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Math-2197 · mathematics · Number theory

Hilbert's Tenth Problem for Number Fields

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Statement

Definitions and mathematical statement

For a number field $K$, does an algorithm exist that takes an arbitrary multivariable polynomial with coefficients in $K$ and decides whether it has a zero in $K$? Determine for which number fields such a decision procedure exists. Solutions are sought in the field $K$ itself.

Short English statement

For a number field $K$, does an algorithm exist that takes an arbitrary multivariable polynomial with coefficients in $K$ and decides whether it has a zero in $K$? Determine for which number fields such a decision procedure exists. Solutions are sought in the field $K$ itself.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2197-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2197

Undecidability over the rational field settles that one field, but this question requests a classification or a wider number-field assertion. No conclusion for every other number field follows from the rational reduction alone.

Hilbert's Tenth Problem for Number Fields

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 004; 2026-09-24), the asserted result is that rational solvability of integer-coefficient polynomials with an unrestricted number of variables is undecidable.

Mathematical statement of the claimed result

Manuscript 1.
With $n$ variable and integral polynomial coefficients explicitly encoded, the rational Diophantine decision claim is \[ \{f\in\mathbb Z[X_1,\ldots,X_n]:\exists x\in\mathbb Q^n,\ f(x)=0\} \quad\text{is undecidable}. \] There is no algorithm giving the correct yes/no answer for every such input; the coefficient ring is $\mathbb Z$, but sought solutions lie in $\mathbb Q$.

Outline of the argument

For manuscript 1 (family 004; 2026-09-24), integer solvability is represented by an effective sequence of finite rational tests. Elliptic indices, odd valuation contacts, and a height argument force an ordinary integer witness when every test succeeds; a hypothetical rational decision procedure would then decide the integer problem by two concurrent searches.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 004; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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