global class field theory

E459561

branch of algebraic number theory mathematical theory

Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.

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All labels observed (3)

Label	Occurrences
class field theory	13
global class field theory canonical	8
Class Field Theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T4597205 — 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: global class field theory
Context triple: [Kronecker–Weber theorem, isPartOf, global class field theory]

A. 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.
B. 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.
C. Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
D. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
E. Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
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: global class field theory
Target entity description: Global class field theory is a branch of algebraic number theory that classifies finite abelian extensions of global fields (such as number fields) in terms of their arithmetic data, particularly via idele class groups and reciprocity maps.

A. 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.
B. 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.
C. Grothendieck topology
A Grothendieck topology is an abstract framework in category theory that generalizes the notion of open covers in topology to define sheaves on arbitrary categories.
D. Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
E. Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	branch of algebraic number theory ⓘ mathematical theory ⓘ
appliesTo	global function fields ⓘ number fields ⓘ
centralTheorem	Artin reciprocity law ⓘ existence theorem for class fields ⓘ
characterizes	maximal abelian extension of a global field ⓘ
describes	Galois group of maximal abelian extension as quotient of idele class group ⓘ abelian extensions via open subgroups of finite index in idele class group ⓘ
developedBy	Claude Chevalley ⓘ Emil Artin ⓘ Helmut Hasse ⓘ Teiji Takagi ⓘ
fieldOfStudy	algebraic number theory ⓘ
frameworkFor	Langlands program (abelian case) ⓘ linked to: Langlands program explicit class field theory ⓘ
generalizes	Hilbert class field theory ⓘ linked to: Hilbert class field Kronecker–Weber theorem ⓘ
hasLocalAnalogue	local class field theory ⓘ
historicalRoot	Kronecker Jugendtraum ⓘ linked to: Hilbert’s twelfth problem
implies	Kronecker–Weber theorem for abelian extensions of the rationals ⓘ linked to: Kronecker–Weber theorem existence of Hilbert class field for any number field ⓘ
relatedConcept	class formation ⓘ cohomology of Galois groups ⓘ global field ⓘ idele topology ⓘ narrow class group ⓘ ray class group ⓘ
relates	abelian Galois groups of global fields to idele class groups ⓘ ideal class groups to abelian extensions ⓘ
studies	abelian extensions of global function fields ⓘ abelian extensions of number fields ⓘ finite abelian extensions of global fields ⓘ
typicalReference	Artin–Tate: Class Field Theory ⓘ Cassels–Fröhlich: Algebraic Number Theory ⓘ Neukirch: Algebraic Number Theory ⓘ
usesConcept	Artin reciprocity map ⓘ linked to: Artin reciprocity law Chebotarev density theorem ⓘ Frobenius automorphism ⓘ linked to: Frobenius element Hilbert class field ⓘ class group of a number field ⓘ global reciprocity law ⓘ linked to: Artin reciprocity law idele class characters ⓘ idele class group ⓘ linked to: idèle class group idele class group modulo connected component ⓘ idele group ⓘ norm map ⓘ ray class field ⓘ

How these facts were elicited