Hilbert’s twelfth problem

E213012

Hilbert problem mathematical problem open problem in number theory

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (6)

Label	Occurrences
Kronecker Jugendtraum	2
Artin reciprocity law	1
Hilbert twelfth problem	1
Hilbert’s 12th problem	1
Hilbert’s twelfth problem canonical	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	Kronecker’s Jugendtraum ⓘ linked to: Hilbert’s twelfth problem
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 as a special case ⓘ linked to: Kronecker–Weber theorem 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’s list of 23 problems ⓘ linked to: Hilbert problems
presentedAt	International Congress of Mathematicians 1900 in Paris ⓘ linked to: International Congress of Mathematicians
proposedBy	David Hilbert ⓘ
relatedTo	Brumer–Stark conjecture ⓘ linked to: Stark conjectures 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