Frobenius element
E790516
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Frobenius element canonical | 1 |
| Frobenius elements | 1 |
| Frobenius endomorphism | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9297039 — 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: Frobenius element Context triple: [Chebotarev density theorem, usesConcept, Frobenius element]
-
A.
Chebotarev density theorem
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.
-
B.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
-
C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
D.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
E.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
- 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: Frobenius element Target entity description: The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
-
A.
Chebotarev density theorem
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.
-
B.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
-
C.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
D.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
E.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
arithmetic object
ⓘ
group-theoretic element ⓘ |
| actsOn |
algebraic integers modulo a prime
ⓘ
residue field ⓘ |
| appearsIn |
Cebotarev-type equidistribution results
ⓘ
Dirichlet density statements for primes ⓘ proofs of prime splitting criteria ⓘ |
| associatedWith |
Galois group
ⓘ
finite Galois extension of number fields ⓘ prime ideal ⓘ unramified prime ⓘ |
| characterizedBy | x ↦ x^N on residue field, where N is residue field size ⓘ |
| context |
finite extension of global fields
ⓘ
unramified places of function fields ⓘ unramified places of number fields ⓘ |
| definedFor | unramified prime ideals ⓘ |
| encodes |
decomposition of primes
ⓘ
inertness of primes ⓘ residue field automorphism ⓘ splitting behavior of primes in extensions ⓘ |
| field |
algebraic number theory
ⓘ
arithmetic geometry ⓘ |
| generalization | Frobenius automorphism NERFINISHED ⓘ |
| hasProperty |
conjugacy class depends only on the prime
ⓘ
order divides size of residue field minus one in abelian case ⓘ well-defined up to conjugacy in the Galois group ⓘ |
| liesIn | decomposition group ⓘ |
| namedAfter | Ferdinand Georg Frobenius NERFINISHED ⓘ |
| notDefinedFor | ramified primes without modification ⓘ |
| projectsTo | generator of Galois group of residue field extension ⓘ |
| relatedTo |
Frobenius endomorphism
NERFINISHED
ⓘ
Weil group element ⓘ arithmetic Frobenius NERFINISHED ⓘ geometric Frobenius NERFINISHED ⓘ |
| usedIn |
Artin L-functions
NERFINISHED
ⓘ
Chebotarev density theorem NERFINISHED ⓘ Galois representations NERFINISHED ⓘ Langlands program NERFINISHED ⓘ Sato–Tate type distributions NERFINISHED ⓘ Weil conjectures NERFINISHED ⓘ class field theory ⓘ global class field theory reciprocity map ⓘ local L-factors ⓘ étale cohomology ⓘ |
| usedToDefine |
Artin symbol
NERFINISHED
ⓘ
Frobenius conjugacy class NERFINISHED ⓘ local factors of zeta functions of varieties over finite fields ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Frobenius element Description of subject: The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Frobenius endomorphism
this entity surface form:
Frobenius elements