Hilbert’s twelfth problem
E213012
Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Artin reciprocity law | 1 |
| Hilbert’s twelfth problem canonical | 1 |
| Kronecker Jugendtraum | 1 |
| Kronecker’s Jugendtraum | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1859186 — 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 twelfth problem Context triple: [Hilbert problems, hasPart, Hilbert’s twelfth problem]
-
A.
Die Theorie der algebraischen Zahlkörper
"Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
-
B.
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.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Hilbert’s fourteenth problem
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.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- 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 twelfth problem Target entity description: Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
-
A.
Die Theorie der algebraischen Zahlkörper
"Die Theorie der algebraischen Zahlkörper" is a foundational mathematical monograph on algebraic number fields, authored by David Hilbert and published as part of his influential Zahlbericht.
-
B.
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.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Hilbert’s fourteenth problem
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.
-
E.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Hilbert problem
ⓘ
mathematical problem ⓘ open problem in number theory ⓘ |
| aimsToDescribe |
abelian extensions via transcendental analytic data
ⓘ
maximal abelian extension of a number field ⓘ |
| asksFor |
explicit construction of all abelian extensions of a given number field
ⓘ
generation of abelian extensions by special values of analytic functions ⓘ |
| concerns |
abelian extensions of number fields
ⓘ
explicit class field theory ⓘ special values of L-functions ⓘ special values of analytic functions ⓘ special values of complex analytic functions ⓘ special values of elliptic functions ⓘ special values of modular functions ⓘ |
| field |
algebraic number theory
ⓘ
analytic number theory ⓘ class field theory ⓘ number theory ⓘ |
| generalizes |
Hilbert’s twelfth problem
self-linksurface differs
ⓘ
surface form:
Kronecker’s Jugendtraum
|
| influenced |
development of modern class field theory
ⓘ
research on special values of L-functions ⓘ |
| involves |
CM fields
ⓘ
Galois groups of abelian extensions ⓘ Hecke characters ⓘ Kronecker–Weber theorem ⓘ
surface form:
Kronecker–Weber theorem as a special case
class field theory reciprocity maps ⓘ idele class characters ⓘ imaginary quadratic fields ⓘ theory of complex multiplication ⓘ totally real fields ⓘ |
| languageOfOriginalFormulation | German ⓘ |
| numberInHilbertList | 12 ⓘ |
| partiallySolvedFor |
CM fields
ⓘ
some abelian extensions of totally real fields ⓘ |
| partOf |
Hilbert problems
ⓘ
surface form:
Hilbert’s list of 23 problems
|
| presentedAt |
International Congress of Mathematicians
ⓘ
surface form:
International Congress of Mathematicians 1900 in Paris
|
| proposedBy | David Hilbert ⓘ |
| relatedTo |
Stark conjectures
ⓘ
surface form:
Brumer–Stark conjecture
Iwasawa theory ⓘ Langlands program ⓘ Shimura varieties ⓘ Stark conjectures ⓘ automorphic forms ⓘ modular forms ⓘ |
| solvedFor |
imaginary quadratic fields via elliptic functions and modular functions
ⓘ
rational number field via roots of unity ⓘ |
| status | unsolved in full generality ⓘ |
| yearProposed | 1900 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hilbert’s twelfth problem Description of subject: Hilbert’s twelfth problem is one of David Hilbert’s famous list of 23 problems, asking for a general explicit class field theory that would generate all abelian extensions of a given number field using special values of analytic functions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kronecker Jugendtraum
this entity surface form:
Artin reciprocity law
this entity surface form:
Kronecker’s Jugendtraum