algebraic number theory
E530317
Algebraic number theory is a branch of mathematics that studies algebraic structures related to algebraic integers and number fields, focusing on properties of integers through tools from abstract algebra.
All labels observed (2)
| Label | Occurrences |
|---|---|
| algebraic number theory canonical | 3 |
| Algebraic Number Theory | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5570639 — 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: algebraic number theory Context triple: [Fermat's theorem on sums of two squares, usedIn, algebraic number theory]
-
A.
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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B.
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.
-
C.
Cassels–Fröhlich: Algebraic Number Theory
Cassels–Fröhlich: Algebraic Number Theory is a classic graduate-level textbook that provides a comprehensive and rigorous introduction to algebraic number theory and its foundational results.
-
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: algebraic number theory Target entity description: Algebraic number theory is a branch of mathematics that studies algebraic structures related to algebraic integers and number fields, focusing on properties of integers through tools from abstract algebra.
-
A.
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.
-
B.
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.
-
C.
Cassels–Fröhlich: Algebraic Number Theory
Cassels–Fröhlich: Algebraic Number Theory is a classic graduate-level textbook that provides a comprehensive and rigorous introduction to algebraic number theory and its foundational results.
-
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 (68)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematics
ⓘ
subfield of number theory ⓘ |
| aimsTo |
generalize properties of integers to number fields
ⓘ
understand arithmetic of algebraic numbers ⓘ |
| centralConcept |
Galois group of a number field
ⓘ
algebraic integer ⓘ class group ⓘ completion of fields ⓘ ideal ⓘ localization of rings ⓘ number field ⓘ prime ideal ⓘ unit group ⓘ |
| fieldOfStudy |
Dedekind domains
ⓘ
linked to:
Dedekind domain
Diophantine equations ⓘ Dirichlet L-functions ⓘ Dirichlet characters ⓘ Galois cohomology ⓘ Galois representations ⓘ Iwasawa theory ⓘ L-functions ⓘ Minkowski theory ⓘ algebraic integers ⓘ arithmetic geometry ⓘ class field theory ⓘ class groups ⓘ elliptic curves over number fields ⓘ global fields ⓘ ideal class groups ⓘ ideal theory ⓘ local fields ⓘ modular forms in number theory ⓘ number fields ⓘ p-adic numbers ⓘ prime decomposition in number fields ⓘ ramification theory ⓘ ring of integers of a number field ⓘ units in number fields ⓘ valuation theory ⓘ zeta functions of number fields ⓘ |
| historicalDevelopment | 19th century ⓘ |
| notableContributor |
André Weil
ⓘ
David Hilbert ⓘ Emil Artin ⓘ Ernst Kummer ⓘ
linked to:
Ernst Eduard Kummer
Helmut Hasse ⓘ John Tate ⓘ Kenku Iwasawa ⓘ Kurt Hensel ⓘ Leopold Kronecker ⓘ Richard Dedekind ⓘ |
| relatedTo |
analytic number theory
ⓘ
arithmetic geometry ⓘ representation theory ⓘ |
| studiesPropertyOf |
algebraic extensions of the rationals
ⓘ
class numbers ⓘ factorization in rings of integers ⓘ integers ⓘ norms and traces in number fields ⓘ prime numbers ⓘ unit groups of number fields ⓘ |
| usesTool |
Galois theory
ⓘ
algebraic geometry ⓘ commutative algebra ⓘ field theory ⓘ group theory ⓘ homological algebra ⓘ module theory ⓘ |
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: algebraic number theory Description of subject: Algebraic number theory is a branch of mathematics that studies algebraic structures related to algebraic integers and number fields, focusing on properties of integers through tools from abstract algebra.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: algebraic number theory
linked to: algebraic number theory
subject linked to:
Euclidean domains