGolod-Shafarevichabstract algebraOpenAIOpenAI mathAI mathematicsAI proofmathsmath shorts3D animation
