Galois theory · arithmetic geometry · finite groups · embedding problems

Inverse Galois Problem over ℚ

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Does every finite abstract symmetry group occur as the Galois group of some finite Galois extension of the rational numbers?

\forall G finite, \exists L / ℚ finite Galois : Gal (L / ℚ) ≅ G

Listed inFrontierMath: Open Problems — Inverse Galois problem for M23

Known results and sources

The inverse Galois problem asks whether every finite symmetry group can be realized exactly by the automorphisms of a finite Galois extension of the rational numbers; the general question remains open.

Research problem

Exact mathematical statement

For every finite group $G$ , there should exist a finite Galois extension $L/\mathbb Q$ whose group of field automorphisms over $\mathbb Q$ is isomorphic to $G$ :

\forall\,G\text{ finite},\quad \exists\,L/\mathbb Q\text{ finite Galois such that }\operatorname{Gal}(L/\mathbb Q)\cong G.

Equivalently, every finite abstract group should occur as the full symmetry group of a finite Galois extension of the rational numbers. The retained source explicitly reports that the general problem remains unresolved; its finite-group, module-cohomology, and local-lifting results do not by themselves supply the required global realization.

Problem infographic

Problem at a glance

A problem-first scientific infographic defines a finite group G, a finite Galois extension L over the rational numbers, and the target isomorphism Gal of L over Q congruent to G; examples of finite symmetries feed into a field tower, while a separate lower band marks known local constructions and the unresolved global realization bridge. — Finite groups encode abstract symmetry, while a Galois group records the rational-field automorphisms of a splitting field. The challenge is to realize every finite group, with local and family-specific advances still short of a general construction.

Current mathematical picture

Where work on Inverse Galois Problem over ℚ stands

Open problem

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureUnrestricted CC(p) refuted; restricted route conditional

Unrestricted controlled central lifting CC(p) is false even for a split extension because the desired twist encounters a Poitou–Tate reciprocity obstruction. The exact split-prime patching criterion and cyclotomic restricted lifting route remain viable, but their global use is still conditional on the current work's stated Poitou–Tate and Hasse inputs.

Route status · Narrowed route

Main reductionChief-factor reduction has two branches

A minimal normal subgroup is characteristically simple, so the current work reduces a lifting step to either a power of a nonabelian finite simple group or an elementary abelian p-group.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeIdentify and verify an exact publication-grade ODD-REL-core theorem, including every fixed-quotient, surjectivity, local-condition, and properness hypothesis used by ODD-SN.Task status · Work already reported in progress

Research-record correctionResearch-record correction

We corrected the cited passages. We clarified what the cited material supports. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Inverse Galois Problem over ℚ in numbers

2.8kretained lines of mathematical investigation2,830 in the current working snapshot

Argument development: 2,345 · 83%
Explored or eliminated routes: 83 · 3%
Computational analysis: 10 · 0%
Open obligations: 124 · 4%
Definitions and setup: 268 · 9%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byDavid Hilbert (specialization method and symmetric/alternating realizations), Emmy Noether (prescribed-group invariant-field formulation)Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove a detector-preserving S_n specialization theorem realizing the embedded inertia and Frobenius data of Section 29.5 while keeping the full group and protected conditions.

Suggested move: Formulate the specialization target with the exact cycle types, generator ordering, residue-class modulus, full-group condition, split places, and disjointness before attempting a Hilbert argument.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeUnrestricted CC(p) refuted; restricted route conditional

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove a detector-preserving S_n specialization theorem realizing the embedded inertia and Frobenius data of Section 29.5 while keeping the full group and protected conditions.Suggested move: Formulate the specialization target with the exact cycle types, generator ordering, residue-class modulus, full-group condition, split places, and disjointness before attempting a Hilbert argument.

Ready to work on

Integrate the specialization with exact modified-real Poitou–Tate duality and a properness argument for the deleted, co-deleted, and heart module embedding problems.Suggested move: Pin the finite-module duality statement including real-place conventions and local conditions, then trace how exclusion of the exceptional dual class yields a weak solution and how the chosen place forces surjectivity.

Ready to work on

Identify and verify an exact publication-grade ODD-REL-core theorem, including every fixed-quotient, surjectivity, local-condition, and properness hypothesis used by ODD-SN.Suggested move: Write a hypothesis-by-hypothesis theorem map for the relative solvable-kernel step and stop at the first assumption not supplied by the constructed S_n quotient or local weak lifts.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen problem

Open in full generality: it is not known whether every finite group occurs as the Galois group of a finite Galois extension of Q. Positive results include all finite solvable groups, all symmetric and alternating groups, 25 of the 26 sporadic simple groups, and multiple Lie-type families. M23 remains the final sporadic simple case, but realizing M23 would not resolve the universal problem.

