Algebraic number theory · explicit class field theory · special values

Hilbert’s Twelfth Problem

Collaboration beta

Can every finite abelian extension of every number field be generated effectively by explicitly prescribed algebraic values of analytic or automorphic functions, with the Artin reciprocity action visible on those values?

K_{m} = K (u_{j} (b)), u (b)^{Art (a)} = u (a^{- 1} b)

Hilbert’s 23 problems — Problem 12

Known results and sources

Hilbert’s twelfth problem asks for effective analytic generators of every abelian extension, with algebraicity and reciprocity visible; the bridge remains open in general.

Research problem

Exact mathematical statement

Let $K$ be any number field and $\mathfrak m$ any ray modulus, with ray class field $K_{\mathfrak m}/K$ . Give a finite effective procedure that specifies analytic or automorphic functions and special arguments whose normalized values or prescribed ratios are algebraic in $K_{\mathfrak m}$ , transform under ray classes by the Artin reciprocity law, and generate the full ray class field. Equivalently, require

K_{\mathfrak m}=K\bigl(u_j(\mathfrak b)\bigr),\qquad u(\mathfrak b)^{\operatorname{Art}(\mathfrak a)}=u(\mathfrak a^{-1}\mathfrak b),

with every modulus, auxiliary ideal, prime, unit basis, cell, smoothing datum, function, normalization, and certified recognition bound computable from $(K,\mathfrak m)$ . Since every finite abelian extension is contained in a ray class field, the construction must cover arbitrary finite abelian extensions, not merely one selected field.

Problem infographic

Problem at a glance

Three-zone diagram of number-field input, analytic special values, and a finite abelian ray class field separated by an explicitly open general bridge. — The required construction must turn finite arithmetic input into algebraic special values that generate the ray class field over the base field and transform by the Artin action; special cases do not supply the general bridge.

Current mathematical picture

Where work on Hilbert’s Twelfth Problem stands

Open problem

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: The statement that the finite local target is completely frozen is too strong when read analytically: the norm-19 factor is only a formal candidate and even the exact rank-one norm-73 factor still lacks source-chain realization. The immediate bridge is to prove the gamma/cone and smoothing realizations together with projective algebraicity, finite-divisor cancellation, Artin covariance, and the real higher Kronecker-limit identity; the general problem additionally requires arbitrary-degree construction and descent.

Route status · Narrowed route

Main reductionConditional unit-generation theorem

Assuming four unresolved projective-column and determinant hypotheses, the current work formally derives norm-one units that generate the selected extension and have subgroup index 20.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeClose the source-chain realization and projective-column theorem.Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Hilbert’s Twelfth Problem in numbers

4.1kretained lines of mathematical investigation4,060 in the current working snapshot

Argument development: 3,347 · 82%
Explored or eliminated routes: 76 · 2%
Computational analysis: 155 · 4%
Open obligations: 156 · 4%
Definitions and setup: 326 · 8%

8selected mapped statements1routes investigated3open questions3contribution-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 HilbertProposed1900Status checkedAug 14, 2026

Source recordedAug 14, 2026

Recommended next task

Close the source-chain realization and projective-column theorem.

Suggested move: Audit the cited gamma-to-cone, smoothing, coefficient-module, boundary, and equivariance conventions, then construct two nonzero projective columns with algebraicity, divisor cancellation, and Artin swapping proved.

What would count as progress

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

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

12 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 routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Close the source-chain realization and projective-column theorem.Suggested move: Audit the cited gamma-to-cone, smoothing, coefficient-module, boundary, and equivariance conventions, then construct two nonzero projective columns with algebraicity, divisor cancellation, and Artin swapping proved.

Ready to work on

Prove the real higher Kronecker-limit determinant identity.Suggested move: Construct one normalized character-twisted Eisenstein two-form and compute its torus and gamma-boundary periods with all local factors, orientations, and universal constants matched exactly.

Ready to work on

Globalize beyond the selected quadratic quartic ray extension.Suggested move: Develop arbitrary-degree norm-one cycles, coherent conductor-adapted cells, all character columns, descent, and an effective algorithm with certified error and height bounds.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026

Current statusOpen problem

Hilbert’s twelfth problem remains unresolved in general. The current authoritative survey reports that only a few relevant extensions are completely understood; modern Stark-type advances do not supply the requested construction for arbitrary number fields.

