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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Observed surface forms (2)

Surface form	Occurrences
Class Field Theory	1
class field theory	1

Statements (48)

Predicate	Object
instanceOf	branch of algebraic number theory ⓘ mathematical theory ⓘ
appliesTo	global function fields ⓘ number fields ⓘ
centralTheorem	Artin reciprocity law NERFINISHED ⓘ 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 NERFINISHED ⓘ Emil Artin NERFINISHED ⓘ Helmut Hasse NERFINISHED ⓘ Teiji Takagi NERFINISHED ⓘ
fieldOfStudy	algebraic number theory ⓘ
frameworkFor	Langlands program (abelian case) NERFINISHED ⓘ explicit class field theory ⓘ
generalizes	Hilbert class field theory NERFINISHED ⓘ Kronecker–Weber theorem NERFINISHED ⓘ
hasLocalAnalogue	local class field theory ⓘ
historicalRoot	Kronecker Jugendtraum NERFINISHED ⓘ
implies	Kronecker–Weber theorem for abelian extensions of the rationals NERFINISHED ⓘ 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 NERFINISHED ⓘ Cassels–Fröhlich: Algebraic Number Theory NERFINISHED ⓘ Neukirch: Algebraic Number Theory NERFINISHED ⓘ
usesConcept	Artin reciprocity map NERFINISHED ⓘ Chebotarev density theorem NERFINISHED ⓘ Frobenius automorphism NERFINISHED ⓘ Hilbert class field NERFINISHED ⓘ class group of a number field ⓘ global reciprocity law NERFINISHED ⓘ idele class characters ⓘ idele class group NERFINISHED ⓘ idele class group modulo connected component ⓘ idele group ⓘ norm map ⓘ ray class field ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kronecker–Weber theorem → isPartOf → global class field theory ⓘ

quadratic reciprocity law → proofMethodsInclude → global class field theory ⓘ

this entity surface form: class field theory

Jürgen Neukirch → notableWork → global class field theory ⓘ

this entity surface form: Class Field Theory

Hasse–Weil zeta function → playsRoleIn → global class field theory ⓘ

Hasse norm theorem → involves → global class field theory ⓘ