Hilbert class field
E459562
The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hilbert class field canonical | 1 |
| Hilbert class fields of imaginary quadratic fields | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4597211 — 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 class field Context triple: [Kronecker–Weber theorem, relatedTo, Hilbert class field]
-
A.
Hilbert’s twelfth problem
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.
-
B.
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.
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C.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
-
D.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
E.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
- 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 class field Target entity description: The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
-
A.
Hilbert’s twelfth problem
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.
-
B.
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.
-
C.
Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
-
D.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
E.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
number field extension ⓘ object in algebraic number theory ⓘ |
| appearsIn |
Hilbert’s 12th problem
NERFINISHED
ⓘ
theory of complex multiplication of elliptic curves ⓘ |
| characterizedBy |
Artin reciprocity isomorphism with the ideal class group
ⓘ
conductor equal to the ring of integers of the base field ⓘ every ideal class of the base field becomes principal in it ⓘ no nontrivial unramified abelian extension of the base field lies strictly above it ⓘ |
| constructedVia |
class field theory
ⓘ
idele class group and Artin map ⓘ ray class fields with trivial modulus ⓘ |
| contains | all finite unramified abelian extensions of the base field ⓘ |
| correspondsTo |
ideal class group of the base field
ⓘ
maximal unramified abelian extension in class field theory ⓘ |
| definedOver | number field ⓘ |
| degreeOverBaseFieldEquals | class number of the base field ⓘ |
| fieldOfStudy |
algebraic number theory
ⓘ
class field theory NERFINISHED ⓘ |
| forBaseField | number field with given ideal class group ⓘ |
| GaloisGroupIsomorphicTo | ideal class group of the base field ⓘ |
| generalizationOf | Hilbert class field of an imaginary quadratic field generated by j-invariants ⓘ |
| hasApplication |
computing class numbers
ⓘ
studying capitulation of ideal classes in extensions ⓘ |
| hasProperty |
Galois over the base number field
ⓘ
abelian over the base number field ⓘ finite extension of the base number field ⓘ its Galois group is isomorphic to the ideal class group of the base field ⓘ maximal unramified abelian extension of a number field ⓘ normal extension of the base number field ⓘ unique up to isomorphism for a given number field ⓘ unramified at all finite places of the base field ⓘ unramified at all finite primes of the base field ⓘ |
| hasSpecialCase |
Hilbert class field of a real quadratic field
ⓘ
Hilbert class field of an imaginary quadratic field ⓘ Hilbert class field of the rational numbers NERFINISHED ⓘ |
| namedAfter | David Hilbert NERFINISHED ⓘ |
| relatedTo |
Hilbert class polynomial
NERFINISHED
ⓘ
Kronecker–Weber theorem in the case of Q NERFINISHED ⓘ maximal unramified extension ⓘ narrow Hilbert class field NERFINISHED ⓘ ray class field ⓘ |
| unramifiedAt | every nonzero prime ideal of the ring of integers of the base field ⓘ |
| usedIn |
complex multiplication theory
ⓘ
construction of unramified extensions ⓘ explicit class field theory ⓘ proofs and formulations of class field theory ⓘ study of ideal class groups ⓘ study of the arithmetic of quadratic fields ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hilbert class field Description of subject: The Hilbert class field of a number field is its maximal unramified abelian extension, central in class field theory as it corresponds to the field’s ideal class group.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hilbert class fields of imaginary quadratic fields