Frobenius element

E790516

arithmetic object group-theoretic element

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.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (2)

Surface form	Occurrences
Frobenius elements	1
Frobenius endomorphism	1

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 ⓘ

Referenced by (3)

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

Chebotarev density theorem → usesConcept → Frobenius element ⓘ

Koblitz curves → relatedTo → Frobenius element ⓘ

this entity surface form: Frobenius endomorphism

Weil group → hasElementType → Frobenius element ⓘ

this entity surface form: Frobenius elements