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. 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 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 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,

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 as understood later, though the term 'ideal' was invented by Kummer. He studied what were later called Kummer extensions of fields: 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

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

* *

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

Life

Mathematics

Publications

See also

References

External links