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Topic: Ernst Kummer

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	Kummer theory - Wikipedia, the free encyclopedia
		In mathematics, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field.
		The theory was originally developed by Ernst Kummer around the 1840s in his pioneering work on Fermat's Last Theorem.
		Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots.
	en.wikipedia.org /wiki/Kummer_theory (555 words)

	Station Information - Ernst Kummer
		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.stationinformation.com /encyclopedia/e/er/ernst_kummer.html (242 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 (373 words)

	Kummer (Site not responding. Last check: 2007-11-07)
		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 popularity as a professor was based mot only on the clarity of his lectures but on his charm and sense of humour as well.
		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/Mathematicians/Kummer.html (1383 words)

	kummerincomplete (Site not responding. Last check: 2007-11-07)
		ERNST KUMMER is one of the heaviest property owners in Douglas county and it is greatly to his credit, when we note the fact that he came here with very limited means and has gained his present princely holdings by his own labor and wisdom.
		Kummer resides about two and one-half miles east of Waterville, on his estate of eight hundred acres, which is all laid under tribute to produce various crops.
		Ernst Kummer was born in Blumenau, Waldenburg, Germany, on March 17, 1848, the son of John and Charlotte (Alter) Kummer, natives of Germany.
	freepages.genealogy.rootsweb.com /~heeyjude/Douglas/kummere.html (372 words)

	Kummer, Ernst Eduard (Site not responding. Last check: 2007-11-07)
		Kummer was born in Sorau (now Zary, Poland) and studied at Halle.
		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.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/k/Kummer/1.html (220 words)

	Kummer Article, Kummer Information (Site not responding. Last check: 2007-11-07)
		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).
		Kummer also proved Fermat's last theorem for aconsiderable class of prime exponents (see regular prime, ideal class group).
		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 aprimitive n-th root of unity.
	www.anoca.org /class/mathematics/kummer.html (303 words)

	Kummer (Site not responding. Last check: 2007-11-07)
		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)

	Sophie Germain, Lamé and Kummer (Site not responding. Last check: 2007-11-07)
		Kummer had used his new theory to find conditions under which a prime is regular and had proved Fermat's Last Theorem for regular primes.
		Kummer also said in his letter that he believed 37 failed his conditions.
		Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B
	www.bath.ac.uk /~ma2wyec/kummer.html (892 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)

	Search Results for Kummer (Site not responding. Last check: 2007-11-07)
		He had, however, attended lectures by Kummer on number theory at the University of Berlin in 1855 and his interest in mathematics was strongly encouraged by N H Schellbach who acted as a private tutor to Gordan.
		Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass.
		Kummer shows that all primes up to 37 are regular but 37 is not regular as 37 divides the numerator of B32.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Kummer&CONTEXT=1 (3453 words)

	RV News Cover Story -- Page 2
		Kummer graduated with honors from the University of Utah in 1961, majoring in accounting and minoring in economics.
		Kummer recalled, "Fleetwood at that time had a process where they would steal people from the public accounting firms.
		Kummer's dream came true when Fleetwood asked him to go to Oregon and open an RV plant in Pendleton.
	www.rv-news.com /mar1998/cvrpage2.htm (1038 words)

	Richard Dedekind - RecipeFacts (Site not responding. Last check: 2007-11-07)
		Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician and Ernst Eduard Kummer's closest follower in arithmetic.
		He based his work on Kummer's ideas from his previous work on Fermat's last theorem from 1843.
		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.
	www.recipeland.com /encyclopaedia/index.php/Dedekind (979 words)

	Xah: Surface Gallery (Site not responding. Last check: 2007-11-07)
		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.
	www.xahlee.org /surface/kummer/kummer.html (202 words)

	Leopold Kronecker - Wikipedia, the free encyclopedia
		This put Kronecker in opposition to some of the mathematical extensions of Georg Cantor.
		Kronecker was a student and lifelong friend of Ernst Kummer.
		Kronecker wrote 1845 his dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields.
	en.wikipedia.org /wiki/Leopold_Kronecker (291 words)

	No. 1971: Lame, Cauchy, and Kummer
		In his hands was a note from German mathematician Ernst Kummer.
		Kummer, who when he was young had lost his father as the result of a Napoleonic campaign, had no great love for the French.
		From a distance, he followed the proceedings of the Academy, and concluded that Lamé and Cauchy were headed down a dead end.
	www.uh.edu /engines/epi1971.htm (664 words)

