Kronecker–Weber theorem

E100232

theorem in algebraic number theory

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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All labels observed (2)

Label	Occurrences
Kronecker–Weber theorem canonical	3
Kronecker–Weber theorem as a special case	1

Statements (43)

Predicate	Object
instanceOf	theorem in algebraic number theory ⓘ
appliesTo	finite Galois extensions of Q with abelian Galois group ⓘ
characterizes	finite abelian extensions of Q as subfields of cyclotomic fields ⓘ maximal abelian extension of Q ⓘ
concerns	abelian Galois extensions ⓘ class field theory over Q ⓘ extensions of the rational number field Q ⓘ
describes	finite abelian extensions of the rational numbers ⓘ
domain	Galois extensions of Q ⓘ number fields ⓘ
equivalentTo	statement that every finite abelian extension of Q has conductor n for some n and is contained in Q(ζ_n) ⓘ
excludes	non-abelian extensions of Q ⓘ
field	algebraic number theory ⓘ
generalizationOf	properties of cyclotomic fields studied by Gauss ⓘ
hasConsequence	classification of finite abelian extensions of Q by conductors ⓘ description of abelian Galois groups of Q as quotients of (Z/nZ)^× ⓘ
historicalPeriod	19th century mathematics ⓘ
implies	Q^ab = ⋃_n Q(ζ_n) ⓘ the maximal abelian extension of Q is the union of all cyclotomic fields ⓘ
involves	cyclotomic fields ⓘ roots of unity ⓘ
isPartOf	global class field theory ⓘ
namedAfter	Heinrich Martin Weber ⓘ Leopold Kronecker ⓘ
originallyFormulatedFor	abelian extensions of Q ⓘ
provedBy	Heinrich Martin Weber ⓘ Leopold Kronecker ⓘ
relatedTo	Hilbert’s twelfth problem ⓘ surface form: Artin reciprocity law Dirichlet characters ⓘ Hilbert class field ⓘ Hilbert’s twelfth problem ⓘ surface form: Kronecker Jugendtraum ray class fields over Q ⓘ
standardReference	Algebraic Number Theory textbooks ⓘ Cassels–Fröhlich: Algebraic Number Theory ⓘ Neukirch: Algebraic Number Theory ⓘ
states	every finite abelian extension of Q is contained in Q(ζ_n) for some n ⓘ every finite abelian extension of the rational numbers is contained in a cyclotomic field ⓘ
usesConcept	Galois group ⓘ abelian group ⓘ conductor of an abelian extension ⓘ cyclotomic polynomial ⓘ local fields at primes of Q ⓘ ramification in number fields ⓘ

Referenced by (4)

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

Leopold Kronecker → notableWork → Kronecker–Weber theorem ⓘ

cyclotomic fields → relatedTo → Kronecker–Weber theorem ⓘ

subject surface form: cyclotomic field

cyclotomic fields → usedInProofOf → Kronecker–Weber theorem ⓘ

subject surface form: cyclotomic field

Hilbert’s twelfth problem → involves → Kronecker–Weber theorem ⓘ

this entity surface form: Kronecker–Weber theorem as a special case