Kronecker–Weber theorem
E100232
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kronecker–Weber theorem canonical | 3 |
| Kronecker–Weber theorem as a special case | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T846896 — 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: Kronecker–Weber theorem Context triple: [Leopold Kronecker, notableWork, Kronecker–Weber theorem]
-
A.
Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
-
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.
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.
-
D.
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.
-
E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- 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: Kronecker–Weber theorem Target entity description: 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.
-
A.
Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
-
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.
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.
-
D.
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.
-
E.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf | theorem in algebraic number theory ⓘ |
| appliesTo | finite Galois extensions of Q with abelian Galois group ⓘ |
| characterizes |
finite abelian extensions of Q as subfields of cyclotomic fields
ⓘ
maximal abelian extension of Q ⓘ |
| concerns |
abelian Galois extensions
ⓘ
class field theory over Q ⓘ extensions of the rational number field Q ⓘ |
| describes | finite abelian extensions of the rational numbers ⓘ |
| domain |
Galois extensions of Q
ⓘ
number fields ⓘ |
| equivalentTo | statement that every finite abelian extension of Q has conductor n for some n and is contained in Q(ζ_n) ⓘ |
| excludes | non-abelian extensions of Q ⓘ |
| field | algebraic number theory ⓘ |
| generalizationOf | properties of cyclotomic fields studied by Gauss ⓘ |
| hasConsequence |
classification of finite abelian extensions of Q by conductors
ⓘ
description of abelian Galois groups of Q as quotients of (Z/nZ)^× ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| implies |
Q^ab = ⋃_n Q(ζ_n)
ⓘ
the maximal abelian extension of Q is the union of all cyclotomic fields ⓘ |
| involves |
cyclotomic fields
ⓘ
roots of unity ⓘ |
| isPartOf | global class field theory ⓘ |
| namedAfter |
Heinrich Martin Weber
ⓘ
Leopold Kronecker ⓘ |
| originallyFormulatedFor | abelian extensions of Q ⓘ |
| provedBy |
Heinrich Martin Weber
ⓘ
Leopold Kronecker ⓘ |
| relatedTo |
Hilbert’s twelfth problem
ⓘ
surface form:
Artin reciprocity law
Dirichlet characters ⓘ Hilbert class field ⓘ Hilbert’s twelfth problem ⓘ
surface form:
Kronecker Jugendtraum
ray class fields over Q ⓘ |
| standardReference |
Algebraic Number Theory textbooks
ⓘ
Cassels–Fröhlich: Algebraic Number Theory ⓘ Neukirch: Algebraic Number Theory ⓘ |
| states |
every finite abelian extension of Q is contained in Q(ζ_n) for some n
ⓘ
every finite abelian extension of the rational numbers is contained in a cyclotomic field ⓘ |
| usesConcept |
Galois group
ⓘ
abelian group ⓘ conductor of an abelian extension ⓘ cyclotomic polynomial ⓘ local fields at primes of Q ⓘ ramification in number fields ⓘ |
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: Kronecker–Weber theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
cyclotomic field
subject surface form:
cyclotomic field
this entity surface form:
Kronecker–Weber theorem as a special case