[4][9]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2022–2026 status
Authoritative summaryAuthoritative current summaries retain the universal problem as wide open. Among sporadic simple groups M23 remains the single unrealized case, while the universal quantifier also ranges far beyond sporadic…[4][9]
2000
Peer reviewedKluners and Malle constructed regular Q(t)-realizations for every transitive group of degree at most 15 and explicit number-field polynomials. This is finite-degree coverage, not a bound reducing the…[6]
1954; complete proof 1998
Peer reviewedShafarevich proved that every finite solvable group occurs as a Galois group over Q. Schmidt and Wingberg supplied a complete proof addressing the historical proof gap.[5][4]
1917
Historical sourceNoether's prescribed-group paper developed the invariant-field rationality route. Rationality of the relevant invariant field supplies a realization, but this requirement is stronger than inverse-Galois…[2][3]

Mathematical neighborhood

Related results and reusable starting points

Current focusInverse Galois problem over Q

Stronger or generalized formRegular inverse Galois problem over Q(t)

A regular Galois realization over Q(t) specializes to a realization over Q by Hilbert irreducibility. The converse is not known in this generality, so regular inverse Galois is the stronger problem.

[3][4]

Dependency or reductionNoether's problem

A positive solution to Noether's rationality problem for a given finite group supplies an inverse-Galois realization, but rationality is not necessary and fails for some groups.

[2][3]

Solved special caseFinite solvable groups

Every finite solvable group is known to occur as a Galois group over Q.

[5]

Related problemExplicit M23 realization over Q

The M23 task asks for one degree-23 polynomial over Q with Galois group M23. Solving it would complete the sporadic-simple-group family but would not settle the universal problem.

[9][4]

Related problemGrunwald and finite embedding problems

Prescribing compatible local behavior introduces Grunwald and embedding-problem constraints beyond mere realization of the abstract finite group.

[3][4]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal statement · statement onlyFormal Conjectures: InverseGalois.lean
At exact repository commit c594af4ba42f58465253b8550545e0132959a78c, the file states the universal Q-realizability theorem but closes it with sorry; its solved-variant declarations are also placeholders rather than checked proofs.
[10]
formal library support · source linked; not reproduced by ProofAtlasMathlib Galois theory
Mathlib defines finite Galois extensions, fixed fields, and the Galois correspondence. This is prerequisite library support and does not prove universal finite-group realizability over Q.
[11]
dataset · not independently reproducedGaloisDB
The maintained database provides defining polynomials and group data for broad finite-degree ranges and reports all transitive groups in its supported range except M23; ProofAtlas did not independently recompute the entries.
[7]
software · not independently reproducedDokchitser inverse-Galois Magma resources
The Bristol page makes explicit families and Magma tools available for constructing and studying Galois groups; this collection pass did not execute or certify them.
[8]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA reusable formal bridge from abstract finite groups to automorphism groups of explicit finite extensions of Q.
Formalization targetFormal Hilbert irreducibility and specialization machinery strong enough to transport regular Q(t)-realizations to Q.
Formalization targetFormal infrastructure for generic polynomials, invariant fields, resolvents, and certified Galois-group computation.
Formalization targetChecked end-to-end realizations for the major known families, with exact coefficient, regularity, and specialization hypotheses.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We clarified what the cited material supports. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected supporting details in the research record. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements3 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma5 of 95

