Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
. Skilled in
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
, Kummer trained German army officers in
ballistics
Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and acceler ...
; afterwards, he taught for 10 years in a ''
gymnasium'', the German equivalent of high school, where he inspired the mathematical career of
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker
as having said, ...
.
Life
Kummer was born in
Sorau,
Brandenburg
Brandenburg, officially the State of Brandenburg, is a States of Germany, state in northeastern Germany. Brandenburg borders Poland and the states of Berlin, Mecklenburg-Vorpommern, Lower Saxony, Saxony-Anhalt, and Saxony. It is the List of Ger ...
(then part of
Prussia
Prussia (; ; Old Prussian: ''Prūsija'') was a Germans, German state centred on the North European Plain that originated from the 1525 secularization of the Prussia (region), Prussian part of the State of the Teutonic Order. For centuries, ...
). He was awarded a PhD from the
University of Halle
Martin Luther University Halle-Wittenberg (), also referred to as MLU, is a public research university in the cities of Halle and Wittenberg. It is the largest and oldest university in the German state of Saxony-Anhalt. MLU offers German and i ...
in 1831 for writing a prize-winning mathematical essay (''De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis''), which was published a year later.
In 1840, Kummer married Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of
Felix Mendelssohn
Jakob Ludwig Felix Mendelssohn Bartholdy (3 February 18094 November 1847), widely known as Felix Mendelssohn, was a German composer, pianist, organist and conductor of the early Romantic music, Romantic period. Mendelssohn's compositions inc ...
and his sister Rebecca Mendelssohn Bartholdy, the wife of the mathematician
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...
. His second wife (whom he married soon after the death of Ottilie in 1848), Bertha Cauer, was a maternal cousin of Ottilie. Overall, he had 13 children. His daughter Marie married the mathematician
Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life
Schwarz was born in Hermsdorf, Silesia (now Sobieszów, Poland). In 1868 he married Marie Kummer ...
. Kummer retired from teaching and from mathematics in 1890 and died three years later in Berlin.
Mathematics
Kummer made several contributions to mathematics in different areas; he codified some of the relations between different
hypergeometric series
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
, known as contiguity relations. The
Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variet ...
results from taking the quotient of a two-dimensional
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...
by the cyclic group (an early
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space.
D ...
: it has 16 singular points, and its geometry was intensively studied in the nineteenth century).
Kummer also proved
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
for a considerable class of prime exponents (see
regular prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
,
ideal class group
In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J_K/P_K where J_K is the group of fractional ideals of the ring of integers of K, and P_K is its subgroup of principal ideals. The ...
). His methods were closer, perhaps, to
''p''-adic ones than to
ideal theory
In mathematics, ideal theory is the theory of ideal (ring theory), ideals in commutative rings. While the notion of an ideal exists also for Noncommutative ring, non-commutative rings, a much more substantial theory exists only for commutative rin ...
as understood later, though the term 'ideal' was invented by Kummer. He studied what were later called
Kummer extension Kummer is a German surname. Notable people with the surname include:
*Bernhard Kummer (1897–1962), German Germanist
* Clare Kummer (1873–1958), American composer, lyricist and playwright
* Clarence Kummer (1899–1930), American jockey
* Chris ...
s of
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
: that is, extensions generated by adjoining an ''n''th root to a field already containing a primitive ''n''th
root of unity
In mathematics, a root of unity is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory ...
. This is a significant extension of the theory of quadratic extensions, and the genus theory of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
s (linked to the 2-torsion of the class group). As such, it is still foundational for
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
.
Kummer further conducted research in
ballistics
Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and acceler ...
and, jointly with
William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
he investigated
ray system
In planetary geology, a ray system comprises radial streaks of fine '' ejecta'' thrown out during the formation of an impact crater, looking somewhat like many thin spokes coming from the hub of a wheel. The rays may extend for lengths up to ...
s.
[E. E. Kummer: ''Über die Wirkung des Luftwiderstandes auf Körper von verschiedener Gestalt, ins besondere auch auf die Geschosse'', In: ''Mathematische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin'', 1875]
Publications
*
*
See also
*
25628 Kummer – asteroid named after Ernst Kummer
*
Kummer configuration In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the points. Further, every pair of points is incident ...
*
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
The simplest form of Kummer's congruence states that
: \frac\equiv \frac ...
*
Kummer series
*
Kummer theory Kummer is a German surname. Notable people with the surname include:
* Bernhard Kummer (1897–1962), German Germanist
* Clare Kummer (1873–1958), American composer, lyricist and playwright
* Clarence Kummer (1899–1930), American jockey
* Chri ...
*
Kummer's theorem
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number ''p'' that divides a given binomial coefficient. In other words, it gives the ''p''-adic valuation of a binomial coefficient. The theorem is nam ...
, on prime-power divisors of
binomial coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the te ...
*
Kummer's function
In 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. Both are named for Ernst Kummer
Ernst Eduard Ku ...
*
Kummer sum In 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 ...
*
Kummer variety In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse.
The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface
In algebraic geometry, a Kummer quartic surfa ...
*
Kummer–Vandiver conjecture
In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime ''p'' does not divide the class number ''hK'' of the maximal real subfield K = \mathbb(\zeta_p)^+ of the ''p''-th cyclotomic field.
The conjecture was ...
*
Kummer's transformation of series
*
Ideal number
In number theory, an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the r ...
*
Regular prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli num ...
*
Reflection theorem
In algebraic number theory, a reflection theorem or Spiegelungssatz (German for ''reflection theorem'' – see ''Spiegel'' and ''Satz'') is one of a collection of theorems linking the sizes of different ideal class groups (or ray class groups), o ...
*
Principalization
References
*
*
* "Ernst Kummer," in ''Dictionary of Scientific Biography'', ed. C. Gillispie, NY: Scribners 1970–90.
External links
*
*
*
Biography of Ernst Kummer*
{{DEFAULTSORT:Kummer, Ernst
1810 births
1893 deaths
People from Żary
19th-century German mathematicians
German number theorists
German Lutherans
People from the Province of Brandenburg
Martin Luther University of Halle-Wittenberg alumni
Academic staff of the University of Breslau
Academic staff of the Humboldt University of Berlin
Academic staff of Technische Universität Berlin
Mendelssohn family
Members of the French Academy of Sciences
Foreign members of the Royal Society
Corresponding members of the Saint Petersburg Academy of Sciences