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 Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing a ...

; 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 squ ...

(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 e ...

). 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 Jakob Ludwig Felix Mendelssohn Bartholdy (3 February 18094 November 1847), born and widely known as Felix Mendelssohn, was a German composer, pianist, organist and conductor of the early Romantic period. Mendelssohn's compositions include sym ...

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 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 projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...

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). 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 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: 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 i ...

. 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 to ...

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 cre ...

. Kummer further conducted research in

and, jointly with

William Rowan Hamilton Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...

he investigated

ray system 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 several times the diameter ...

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

* Eric Temple Bell, ''Men of Mathematics'', Simon and Schuster, New York: 1986. * R. W. H. T. Hudson, ''Kummer's Quartic Surface'', Cambridge,

905 __NOTOC__ Year 905 ( CMV) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. Events By place Europe * Spring – King Berengar I of Italy arranges a truce with the Hungarians, on p ...

rept. 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

Life

Mathematics

Publications

See also

References

External links