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementEvery finite group should arise as an exact symmetry group of a finite Galois extension of ℚ.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementChief-factor reduction has two branchesintermediate
retained route statementInterior subset hearts have an explicit finite obstruction packageintermediate
retained route statementCarry detectors lift through every corresponding extensionintermediate
retained route statementOdd kernels have an exact local weak liftintermediate
retained route statementA split-fiber symmetric cover is constructedintermediate
retained route statementEmbedded detector specialization remains openintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetIdentify and verify an exact publication-grade ODD-REL-core theorem, including every fixed-quotient, surjectivity, local-condition, and properness hypothesis used by ODD-SN.in progress reported
Research targetProve a detector-preserving S_n specialization theorem realizing the embedded inertia and Frobenius data of Section 29.5 while keeping the full group and protected conditions.open
Research targetIntegrate the specialization with exact modified-real Poitou–Tate duality and a properness argument for the deleted, co-deleted, and heart module embedding problems.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureUnrestricted controlled central lifting CC(p)reported failure
ComputationSource-reported normalized-bar computations for the minimum Klein-four detector restrictions.The source reports dimensions 2 for s=1 and 8 for s=3, with zero normalizer-fixed part in both cases. These checks support vanishing of the restricted classes; they do not assert H^2(U,E)=0. · reported unreproduced
Narrowed routeUnrestricted CC(p) refuted; restricted route conditionalUnrestricted controlled central lifting CC(p) is false even for a split extension because the desired twist encounters a Poitou–Tate reciprocity obstruction. The exact split-prime patching criterion and cyclotomic restricted lifting route remain viable, but their global use is still conditional on the current work's stated Poitou–Tate and Hasse inputs.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeIdentify and verify an exact publication-grade ODD-REL-core theorem, including every fixed-quotient, surjectivity, local-condition, and properness hypothesis used by ODD-SN.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointProve a detector-preserving S_n specialization theorem realizing the embedded inertia and Frobenius data of Section 29.5 while keeping the full group and protected conditions.

Inverse Galois Problem over ℚ · ready to start

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Research contextPrepared context for any AI agent

Does every finite abstract symmetry group occur as the Galois group of some finite Galois extension of the rational numbers?

Exact question and boundaries
Current routes and known obstacles
What a useful result should report

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Sources and references12 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficientenoriginal source · David Hilbert · Journal für die reine und angewandte Mathematik · 1892 · DOI 10.1515/crll.1892.110.104 · accessed Aug 7, 2026
2
Gleichungen mit vorgeschriebener Gruppeoriginal source · Emmy Noether · Mathematische Annalen · 1917 · DOI 10.1007/BF01457099 · accessed Aug 7, 2026
3
Inverse Galois Theorysurvey or monograph · Gunter Malle, B. Heinrich Matzat · Springer · 1999 · DOI 10.1007/978-3-662-12123-8 · accessed Aug 7, 2026
4
Around the Inverse Galois Problemsurvey or monograph · Olivier Wittenberg · Institute for Advanced Study / Park City Mathematics Institute · 2022 · accessed Aug 7, 2026
5
Šafarevič's theorem on solvable groups as Galois groupspeer reviewed result · Alexander Schmidt, Kay Wingberg · Journal of Algebra · 2000 · ARXIV math/9809211 · accessed Aug 7, 2026
6
Explicit Galois Realization of Transitive Groups of Degree up to 15peer reviewed result · Jürgen Klüners, Gunter Malle · Journal of Symbolic Computation · 2000 · DOI 10.1006/jsco.2000.0378 · accessed Aug 7, 2026
7
GaloisDBsoftware or dataset · Paderborn University · accessed Aug 7, 2026
8
Inverse Galois Problem: course and Magma resourcessoftware or dataset · Tim Dokchitser · University of Bristol · accessed Aug 7, 2026
9
The Inverse Galois Problem for the Mathieu Group M23maintained problem list · Epoch AI · 2026 · accessed Aug 7, 2026
10
FormalConjectures/Wikipedia/InverseGalois.lean at c594af4ba42f58465253b8550545e0132959a78cformalization · The Formal Conjectures Authors · Google DeepMind · accessed Aug 7, 2026
11
Mathlib.FieldTheory.Galois.Basicformalization · The Mathlib Contributors · Mathlib · accessed Aug 7, 2026
12
Inverse Galois problemencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

The record covers the universal realization problem for finite groups over the rational numbers. It does not identify that statement with the stronger regular problem over Q(t), Noether's rationality problem, Grunwald problems, or one explicit-polynomial construction task.
The proposed year is null because the modern universal problem emerged through Hilbert's 1892 specialization work and Noether's 1917 prescribed-group and invariant-field formulation rather than one uncontested proposal event.
The list of known groups and milestones is representative, not a complete realization bibliography; many further families and refinements concerning ramification or local conditions are omitted.
The current M23 FrontierMath task is one construction instance inside the universal problem. Its open status does not imply that M23 is the only obstruction to the universal problem.
GaloisDB and the Bristol Magma resources were inspected as public computational resources, but their calculations were not rerun or independently certified in this collection pass.
The scoped formalization search covered the exact current Formal Conjectures tree and current Mathlib documentation. It does not establish absence from all proof assistants, branches, or private developments.
No unreviewed source material, contributor claim, packet computation, or unpublished submission was inspected or used as external authority. This record grants no proof, novelty, review, acceptance, or publication authority.

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