Hilbert’s fourteenth problem
E208850
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hilbert’s fourteenth problem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1859188 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s fourteenth problem Context triple: [Hilbert problems, hasPart, Hilbert’s fourteenth problem]
-
A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
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C.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
D.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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E.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s fourteenth problem Target entity description: Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
D.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
-
E.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
Hilbert problem
ⓘ
mathematical problem ⓘ |
| asksWhether | certain algebras of invariants are finitely generated ⓘ |
| concerns |
finite generation of algebras of invariants
ⓘ
finite generation of subalgebras of polynomial rings ⓘ rings of invariants under group actions ⓘ |
| counterexampleInvolves |
action of an algebraic group on affine space
ⓘ
non-finitely generated ring of invariants ⓘ |
| difficulty | open in many specific cases despite general counterexamples ⓘ |
| falseFor |
certain additive group actions
ⓘ
some non-reductive algebraic groups ⓘ |
| field |
algebraic geometry
ⓘ
invariant theory ⓘ |
| hasAlternativeFormulation | finite generation of intersections of polynomial rings with subfields of their fraction fields ⓘ |
| hasCounterexampleProvidedBy | Masayoshi Nagata ⓘ |
| hasCounterexampleYear | 1958 ⓘ |
| hasModernFormulation | finite generation of k[V]^G for algebraic group G acting on affine variety V ⓘ |
| influenced |
development of modern invariant theory
ⓘ
research on finite generation of rings ⓘ study of algebraic transformation groups ⓘ |
| languageOfOriginalStatement | German ⓘ |
| numberInHilbertList | 14 ⓘ |
| originalFormulationContext | invariant theory of algebraic transformation groups ⓘ |
| partOf |
Hilbert’s list of 23 problems
ⓘ
linked to:
Hilbert problems
|
| posedAt |
International Congress of Mathematicians in Paris
ⓘ
linked to:
International Congress of Mathematicians
|
| posedIn | 1900 ⓘ |
| relatedConcept |
finitely generated k-algebra
ⓘ
polynomial ring over a field ⓘ rational function field ⓘ ring of invariants under group action ⓘ subalgebra of a polynomial ring ⓘ |
| relatedTo |
Hilbert’s basis theorem
ⓘ
linked to:
Hilbert basis theorem
Noetherian rings ⓘ algebraic transformation groups ⓘ ring of polynomial functions on affine space ⓘ subfields of rational function fields ⓘ |
| statedBy | David Hilbert ⓘ |
| status | in general answered in the negative ⓘ |
| studiedIn |
algebraic group theory
ⓘ
birational geometry ⓘ commutative algebra ⓘ |
| trueFor |
certain classes of groups and representations
ⓘ
finite groups ⓘ linearly reductive groups over fields of characteristic zero ⓘ reductive algebraic groups acting linearly ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hilbert’s fourteenth problem Description of subject: Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.