Chebotarev density theorem

E223663

result in analytic number theory theorem in algebraic number theory

The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.

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All labels observed (7)

Label	Occurrences
Chebotarev density theorem canonical	18
Chebotarev density theorem (contextual)	1
Chebotarev density theorem under Generalized Riemann Hypothesis	1
Chebotarev sets of primes	1
Chebotaryov theorem on roots of polynomials	1
Frobenius density theorem	1
effective Chebotarev density theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T1994284 — 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: Chebotarev density theorem
Context triple: [Hilbert’s irreducibility theorem, relatedTo, Chebotarev density theorem]

A. 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.
B. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
C. Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
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. Kronecker–Weber theorem
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.
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: Chebotarev density theorem
Target entity description: The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.

A. 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.
B. Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
C. Hasse norm theorem
The Hasse norm theorem is a fundamental result in algebraic number theory that characterizes when an element of a global field is a norm from a cyclic extension by relating this property to its behavior in all completions of the field.
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. Kronecker–Weber theorem
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.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	result in analytic number theory ⓘ theorem in algebraic number theory ⓘ
appliesTo	finite Galois extensions of number fields ⓘ
assumes	finite Galois extension of number fields K over k ⓘ
characterizes	Galois group via splitting behavior of primes ⓘ
concerns	Frobenius conjugacy classes in Galois groups ⓘ distribution of prime ideals in number fields ⓘ
dependsOn	analytic properties of Artin L-functions ⓘ zero-free regions for L-functions ⓘ
field	algebraic number theory ⓘ
generalizes	prime number theorem ⓘ
givesDensityOf	set of unramified primes with given Frobenius conjugacy class ⓘ
hasConsequence	every finite group occurs as a Galois group over Q under suitable conditions ⓘ information about splitting, inertia, and decomposition of primes ⓘ
hasSpecialCase	prime number theorem for cyclotomic fields ⓘ prime number theorem for splitting of primes in quadratic fields ⓘ
hasVariant	Chebotarev density theorem under Generalized Riemann Hypothesis ⓘ linked to: Chebotarev density theorem effective Chebotarev density theorem ⓘ linked to: Chebotarev density theorem
implies	Dirichlet's theorem on arithmetic progressions ⓘ existence of infinitely many primes with any given Frobenius conjugacy class ⓘ infinitude of primes splitting in a given way in a number field ⓘ prime number theorem for arithmetic progressions ⓘ
involves	ramified primes being excluded from the density statement ⓘ unramified primes ⓘ
isAnalogOf	equidistribution theorems in harmonic analysis ⓘ
isAnalogousTo	equidistribution of Frobenius elements in étale cohomology ⓘ
isCentralTo	modern algebraic number theory ⓘ
isToolFor	Galois representations ⓘ Langlands program ⓘ arithmetic statistics ⓘ class field theory ⓘ inverse Galois problem ⓘ study of L-functions ⓘ
namedAfter	Nikolai Chebotaryov ⓘ
publicationYear	1926 ⓘ
relates	conjugacy classes in the Galois group of the extension ⓘ prime ideals of base field ⓘ
statesThat	density of primes with Frobenius in a conjugacy class C equals \|C\| / \|G\| where G is the Galois group ⓘ
strengthens	Frobenius density theorem ⓘ linked to: Chebotarev density theorem
usedInProofOf	existence of infinitely many primes in given conjugacy classes of Galois groups ⓘ
usesConcept	Dirichlet density ⓘ Frobenius conjugacy class ⓘ Frobenius element ⓘ Galois group ⓘ natural density ⓘ prime ideal ⓘ
wasProvedBy	Nikolai Chebotaryov ⓘ

How these facts were elicited

Referenced by (24)

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

Hilbert’s irreducibility theorem → relatedTo → Chebotarev density theorem ⓘ

Deuring–Heilbronn phenomenon → relatedTo → Chebotarev density theorem ⓘ

Chebotarev density theorem → hasVariant → effective Chebotarev density theorem ⓘ

linked to: Chebotarev density theorem

Chebotarev density theorem → hasVariant → Chebotarev density theorem under Generalized Riemann Hypothesis ⓘ

linked to: Chebotarev density theorem

Chebotarev density theorem → strengthens → Frobenius density theorem ⓘ

linked to: Chebotarev density theorem

Basic Number Theory → hasTopic → Chebotarev density theorem (contextual) ⓘ

linked to: Chebotarev density theorem

Dirichlet L-functions → usedIn → Chebotarev density theorem ⓘ

prime number theorem → generalizedTo → Chebotarev density theorem ⓘ

Dedekind zeta function → relatedTo → Chebotarev density theorem ⓘ

subject linked to: Dedekind zeta functions

global class field theory → usesConcept → Chebotarev density theorem ⓘ

Cassels–Fröhlich: Algebraic Number Theory → topic → Chebotarev density theorem ⓘ

Dirichlet's theorem on arithmetic progressions → relatedTo → Chebotarev density theorem ⓘ

analytic number theory → centralTheorem → Chebotarev density theorem ⓘ

Artin reciprocity law → relatedTo → Chebotarev density theorem ⓘ

Linnik’s theorem on the least prime in an arithmetic progression → isRelatedTo → Chebotarev density theorem ⓘ

Sato–Tate distribution (for families of elliptic curves) → relatedConcept → Chebotarev density theorem ⓘ

Frobenius element → usedIn → Chebotarev density theorem ⓘ

Frobenius conjugacy class → usedIn → Chebotarev density theorem ⓘ

Dirichlet density → appliesTo → Chebotarev sets of primes ⓘ

linked to: Chebotarev density theorem

Dirichlet density → relatedTo → Chebotarev density theorem ⓘ

Nikolai Chebotaryov → notableFor → Chebotarev density theorem ⓘ

Nikolai Chebotaryov → hasTheoremNamedAfter → Chebotarev density theorem ⓘ

Nikolai Chebotaryov → hasTheoremNamedAfter → Chebotaryov theorem on roots of polynomials ⓘ

linked to: Chebotarev density theorem

Artin L-functions → usedToStudy → Chebotarev density theorem ⓘ

All labels observed (7)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (24)

How this entity was disambiguated

How these facts were elicited