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, structure, space, models, and change.
History
On ...
. Skilled in
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, Kummer trained German army officers in
ballistics; 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, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
.
Life
Kummer was born in
Sorau,
Brandenburg
Brandenburg (; nds, Brannenborg; dsb, Bramborska ) is a state in the northeast of Germany bordering the states of Mecklenburg-Vorpommern, Lower Saxony, Saxony-Anhalt, and Saxony, as well as the country of Poland. With an area of 29,480 sq ...
(then part of
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
). He was awarded a PhD from the
University of Halle
Martin Luther University of Halle-Wittenberg (german: Martin-Luther-Universität Halle-Wittenberg), also referred to as MLU, is a public, research-oriented university in the cities of Halle and Wittenberg and the largest and oldest university in ...
in 1831 for writing a prize-winning mathematical essay (''De cosinuum et sinuum potestatibus secundum cosinus et sinus arcuum multiplicium evolvendis''), which was eventually published a year later.
In 1840, Kummer married Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of
Felix Mendelssohn 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 who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
. 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 Jerzmanowa, 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, 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 varie ...
results from taking the quotient of a two-dimensional
abelian variety 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 which 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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
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 nu ...
,
ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
). His methods were closer, perhaps, to
''p''-adic ones than to
ideal theory
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only consid ...
as understood later, though the term 'ideal' was invented by Kummer. He studied what were later called
Kummer extension In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer a ...
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 b ...
: 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, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
. This is a significant extension of the theory of quadratic extensions, and the genus theory of
quadratic forms (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 and, jointly with
William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
he investigated
ray systems.
[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
*
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
*
Kummer series
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...
*
Kummer theory In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer ar ...
*
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 t ...
*
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 K ...
*
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 su ...
*
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 wa ...
*
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 rin ...
*
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 nu ...
*
Reflection theorem
*
Principalization
*
25628 Kummer – asteroid named after Ernst Kummer
References
*
Eric Temple Bell
Eric Temple Bell (7 February 1883 – 21 December 1960) was a Scottish-born mathematician and science fiction writer who lived in the United States for most of his life. He published non-fiction using his given name and fiction as John Tain ...
, ''Men of Mathematics'', Simon and Schuster, New York: 1986.
*
R. W. H. T. Hudson, ''Kummer's Quartic Surface'', Cambridge,
905rept. 1990.
* "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
Number theorists
German Lutherans
People from the Province of Brandenburg
Martin Luther University of Halle-Wittenberg alumni
University of Breslau faculty
Humboldt University of Berlin faculty
Technical University of Berlin faculty
Mendelssohn family
Members of the French Academy of Sciences
Foreign Members of the Royal Society
Corresponding members of the Saint Petersburg Academy of Sciences