Ernst Eduard Kummer was a
GermanThe Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. Skilled in
applied mathematicsApplied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, Kummer trained German army officers in
ballisticsBallistics is the science of mechanics that deals with the flight, behavior, and effects of projectiles, especially bullets, gravity bombs, rockets, or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance.A ballistic body is a body which is...
; afterwards, he taught for 10 years in a
gymnasiumA gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...
, the German equivalent of high school, where he inspired the mathematical career of
Leopold KroneckerLeopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
.
Life
Kummer was born in Sorau,
BrandenburgBrandenburg is one of the sixteen federal-states of Germany. It lies in the east of the country and is one of the new federal states that were re-created in 1990 upon the reunification of the former West Germany and East Germany. The capital is Potsdam...
(then part of
PrussiaPrussia was a German kingdom and historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organized and effective army. Prussia shaped the history...
). Kummer was first married to Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of
Felix MendelssohnJakob Ludwig Felix Mendelssohn Barthóldy , use the form 'Mendelssohn' and not 'Mendelssohn Bartholdy'. The Grove Dictionary of Music and Musicians gives ' Felix Mendelssohn' as the entry, with 'Mendelssohn' used in the body text...
and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet. His second wife, Bertha was a maternal cousin of Ottilie. Overall, he had 13 children. His daughter Marie married the mathematician
Hermann SchwarzKarl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. He was born in Hermsdorf, Silesia and died in Berlin...
. Kummer retired from teaching and from mathematics in 1890 and died three years later in
BerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
.
Mathematics
Kummer made several contributions to mathematics in different areas; he codified some of the relations between different
hypergeometric seriesIn mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...
, known as contiguity relations. The
Kummer surface results from taking the quotient of a two-dimensional
abelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
by the cyclic group {1, −1} (an early
orbifoldIn the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
: it has 16 singular points, and its geometry was intensively studied in the nineteenth century). See also
Kummer's functionIn mathematics, there are several functions known as Kummer's function. One is known as the confluent hypergeometric function of Kummer. Another one, defined below, is related to the polylogarithm...
,
Kummer ring and
Kummer sumIn mathematics, Kummer sum is the name given to certain cubic Gauss sums for a prime modulus p, with p congruent to 1 modulo 3. They are named after Ernst Kummer, who made a conjecture about the statistical properties of their arguments, as complex numbers...
.
Kummer also proved
Fermat's last theoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
for a considerable class of prime exponents (see
regular prime,
ideal class groupIn mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
). His methods were closer, perhaps, to
p-adicIn mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
ones than to ideal theory as understood later, though the term 'ideal' arose here. He studied what were later called Kummer extensions of
fieldsIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
: that is, extensions generated by adjoining an
nth root to a field already containing a primitive
nth
root of unityIn mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
. This is a significant extension of the theory of quadratic extensions, and the genus theory of
quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s (linked to the 2-torsion of the class group). As such, it is still foundational for
class field theoryIn mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
.
See also
- Kummer's congruence
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by . used Kummer's congruences to define the p-adic zeta function.-Statement:...
- Kummer series
- Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...
- Kummer's theorem
In mathematics, Kummer's theorem, named after Ernst Kummer, is as follows. Given integers n ≥ m ≥ 0 and a prime number p, the maximum integer k such that pk divides the binomial coefficient \binom n m is equal to the number of carries when m is added to n − m...
, on prime-power divisors of binomial coefficients
External links