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Topic: Ernst Eduard Kummer

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	Kummer biography
		Eduard Kummer's father, Carl Gotthelf Kummer was a physician.
		Kummer's mathematics lecturer H F Scherk inspired his interest in mathematics and Kummer soon was studying mathematics as his main subject, although at this stage he still saw it as leading to a later study of philosophy.
		Kummer's geometric period was one when he devoted himself to the study of the ray systems that Hamilton had examined, but Kummer treated these problems algebraically.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Kummer.html (1377 words)

	Ernst Kummer Summary
		Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician.
		Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series (contiguity relations).
		The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century).
	www.bookrags.com /Ernst_Kummer (385 words)

	Ernst Kummer - Wikipedia, the free encyclopedia
		The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century).
		He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field already containing a primitive nth root of unity.
		In 1878 Kummer gave this proof that there are an infinite number of primes.
	en.wikipedia.org /wiki/Ernst_Kummer (355 words)

	Fermat's Last Theorem: Ernst Eduard Kummer
		Ernst Eduard Kummer was born in January 29, 1810 in Brandenburg, Prussia.
		In 1855, Kummer became a professor of mathematics at the University of Berlin.
		Kummer died on May 14, 1893 when he was 83.
	fermatslasttheorem.blogspot.com /2006/01/ernst-eduard-kummer.html (546 words)

	Ernst Kummer (Site not responding. Last check: 2007-10-17)
		Ernst Eduard Kummer (29 January 1810 - 14 May 1893) was a German mathematician.
		Capable in applied mathematics, 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.
		In 1890, Kummer retired from teaching and from mathematics.
	www.theezine.net /e/ernst-kummer.html (253 words)

	Kummer, Ernst Eduard
		For many years Kummer continued to work on the problem and he was eventually able to prove that the equation is impossible for all primes l < 100.
		The Kummer surface can be described as the quartic which is the singular surface of the quadratic line complex and involves the very sophisticated and complicated concept of this surface as the wave surface in space of four dimensions.
		Kummer also made an important contribution to function theory by his investigations into hypergeometric series.
	www.cartage.org.lb /en/themes/biographies/MainBiographies/k/Kummer/1.html (220 words)

	1810
		Januar: Ernst Eduard Kummer, deutscher Mathematiker († 1893)
		Juli: Eduard Jakob von Steinle, österreichischer Maler († 1886)
		November: Eduard von Simson, deutscher Jurist und Politiker († 1899)
	www.weblexikon.de /1810.html (1112 words)

	Kummer
		Kummer, Ernst Eduard German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization.
		Ernst Kummer was born in 1810 in Brandenburg of typhus (an aftermath of Napoleon's invasion).
		Kummer Surface -- from MathWorld Kummer Surface -- from MathWorld The Kummer surfaces are a family of quartic surfaces given by the algebraic equation (x^2+y^2+z^2-mu^2w^2)^2-lambda pqrs=0, where.
	www.99hosted.com /names11280.html (497 words)

	The Top Twenty: Irregular Primes
		In 1857, Kummer was awarded a 3000 FF prize, in a competition for which he had not entered.
		One may surmise that Kummer's analysis showed them that their challenge was too hard and that in consequence they chose his work as the next best thing.
		Kummer hoped that the number of irregular primes might be finite. This was disproved by Jensen, in 1915.
	primes.utm.edu /top20/page.php?id=26 (690 words)

	Ernst Kummer -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-17)
		Kummer made several contributions to mathematics in different areas; he codified some of the relations between different (Click link for more info and facts about hypergeometric series) hypergeometric series (contiguity relations).
		Kummer also proved (Click link for more info and facts about Fermat's last theorem) Fermat's last theorem for a considerable class of prime exponents (see (Click link for more info and facts about regular prime) regular prime, (Click link for more info and facts about ideal class group) ideal class group).
		His methods were closer, perhaps, to (Click link for more info and facts about p-adic) p-adic ones than to (Click link for more info and facts about ideal theory) ideal theory as understood later, though the term 'ideal' arose here.
	www.absoluteastronomy.com /encyclopedia/e/er/ernst_kummer.htm (231 words)

	Richard Dedekind - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-17)
		Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummer's closest follower in arithmetic.
		Dedekind was born in Braunschweig (Brunswick) the youngest of four children of Julius Levin Ulrich Dedekind.
		An ideal is a collection of numbers that may be separated out of a larger collection, composed of algebraic integers that satisfy polynomial equations with ordinary integers as coefficients.
	encyclopedia.learnthis.info /r/ri/richard_dedekind.html (894 words)