[2]

External progress

What the literature has established

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

2026
Authoritative summaryA current Simons Foundation survey lists Hilbert’s twelfth problem as unresolved and says only a few of the relevant extensions are completely understood.[2]
2023
Peer reviewedDasgupta and Kakde proved the Brumer–Stark conjecture away from 2 and derived Rubin’s higher-rank version away from 2, a related explicit-class-field advance rather than the general Hilbert problem.[3]
1900–1902
Historical sourceHilbert presented the problem in 1900; the English translation appeared in the Bulletin of the American Mathematical Society in 1902.[1]

Mathematical neighborhood

Related results and reusable starting points

Current focusHilbert’s twelfth problem

Related problemBrumer–Stark and higher-rank Brumer–Stark conjectures

Modern Stark-type results concern special values, units, and abelian extensions in important scopes, but the cited theorem does not supply a general effective analytic construction of every ray class field.

[3]

Formalization opportunities

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

Formalization targetAn exact formal statement must encode number fields, ray moduli, ray class groups and fields, Artin reciprocity, algebraicity, field generation, and the intended finite effectivity contract.
Formalization targetA proof-level formalization would require substantial analytic or automorphic function theory and certified special-value recognition, not merely abstract class field theory.
Formalization targetNo scoped public problem-level formalization was identified.

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 removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

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 statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
reduction3 of 83
lemma2 of 82
negative result2 of 82

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included

retained route statementExplicit analytic values should generate every ray class field and realize Artin reciprocity effectively
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementSelected quartic ray fieldintermediate
retained route statementRank-one norm-73 quotientintermediate
retained route statementCandidate factor remains conditionalintermediate
retained route statementConditional unit-generation theoremintermediate
retained route statementSelected model does not globalizeintermediate
Recorded relationshipThe source 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 source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSource-reported limitationreported failure
Research targetClose the source-chain realization and projective-column theorem.open
Research targetProve the real higher Kronecker-limit determinant identity.open
Research targetGlobalize beyond the selected quadratic quartic ray extension.open
Research targetArbitrary-degree globalizationsuperseded
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: The statement that the finite local target is completely frozen is too strong when read analytically: the norm-19 factor is only a formal candidate and even the exact rank-one norm-73 factor still lacks source-chain realization. The immediate bridge is to prove the gamma/cone and smoothing realizations together with projective algebraicity, finite-divisor cancellation, Artin covariance, and the real higher Kronecker-limit identity; the general problem additionally requires arbitrary-degree construction and descent.

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 bridgeClose the source-chain realization and projective-column theorem.

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 pointClose the source-chain realization and projective-column theorem.

Hilbert’s Twelfth Problem · ready to start

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Research contextPrepared context for any AI agent

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Current routes and known obstacles
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

1
Mathematical Problemsoriginal source · David Hilbert · Bulletin of the American Mathematical Society · 1900 address; English translation 1902 · DOI 10.1090/S0002-9904-1902-00923-3 · accessed Aug 14, 2026
2
Solved, Unsolved and Unsolvable: The Status of Hilbert’s 23 Problems in Mathematicsauthoritative webpage · Evelyn Lamb · Simons Foundation · 2026-06-18 · accessed Aug 14, 2026
3
On the Brumer–Stark conjecturepeer reviewed result · Samit Dasgupta, Mahesh Kakde · Annals of Mathematics 197(1), 289–388 · 2023 · DOI 10.4007/annals.2023.197.1.5 · MR MR4513146 · accessed Aug 14, 2026

Important qualifications

Hilbert’s historical formulation is broader and less contract-like than the workspace’s ray-class-field effectivity conventions; the exact workspace statement remains recorded separately.
The Simons Foundation status page is an authoritative current survey, not a peer-reviewed theorem.
The Dasgupta–Kakde Brumer–Stark theorem is a related advance and is not a solution of Hilbert’s twelfth problem.
The scoped search found no public problem-level formal statement or proof; this does not establish nonexistence across all proof assistants or repositories.
The submitted packet and its URLs were excluded from external authority and were not fetched.

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