Kummer congruences

E463786

mathematical concept number-theoretic congruence result in algebraic number theory

Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.

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Observed surface forms (1)

Surface form	Occurrences
Kummer’s criterion for Fermat’s Last Theorem	1

Statements (48)

Predicate	Object
instanceOf	mathematical concept ⓘ number-theoretic congruence ⓘ result in algebraic number theory ⓘ
appearsIn	proofs of properties of Bernoulli numbers modulo primes ⓘ theory of regular and irregular primes ⓘ
appliesTo	Bernoulli numbers B_n NERFINISHED ⓘ special values of Dirichlet L-functions at non-positive integers ⓘ special values of the Riemann zeta function at negative integers ⓘ
characterizes	p-adic interpolation of Bernoulli numbers ⓘ p-adic interpolation of special L-values ⓘ
describes	behavior of Bernoulli numbers modulo p^n ⓘ congruence relations between Bernoulli numbers modulo powers of primes ⓘ p-adic continuity of special values of L-functions ⓘ
field	number theory ⓘ
formalizes	compatibility of special L-values in p-adic families ⓘ
generalizedBy	Iwasawa theory congruences ⓘ p-adic interpolation theorems ⓘ
hasConsequence	criteria for irregular primes ⓘ relations among class numbers of cyclotomic fields at different levels ⓘ
historicalPeriod	19th century mathematics ⓘ
holdsFor	even indices of Bernoulli numbers ⓘ
introducedBy	Ernst Eduard Kummer NERFINISHED ⓘ
involves	congruences modulo p^k ⓘ p-adic valuations ⓘ powers of primes p^n ⓘ prime numbers p ⓘ
motivation	study of Fermat's Last Theorem for regular primes ⓘ study of class numbers of cyclotomic fields ⓘ
namedAfter	Ernst Eduard Kummer NERFINISHED ⓘ
relatedTo	Bernoulli numbers NERFINISHED ⓘ Dirichlet L-functions NERFINISHED ⓘ Fermat's Last Theorem NERFINISHED ⓘ Iwasawa main conjecture NERFINISHED ⓘ Kubota–Leopoldt p-adic L-functions NERFINISHED ⓘ class number formula ⓘ cyclotomic fields ⓘ ideal class groups ⓘ irregular primes ⓘ p-adic L-functions ⓘ special values of L-functions ⓘ
subfield	Iwasawa theory NERFINISHED ⓘ algebraic number theory ⓘ p-adic number theory ⓘ
usedIn	Iwasawa theory of cyclotomic fields ⓘ construction of p-adic L-functions ⓘ proofs of properties of p-adic zeta functions ⓘ study of class groups of cyclotomic fields ⓘ study of cyclotomic extensions of Q ⓘ

Referenced by (2)

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

Ernst Eduard Kummer → knownFor → Kummer congruences ⓘ

Bernoulli numbers → connectedTo → Kummer congruences ⓘ

this entity surface form: Kummer’s criterion for Fermat’s Last Theorem