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Kurosh's problem for division rings: are algebraic division rings locally finite?

Claims Theorem 1.1: there is a countable division ring $D$ of characteristic zero with center $F$ and $x,y\in D$ such that $D=F[x,y]=F(x,y)$ , $[D:F]=\infty$ , and every element of $D$ is algebraic over $F$ (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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Kurosh's problem for division rings: are algebraic division rings locally finite? · Kurosh problem for division rings

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VibeMathed record: Kurosh's problem for division rings: are algebraic division rings locally finite?

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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 $F$ , with $D=F[x,y]$ for two elements and $[D:F]=\infty$ . Since $F$ 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.