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 (5)

Label	Occurrences
Kronecker–Weber theorem canonical	8
Kronecker–Weber theorem as a special case	1
Kronecker–Weber theorem for abelian extensions of the rationals	1
Kronecker–Weber theorem in the case of Q	1
Kronecker–Weber theorem over the rationals	1

How this entity was disambiguated

This entity first appeared as the object of triple T846896 — 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: Kronecker–Weber theorem
Context triple: [Leopold Kronecker, notableWork, Kronecker–Weber theorem]

A. Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
B. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
E. 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.
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: Kronecker–Weber theorem
Target entity description: 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.

A. Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
B. Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
C. Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
D. Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
E. 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.
F. None of above. chosen

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	Artin reciprocity law ⓘ linked to: Hilbert’s twelfth problem Dirichlet characters ⓘ Hilbert class field ⓘ Kronecker Jugendtraum ⓘ linked to: Hilbert’s twelfth problem 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 ⓘ

How these facts were elicited

Referenced by (12)

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

Leopold Kronecker → notableWork → Kronecker–Weber theorem ⓘ

cyclotomic field → relatedTo → Kronecker–Weber theorem ⓘ

subject linked to: cyclotomic fields

cyclotomic field → usedInProofOf → Kronecker–Weber theorem ⓘ

subject linked to: cyclotomic fields

Hilbert’s twelfth problem → involves → Kronecker–Weber theorem as a special case ⓘ

linked to: Kronecker–Weber theorem

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

global class field theory → implies → Kronecker–Weber theorem for abelian extensions of the rationals ⓘ

linked to: Kronecker–Weber theorem

Hilbert class field → relatedTo → Kronecker–Weber theorem in the case of Q ⓘ

linked to: Kronecker–Weber theorem

Neukirch: Algebraic Number Theory → subject → Kronecker–Weber theorem ⓘ

Cassels–Fröhlich: Algebraic Number Theory → topic → Kronecker–Weber theorem ⓘ

Artin reciprocity law → implies → Kronecker–Weber theorem over the rationals ⓘ

linked to: Kronecker–Weber theorem

local class field theory → relatedTo → Kronecker–Weber theorem ⓘ

reciprocity conjecture → historicalRoot → Kronecker–Weber theorem ⓘ

All labels observed (5)

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Statements (43)

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Referenced by (12)

How this entity was disambiguated

How these facts were elicited