Frobenius conjugacy class

E790517

conjugacy class mathematical concept object in algebraic number theory

A Frobenius conjugacy class is the set of all conjugates of a Frobenius element in a Galois group, encapsulating how a prime ideal splits in a given Galois extension.

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Statements (48)

Predicate	Object
instanceOf	conjugacy class ⓘ mathematical concept ⓘ object in algebraic number theory ⓘ
appearsIn	Euler products of Artin L-functions ⓘ definition of Artin reciprocity map ⓘ statement of the Chebotarev density theorem ⓘ
arisesFrom	Frobenius automorphism at a prime NERFINISHED ⓘ
associatedTo	Frobenius element NERFINISHED ⓘ
characterizedBy	action on residue fields at primes ⓘ cycle structure on roots of polynomials modulo primes ⓘ
definedFor	unramified prime ideals ⓘ
definedInContextOf	Galois representations NERFINISHED ⓘ Galois theory NERFINISHED ⓘ algebraic number theory ⓘ
dependsOn	choice of prime ideal above a rational prime up to conjugacy ⓘ
domain	Galois extension of function fields ⓘ Galois extension of global fields ⓘ Galois extension of number fields ⓘ
encodes	cycle type of Frobenius action on embeddings ⓘ decomposition of primes in number fields ⓘ splitting behavior of a prime ideal in a Galois extension ⓘ
generalizes	classical Frobenius element in finite field extensions ⓘ
hasDefiningGroup	Galois group ⓘ
hasElementType	automorphisms of a field extension ⓘ
hasNotation	Frob_p ⓘ Frob_𝔭 ⓘ
hasProperty	conjugacy-invariant subset of a Galois group ⓘ finite subset when the Galois group is finite ⓘ
independentOf	choice of prime above a given unramified prime (up to conjugacy) ⓘ
isInvariantUnder	inner automorphisms of the Galois group ⓘ
isSetOf	conjugates of a Frobenius element ⓘ
playsRoleIn	classification of primes by splitting type ⓘ comparison of different Galois representations via traces of Frobenius ⓘ equidistribution of primes in Galois extensions ⓘ
relatedTo	Artin symbol NERFINISHED ⓘ Frobenius automorphism NERFINISHED ⓘ decomposition group ⓘ inertia group ⓘ
undefinedFor	ramified prime ideals without additional choices ⓘ
usedIn	Artin L-functions NERFINISHED ⓘ Chebotarev density theorem NERFINISHED ⓘ Langlands program NERFINISHED ⓘ Sato–Tate type equidistribution statements ⓘ Serre’s modularity conjecture NERFINISHED ⓘ proofs of equidistribution results for primes ⓘ study of Galois representations of number fields ⓘ
usedToDefine	Artin conductor contributions at unramified primes ⓘ local factors of L-functions ⓘ

Referenced by (1)

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

Chebotarev density theorem → usesConcept → Frobenius conjugacy class ⓘ