Kummer theory
E463784
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kummer theory canonical | 4 |
| Kummer extensions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4706375 — 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: Kummer theory Context triple: [Ernst Eduard Kummer, knownFor, Kummer theory]
-
A.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
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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.
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.
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D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
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E.
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.
- 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: Kummer theory Target entity description: Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
-
A.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
-
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.
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.
-
D.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
branch of algebraic number theory
ⓘ
theory in field theory ⓘ |
| appliesTo |
global fields
ⓘ
local fields ⓘ number fields ⓘ |
| assumes | base field contains enough roots of unity ⓘ |
| characterizes |
Galois group as subgroup of multiplicative group modulo n-th powers
ⓘ
abelian extensions via n-th power classes ⓘ |
| describes | structure of abelian extensions of prime power degree ⓘ |
| extends | Gauss’s theory of quadratic residues NERFINISHED ⓘ |
| focusesOn | abelian extensions of exponent n ⓘ |
| formalism | often expressed using Galois cohomology groups H^1 and H^2 ⓘ |
| generalizes | classical theory of cyclotomic fields ⓘ |
| hasGeneralization |
Artin reciprocity
NERFINISHED
ⓘ
Iwasawa theory NERFINISHED ⓘ Lubin–Tate theory NERFINISHED ⓘ |
| hasVariant |
global Kummer theory
ⓘ
local Kummer theory ⓘ |
| historicalContext | developed in the 19th century ⓘ |
| influenced |
development of class field theory
ⓘ
development of ideal theory ⓘ |
| isPartOf |
abelian extension theory
ⓘ
algebraic number theory ⓘ |
| namedAfter | Ernst Kummer NERFINISHED ⓘ |
| originallyMotivatedBy |
study of Fermat's Last Theorem
ⓘ
study of cyclotomic fields ⓘ |
| prefigures | global class field theory NERFINISHED ⓘ |
| relatedConcept |
Kummer character
ⓘ
Kummer extension NERFINISHED ⓘ Kummer radical ⓘ Kummer surface (terminologically related but different area) ⓘ |
| relatesTo |
Galois cohomology
NERFINISHED
ⓘ
Hilbert class field NERFINISHED ⓘ Kummer map NERFINISHED ⓘ Kummer pairing NERFINISHED ⓘ Kummer sequence NERFINISHED ⓘ class field theory NERFINISHED ⓘ ideal class group ⓘ ray class group ⓘ |
| requires | primitive n-th root of unity in the base field ⓘ |
| studies |
abelian extensions of fields
ⓘ
cyclotomic extensions ⓘ radical extensions ⓘ |
| typicalAssumption | characteristic of the field does not divide n ⓘ |
| uses |
ideal class groups
ⓘ
local–global principles ⓘ multiplicative group of a field ⓘ n-th power residue symbol ⓘ roots of unity ⓘ |
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: Kummer theory Description of subject: Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kummer extensions
subject surface form:
cyclotomic field