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Math-2519 · mathematics · Algebra

Finite central dimension of algebraic finitely generated division rings

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If a division ring D is generated as an algebra by finitely many elements over its center F and every element of D is algebraic over F, must [D:F] be finite?

Short English statement

If a division ring D is generated as an algebra by finitely many elements over its center F and every element of D is algebraic over F, must [D:F] be finite?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2519-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2519

The theorem gives D=F[x,y], every element algebraic over the center F, and [D:F]=infinity. Its two algebra generators meet the finite generation hypothesis and refute the required finite-dimensional conclusion. Within these assumptions, this is a claimed negative answer to the stated question.

Finite central dimension of algebraic finitely generated division rings

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. An algebraic division ring is claimed to be generated as an algebra by two elements while having infinite dimension over its actual center.

Mathematical statement of the claimed result

Result 1.
There is a division ring $D$, countable as a set, with $\operatorname{char}D=0$ and $Z(D)=F$, generated by two elements over its center, such that \[ D=F[x,y]=F(x,y),\qquad[D:F]=\infty,\qquad \forall z\in D\ \exists0\ne f_z\in F[t]:\ f_z(z)=0. \] Algebraicity of each element therefore coexists with infinite central dimension in this claimed finitely generated example. The notation $F[x,y]$ here means the unital algebra generated by $x,y$ over the centre $F$, permitting noncommutative words.

Outline of the argument

A countable schedule imposes algebraic relations on successive polynomial expressions. Formal-series lifting inside finite-dimensional division-algebra models retains two independent deformation directions; trace constraints and conjugation calculations preserve generation, and a separate descent argument prevents the final center from making the dimension finite.

Formal verification

Family 201 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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