Kummer

Kummer is a surname associated with several mathematical concepts and with the German mathematician Ernst Eduard Kummer (1810–1893). His work in number theory and algebra earned a lasting place in the field, and several objects bear his name.

Ernst Eduard Kummer contributed to the foundations of modern number theory. He introduced the concept of ideal numbers, an early step toward the ideal theory developed by Dedekind. He also studied primes in cyclotomic fields and developed results on p-adic properties of binomial coefficients, a line of inquiry that culminated in what is now known as Kummer’s theorem.

In number theory, Kummer's theorem concerns the p-adic valuation of binomial coefficients. It states that the exponent of a prime p dividing the binomial coefficient binom(n, m) equals the number of carries when adding m and n − m in base p. This result connects binomial arithmetic with p-adic properties and carries a lasting influence in combinatorics and algebraic number theory. Kummer also investigated irregular primes and the limits of certain approaches to Fermat's Last Theorem within his era.

In algebraic number theory, Kummer theory describes abelian extensions of number fields obtained by adjoining nth roots of elements, provided the field contains the nth roots of unity. This framework, known as Kummer extensions, helped shape subsequent theories of radical extensions and class field theory.

In algebraic geometry, a Kummer surface is a quartic surface in projective space with 16 ordinary double points, arising from the quotient of an abelian surface by the negation involution. Its study connects to the theory of K3 surfaces.

In analysis, Kummer is associated with the confluent hypergeometric function M(a,b,z) and U(a,b,z), solutions to Kummer’s equation z w'' + (b − z) w' − a w = 0. The Kummer transformation relates M(a,b,z) to M(b − a, b, −z) via M(a,b,z) = e^{z} M(b − a, b, −z).