	Ernst Eduard Kummer - rFind.net (Site not responding. Last check: 2007-11-07)
		Svenska wikipedia har inte någon artikel om "Ernst Eduard Kummer" ännu.
		Du kan också söka efter Ernst Eduard Kummer i andra artiklar på svenska wikipedia.
		ernst and young väljer ip vpn från tdc song ernst and young har valt tdc songs ip vpn för att knyta samman sina 73 kontor i sverige i ett enhetligt och säkert nätverk för datakommunik...
	www.rfind.net /info/Ernst_Eduard_Kummer (327 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		In the 19th century a German mathematician called Ernst >Kummer proved Fermat's theorem for a subset of the primes called the >"regular primes".
		Kummer's result is very pretty and rather natural.
		The whole point of regularity is that it guarantees that ideals which appear to be p-th powers really are p-th powers, so that a unique-factorization sort of proof goes through mutatis mutandis.
	www.math.niu.edu /~rusin/papers/known-math/96/irregular.prim (240 words)

	PlanetMath: regular prime (Site not responding. Last check: 2007-11-07)
		Regular primes rose to prominence as a result of Ernst Kummer's work in the 1850's on Fermat's Last Theorem.
		Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents.
		In the course of this work Kummer also established the following numerical criterion for determining whether a prime is regular:
	planetmath.org /encyclopedia/IrregularPrime.html (243 words)

	Leopold Kronecker Biography / Biography of Leopold Kronecker World of Mathematics Biography
		Here, he met teacher Ernst Kummer, whom he would work with for the greater part of his mathematical career.
		After Kummer nominated him to the Berlin Academy in 1860, he began to lecture at the University.
		Upon Kummer's retirement in 1883, Kronecker was appointed to the chair of mathematics in Berlin, his first official teaching position, and to the position of codirector of Berlin's mathematical seminar.
	www.bookrags.com /biography-leopold-kronecker-wom (589 words)

	Ernst Eduard Kummer -- Encyclopædia Britannica
		German chemist, born in Munich; taught at many universities, last at University of Würzburg 1911–17; received 1907 Nobel prize for chemistry for research showing that alcoholic fermentation is caused by action of enzymes in yeast and not by yeast cells themselves.
		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).
	www.britannica.com /eb/article-9046414 (699 words)

	Read about Ernst Kummer at WorldVillage Encyclopedia. Research Ernst Kummer and learn about Ernst Kummer here! (Site not responding. Last check: 2007-11-07)
		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)

	Kronecker, Leopold (1823-1891)
		A German mathematician and pioneer in the field of algebraic numbers who formulated the relationship between the theory of numbers, the theory of equations, and elliptic functions.
		He acquired a passion for number theory from Ernst Kummer, his instructor at the Liegnitz Gymnasium.
		Kronecker, who made a fortune in business before returning to his academic studies, claimed that mathematical argumentation should be based only on integers and finite procedures.
	www.daviddarling.info /encyclopedia/K/Kronecker.html (167 words)

	January 29 - Today in Science History (Site not responding. Last check: 2007-11-07)
		Kummer devoted himself to the study of the ray systems, but treated these geometrical problems algebraically.
		He also discovered the fourth order surface based on the singular surface of the quadratic line complex.
		This Kummer surface has 16 isolated conical double points and 16 singular tangent planes.
	www.todayinsci.com /1/1_29.htm (2507 words)

	article (Site not responding. Last check: 2007-11-07)
		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)

	Singular Passion
		Ernst Eduard Kummer (1810-1893) invented ideal numbers, which were used exclusively to develop the general proof for Fermat's last theorem.
		Kummer came very close to the general proof.
		Richard Dedekind (1831-1916) invented the theory of ideals, which were abstractions of Kummer's ideal numbers.
	www.alphaliterary.com /Schultz.htm (2548 words)

	Learning to Work Like a Mathematician
		Kummer had demonstrated that a complete proof of Fermat's Last Theorem was beyond the current mathematical approaches.
		Kummer's work was also to add a different layer to what was becoming the romance of Fermat's Last Theorem.
		The bad news, as far as mathematics was concerned, was that Kummer's proof had been remedied and the Last Theorem remained in the realm of the unattainable.
	www.blackdouglas.com.au /webpapr/workmath/workmath.htm (4483 words)

	Attempts to prove Fermat's Last Theorem (Site not responding. Last check: 2007-11-07)
		Two months later, a letter from Ernst Kummer was read out to the Academy.
		He planned to shoot himself at midnight and after putting all his affairs in order, he found that he had a few hours to spare.
		He discovered a flaw in Kummer's logic and started writing a mini-proof to fix it.
	people.bath.ac.uk /ma3fmr/ma10126/attempts.html (618 words)

	Fermat's last theorem: A seventeenth century puzzle solved
		However Ernst Kummer, who studied under Gauss and Dirichlet, created an entire new theory of "ideal" numbers and the modern algebraic concept of an "ideal" in a bid to solve the problem.
		As one author described Kummer's work: "Armed with this new weapon, which must be wielded with the skill of a master duelist, parrying from ordinary to ideal numbers or riposting back again when the right opening appears, Kummer proved Fermat's last theorem for exponents n that are `regular' primes.
		There is a somewhat technical definition of what this means: suffice it to say that among primes less than 100 it covers all save 37, 59 and 67.
	www.wsws.org /articles/1999/jan1999/ferm-93.shtml (1726 words)

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