[#P3102] Decidability of the first-order theory of F_p((t))

Problem. For a fixed prime \(p\), is the complete first-order theory of the Laurent-series field \(\mathbb F_p((t))\) in the language of rings decidable?

First-order formulas over a Laurent-series field. — A structural automaton diagram of the statement's mathematical objects.

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1Context

Known frontier: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Open boundary: Unconditional decidability in the pure ring language remains open in the checked sources.

2Problem setup

Definition 1 (F_p((t))). Formal Laurent series with coefficients in the finite field F_p.

Definition 2 (decidable theory). There is an algorithm deciding truth of every first-order sentence in the specified structure.

Remark 1. A decision procedure must halt on every ring-language sentence and determine whether it holds in F_p((t)). The valued-field structure is highly organized, but positive characteristic creates wild additive phenomena.

3What counts as a solution

Give a terminating correct decision algorithm for all ring-language sentences.
Or prove the theory undecidable by an effective interpretation or reduction.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Exact unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Exact unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Exact unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Decidability of the first-order theory of F_p((t)). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. The exact unresolved remainder is: Unconditional decidability in the pure ring language remains open in the checked sources. A complete resolution must meet the following acceptance conditions: - Give a terminating correct decision algorithm for all ring-language sentences. - Or prove the theory undecidable by an effective interpretation or reduction. Open full record →
Conditional quantifier-elimination routes and decidability results for related extensions and languages are known.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Conditional quantifier-elimination routes and decidability results for related extensions and languages are known.^[1][2] Relevance to this problem Locates the present research frontier immediately below Decidability of the first-order theory of F_p((t)). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. This leaves the following boundary unresolved: Unconditional decidability in the pure ring language remains open in the checked sources. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.^[1][2] Relevance to this problem Documents why Decidability of the first-order theory of F_p((t)) was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - decidability first order theory F_p((t)) open problem 2026 - quantifier elimination Laurent series field positive characteristic decidability The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources. Related: Claim 1 Open full record →
Unconditional decidability in the pure ring language remains open in the checked sources.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Unconditional decidability in the pure ring language remains open in the checked sources. Relevance to this problem Turns the remaining uncertainty in Decidability of the first-order theory of F_p((t)) into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Unconditional decidability in the pure ring language remains open in the checked sources. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Give a terminating correct decision algorithm for all ring-language sentences. - Or prove the theory undecidable by an effective interpretation or reduction. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. Exact unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.

Equivalent-formulation queries: decidability first order theory F_p((t)) open problem 2026; quantifier elimination Laurent series field positive characteristic decidability
Strongest checked neighboring result: Conditional quantifier-elimination routes and decidability results for related extensions and languages are known.
Exact unresolved remainder: Unconditional decidability in the pure ring language remains open in the checked sources.

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ProblemDecidability of the first-order theory of F_p((t))

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Plain text

“Decidability of the first-order theory of F_p((t)).” TheoremDB. P3102. Problem statement; statement text SHA-256 988bdaa67f4aee136645e1f6e179a7c10a5874efb70c770a76170218ff5fa426. https://theoremdb.org/statement/?ref=P3102

BibTeX

@misc{theoremdb-problem-988bdaa67f4aee136645e1f6e179a7c10a5874efb70c770a76170218ff5fa426,
  title = {{Decidability of the first-order theory of F\_p((t))}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 988bdaa67f4aee136645e1f6e179a7c10a5874efb70c770a76170218ff5fa426},
  url = {https://theoremdb.org/statement/?ref=P3102}
}

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This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Sylvy Anscombe and Arno Fehm, “The existential theory of equicharacteristic henselian valued fields”. Algebra & Number Theory 10(3) (2016), 665-683. DOI 10.2140/ant.2016.10.665. abstract and main existential Ax–Kochen–Ershov theorem. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves unconditional decidability of the existential theory of F_q((t)), a strict fragment of the complete first-order target.Also cited at S. Anscombe and A. Fehm, “The existential theory of equicharacteristic henselian valued fields,” Algebra & Number Theory 10(3) (2016), 665–683. abstract and main existential Ax–Kochen–Ershov theorem.Source used to assess the problem's recorded status.For Decidability of the first-order theory of F_p((t)): This is the dated publication status for the canonical target Decidability of the first-order theory of F_p((t)).Source named by the research packet.
S. Anscombe, Decidability in extensions of F_p((t)), Oxford thesis (2021). abstract and conditional quantifier-elimination result. ↗thesis · primary source · checked 2026-08-01Source use: original summary.Records a conditional route to quantifier elimination and decidability for F_p((t)).Source used to assess the problem's recorded status.For Decidability of the first-order theory of F_p((t)): Records a conditional route to quantifier elimination and decidability for F_p((t)).

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3102] Decidability of the first-order theory of F_p((t))

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