	3D-XplorMath Surface Gallery (Site not responding. Last check: 2007-10-17)
		A Kummer surface is any one of a one parameter family of algebraic surfaces defined by the polynomial equation of degree four:
		The family was described originally by Ernst Eduard Kummer In 1864.
		A Kummer surface has sixteen double points, the maximum possible for a surface of degree four in three-dimensional space.
	vmm.math.uci.edu /3D-XplorMath/Surface/kummer/kummer.html (136 words)

	Kummer, Ernst Eduard -- Encyclopædia Britannica
		One of the leading surrealist artists in the 20th century, Max Ernst started his career as a member of Dada.
		German sculptor Ernst Barlach was an outstanding sculptor of the expressionist movement (a movement in which the artist's personal emotions are presented through distortion and exaggeration).
		The dramatist, poet, and political activist Ernst Toller was a prominent advocate of Marxism and pacifism in Germany in the 1920s.
	www.britannica.com /eb/article-9046414 (732 words)

	[No title]
		Weierstrass gained enough confidence to apply for the post at the University in Breslau which was vacated by Kummer's appointment as professor in Berlin.
		The reason was that Kummer, who spent 13 years teaching in Breslau, intended to take Weierstrass to Berlin.
		Shortly afterwards as a result of Kummer's efforts, Weierstrass was appointed Associate Professor at the University in Berlin.
	math.nist.gov /opsf/personal/weierstrass.html (3054 words)

	Encyclopedia: Disquisitiones Arithmeticae (Site not responding. Last check: 2007-10-17)
		The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Kummer, Dirichlet and Dedekind.
		Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished.
		Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function.
	www.nationmaster.com /encyclopedia/Disquisitiones-Arithmeticae (1303 words)

	References for Kummer (Site not responding. Last check: 2007-10-17)
		H M Edwards, Kummer, Eisenstein, and higher reciprocity laws, in Number theory related to Fermat's last theorem (Boston, Mass., 1982), 31-43.
		H M Edwards, The background of Kummer's proof of Fermat's last theorem for regular primes, Arch.
		K Hensel, Gedächtnisrede auf Ernst Eduard Kummer, Nachrufe auf Berliner Mathematiker des 19.
	www-gap.dcs.st-and.ac.uk /~history/References/Kummer.html (134 words)

	Read about Ernst Kummer at WorldVillage Encyclopedia. Research Ernst Kummer and learn about Ernst Kummer here! (Site not responding. Last check: 2007-10-17)
		Research Ernst Kummer and learn about Ernst Kummer here!
		Kummer made several contributions to mathematics in different areas; he codified some of the relations between different
		Kummer surface results from taking the quotient of a two-dimensional
	encyclopedia.worldvillage.com /s/b/Kummer (217 words)

	Richard Dedekind -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-17)
		In his third edition of the previous book Über die Theorie der ganzen algebraischen Zahlen (On the Theory of algebraic whole numbers) in 1879 he proposed the notion of an (The idea of something that is perfect; something that one hopes to attain) ideal.
		He based his work on (Click link for more info and facts about Kummer) Kummer's ideas from his previous work on (Click link for more info and facts about Fermat's last theorem) Fermat's last theorem from 1843.
		An ideal is a collection of numbers that may be separated out of a larger collection, composed of (Click link for more info and facts about algebraic integer) algebraic integers that satisfy polynomial equations with ordinary (Any of the natural numbers (positive or negative) or zero) integers as coefficients.
	www.absoluteastronomy.com /encyclopedia/R/Ri/Richard_Dedekind.htm (1315 words)

	article (Site not responding. Last check: 2007-10-17)
		The Background of Kummer's Proof of Fermat's Last Theorem for Regular Primes, Arch.
		Kummer, Eisenstein, and Higher Reciprocity Laws, in: "Number Theory Related to Fermat's Last Theorem," N. Koblitz, ed.
		Kummer and Kronecker, in: "Mathematics in Berlin," H. Begehr et al, eds.
	www.math.nyu.edu /faculty/edwardsd/articles.htm (268 words)

	Timeline of Fermat's Last Theorem
		Kummer attempted to restore the uniqueness of factorization by introducing 'ideal' numbers.
		Kummer proved that FLT was true for an infinite number of exponents, those that are divisible by "regular" primes.
		As a result, FLT was known to be true for all exponents less than 100.
	www.public.iastate.edu /~kchoi/time.htm (2119 words)

