Furtwängler’s theorem in class field theory
E713105
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Furtwängler’s theorem in class field theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8144792 — 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: Furtwängler’s theorem in class field theory Context triple: [Philipp Furtwängler, knownFor, Furtwängler’s theorem in class field theory]
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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.
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B.
Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
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C.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
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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.
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E.
Neukirch: Algebraic Number Theory
"Neukirch: Algebraic Number Theory" is a widely respected graduate-level textbook that provides a rigorous, modern introduction to algebraic number theory, including class field theory and foundational results such as the Kronecker–Weber theorem.
- 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: Furtwängler’s theorem in class field theory Target entity description: Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
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.
Artin reciprocity law
The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
-
C.
Artin’s conjecture on L-functions
Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
-
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.
Neukirch: Algebraic Number Theory
"Neukirch: Algebraic Number Theory" is a widely respected graduate-level textbook that provides a rigorous, modern introduction to algebraic number theory, including class field theory and foundational results such as the Kronecker–Weber theorem.
- F. None of above. chosen
Statements (27)
| Predicate | Object |
|---|---|
| instanceOf |
result in class field theory
ⓘ
theorem in algebraic number theory ⓘ |
| appliesTo |
Hilbert class fields
ⓘ
linked to:
Hilbert class field
number fields ⓘ |
| concerns |
abelian extensions of number fields
ⓘ
capitulation of ideal classes ⓘ ideal class groups ⓘ |
| context |
Galois theory of number fields
ⓘ
maximal unramified abelian extensions ⓘ |
| describes | conditions under which ideal classes become principal in extensions ⓘ |
| field |
algebraic number theory
ⓘ
class field theory ⓘ |
| hasImportance |
fundamental in the study of capitulation in class field theory
ⓘ
refines understanding of principalization of ideals ⓘ |
| implies | capitulation of certain ideal classes in the Hilbert class field ⓘ |
| namedAfter | Philipp Furtwängler ⓘ |
| refines |
principal ideal theorem
ⓘ
linked to:
Krull’s principal ideal theorem
|
| relatedTo |
Hilbert class field
ⓘ
class field theory reciprocity laws ⓘ ideal class capitulation ⓘ principal ideal theorem ⓘ |
| studiedIn |
advanced texts on algebraic number theory
ⓘ
treatises on global class field theory ⓘ |
| usesConcept |
Artin reciprocity
ⓘ
linked to:
Artin reciprocity law
class field ⓘ ideal class group ⓘ norm map on ideal class groups ⓘ |
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: Furtwängler’s theorem in class field theory Description of subject: Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.