local class field theory

E753158

branch of number theory class field theory mathematical theory

Local class field theory is a branch of number theory that describes the abelian extensions of local fields (such as p-adic fields) in terms of their multiplicative groups via reciprocity maps.

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Statements (50)

Predicate	Object
instanceOf	branch of number theory ⓘ class field theory ⓘ mathematical theory ⓘ
appliesTo	finite extensions of F_p((T)) ⓘ finite extensions of Q_p ⓘ non-archimedean local fields ⓘ
characterizedBy	compatibility with norm maps and restriction maps ⓘ description of higher ramification groups via unit filtration ⓘ existence theorem for abelian extensions of local fields ⓘ functorial correspondence between open subgroups of multiplicative group and finite abelian extensions ⓘ isomorphism between profinite completion of multiplicative group and abelianized Galois group ⓘ
describes	abelian extensions of local fields in terms of multiplicative groups ⓘ
developedBy	Claude Chevalley NERFINISHED ⓘ Emil Artin NERFINISHED ⓘ Helmut Hasse NERFINISHED ⓘ Shokichi Iyanaga NERFINISHED ⓘ Teiji Takagi NERFINISHED ⓘ
fieldOfStudy	number theory ⓘ
formalizedUsing	Galois cohomology NERFINISHED ⓘ profinite groups ⓘ topological groups ⓘ
generalizes	properties of cyclotomic extensions of Q_p ⓘ
hasApplication	classification of finite abelian extensions of local fields ⓘ computation of local Artin L-factors ⓘ study of ramification in local Galois extensions ⓘ
hasTheorem	Hasse–Arf theorem (in the abelian case) NERFINISHED ⓘ existence theorem of local class field theory ⓘ local reciprocity law ⓘ
influenced	local Langlands program NERFINISHED ⓘ modern algebraic number theory ⓘ
relatedTo	Artin reciprocity NERFINISHED ⓘ Kronecker–Weber theorem NERFINISHED ⓘ Lubin–Tate theory NERFINISHED ⓘ global class field theory NERFINISHED ⓘ local Langlands correspondence NERFINISHED ⓘ
studies	abelian extensions of local fields ⓘ
usesConcept	Brauer group NERFINISHED ⓘ Galois group of an abelian extension ⓘ Hasse invariant NERFINISHED ⓘ finite extensions of F_p((T)) ⓘ finite extensions of Q_p ⓘ idèle group of a local field ⓘ inertia group ⓘ local fields ⓘ local reciprocity map ⓘ multiplicative group of a local field ⓘ norm map ⓘ p-adic fields ⓘ ramification group ⓘ reciprocity map ⓘ

Referenced by (2)

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

Hasse–Arf theorem → context → local class field theory ⓘ

Hasse norm theorem → involves → local class field theory ⓘ