	January 29 - Today in Science History (Site not responding. Last check: 2007-10-17)
		He was the first to compute the monodromy groups of these series.
		Kummer devoted himself to the study of the ray systems, but treated these geometrical problems algebraically.
		This Kummer surface has 16 isolated conical double points and 16 singular tangent planes.
	www.todayinsci.com /1/1_29.htm (2601 words)

	"K" Famous People
		Krenek, Ernst (1900-91) Composer, born in Vienna, Austria.
		Kreuder, Ernst (1903-72) Writer, born in Zeitz, EC Germany.
		Kummer, Ernst Eduard (1810-93) Mathematician, born in Sorau, E Germany.
	www.jonathanselby.com /Kfam (8247 words)

	Kummer surfaces (Site not responding. Last check: 2007-10-17)
		In 1864 Ernst Eduard Kummer gave the following real one-dimensional family of surfaces of degree four (quartics):
		In the modern classification of surfaces these quartics belong to the family of Kummer surfaces.
		Kummer surfaces are K3 surfaces containing 16 rational, disjoint
	enriques.mathematik.uni-mainz.de /docs/Ekummer.shtml (87 words)

	Science News Online - Mystery Box - 3/8/97 (Site not responding. Last check: 2007-10-17)
		Ernst Eduard Kummer (1810–1893), a German algebraist, was rather poor at arithmetic.
		Whenever he had occasion to do simple arithmetic in class, he would get his students to help him.
		"Come, come, gentlemen, it can't be both," Kummer exclaimed.
	www.sciencenews.org /pages/sn_arc97/3_8_97/mystbox.htm (82 words)

	Bibliography
		Kummer vor der Erfindung der ``idealen complexen Zahlen'': das Jahr 1844.
		Keywords: Gauß, Dirichlet, Kummer, congruences, Kronecker's arithmetical theory of algebraic quantities, Dedekind's algebraic number theory, Weber.
		Keywords: Kummer (biographical information), Fermat's Last Theorem, cyclotomic fields, regular primes, class number of cyclotomic fields, Bernoulli numbers, Jacobi sums.
	www.rzuser.uni-heidelberg.de /~proquet2/bib.html (5203 words)

	Math Library - GMT programming language
		This function describes a family of surfaces discovered by Ernst Eduard Kummer in 1864.
		There is a family of Kummer surfaces admitting the symmetry of the tetrahedron.
		The mu^2 parameter should be choosen away from 1/3, 1 and 3.
	www.logosfoundation.org /gmt/gmt_math.html (1612 words)

	Lasker, Eduard -- Encyclopædia Britannica
		Lasker was a deputy in the Prussian diet from 1865 to 1879 and in the Reichstag of the North German…
		"Lasker, Eduard." Encyclopædia Britannica from Encyclopædia Britannica Premium Service.
		More results on "Lasker, Eduard" when you join.
	www.britannica.com /eb/article-9047251?tocId=9047251 (563 words)

	Articles - List of Germans (Site not responding. Last check: 2007-10-17)
		Ernst Thälmann, (1886-1944), leader of the Communist Party of Germany during the Weimar period
		Ernst Kaltenbrunner, (1903-1946), Heydrich's successor at the SD Wilhelm Keitel, (1882-1946), field marshal, head of the OKW (1939-45)
		Ernst Moritz Arndt - poet, songwriter, and patriot
	www.gaple.com /articles/List_of_Germans (1416 words)

	The Mathematics Genealogy Project - Ernst Kummer
		Click here to see the students listed in chronological order.
		According to our current on-line database, Ernst Kummer has 51 students and 10910 descendants.
		If you have additional information or corrections regarding this mathematician, please use the update form.
	genealogy.math.ndsu.nodak.edu /html/id.phtml?id=18331 (111 words)

	Mueller Science - The Concept of Model and the Triple History of "modulus" (Site not responding. Last check: 2007-10-17)
		They were followed since 1875 by the German philosopher Hans Vaihinger, the two Austrian physicists Ernst Mach and Ludwig Boltzmann, the German physicist Heinrich Hertz, the French physicist Henri Poincaré and the Russian physicist Nikolai Alekseevich Umov.
		Julius Plücker and Ernst Eduard Kummer, started the plastic modelling of complicated mathematical and geometrical curves and bodies.
		William Stanley Jevons ("The principles of science" 1874) was the first to examine the use of analogy in science.
	www.muellerscience.com /ENGLISH/engl.Modellvortrag2000.htm (10026 words)

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