of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in 1852 (Kummer 1852). Kummer's theorem states that for given integers...

z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for Ernst Kummer: 2 F 1 ( a , b ; 1 + a − b ;...

the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that...

Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Building on Kummer's work...

In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure. It...

Ankeny–Artin–Chowla. The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem, which says: If the odd prime p does not divide any of the...

MR 0465761 25628 Kummer – asteroid named after Ernst Kummer Kummer configuration Kummer's congruence Kummer series Kummer theory Kummer's theorem, on prime-power...

version of Kummer's test was established by Tong. See also for further discussions and new proofs. The provided modification of Kummer's theorem characterizes...

none existed, that valuation would exceed r). This is an instance of Kummer's theorem (since in base p notation the number |G| ends with precisely k + r...

multinomial coefficient may be computed using a generalization of Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's...

Generalizations of Lucas's theorem for higher prime powers pk are also given by Davis and Webb (1990) and Granville (1997). Kummer's theorem asserts that the largest...

Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory....

Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer (1851). Kubota & Leopoldt (1964) used Kummer's congruences...

hypergeometric functions: Kummer's (confluent hypergeometric) function M(a, b, z), introduced by Kummer (1837), is a solution to Kummer's differential equation...

\end{aligned}}} Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive...

was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the...

formula to the product formula for binomial coefficients produces Kummer's theorem, a similar result on the exponent of each prime in the factorization...

modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected...

abandoned this experiment, though it remains widely used.[citation needed] Kummer's theorem states that the number of carries involved in adding two numbers in...

mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer...

B_{2n}=I_{n}-\sum _{(p-1)|2n}{\frac {1}{p}},} where In is an integer, as desired. Kummer's congruence H. Rademacher, Analytic Number Theory, Springer-Verlag, New...

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit...

6, ..., p − 3. Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain...

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}...

|p|<M_{k}.} page 72 These decompositions can be found using the Dedekind–Kummer theorem. A classical example of the construction of a quadratic field is to...

countably generated is prime. Radical ideal Maximal ideal Dedekind–Kummer theorem Residue field Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra...

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to...

primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gras conjecture...

divisorial ideals is called a Mori domain. Divisorial sheaf Dedekind-Kummer theorem Childress, Nancy (2009). Class field theory. New York: Springer....

exponents p less than 100, except for the irregular primes 37, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was...