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

- ID: `P3102`
- Reference: `fp-laurent-series-theory-decidability`
- Page: https://theoremdb.org/statements/P3102
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (F_p((t))).** Formal Laurent series with coefficients in the finite field F_p.
- **Definition (decidable theory).** There is an algorithm deciding truth of every first-order sentence in the specified structure.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Conditional quantifier-elimination routes and decidability results for related extensions and languages are known. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Unconditional decidability in the pure ring language remains open in the checked sources.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `fp-laurent-series-theory-decidability`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://doi.org/10.2140/ant.2016.10.665
   - 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
   - journal_article; primary source; checked 2026-08-01
   - Open copy: https://arxiv.org/abs/1501.04522
   - Source use: original_summary
   - Proves unconditional decidability of the existential theory of F_q((t)), a strict fragment of the complete first-order target.
   - 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.
2. <a id="reference-2"></a>S. Anscombe, Decidability in extensions of F_p((t)), Oxford thesis (2021). abstract and conditional quantifier-elimination result https://ora.ox.ac.uk/objects/uuid:37283d4a-148f-48c3-8622-b70dcf791c71
   - thesis; primary source; checked 2026-08-01
   - Source 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)).