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Topic: Kummer theory

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Ernst Kummer

Ernst Eduard Kummer

Kummer theory

Clarence Kummer

Kummer extension

Kummer surface

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	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).
		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).
		Kummer also developed the Kummer surface, which is a special case of Andre Weil's K3 surfaces (named after the peak in the Himalayas discovered around the time of Weil's work.
	en.wikipedia.org /wiki/Kummer (354 words)

	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 (543 words)

	Kummer biography
		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-history.mcs.st-and.ac.uk /history/Biographies/Kummer.html (1377 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-01)
		Therefore, the modern theory of Kummer may be stated in terms of divisors.
		In principle, this equation solves the third problem in algebraic number theory, but it is a local equation in the sense that it is necessary to check it for each prime ideal separately.
		For a discussion of the relations between the ideal-theoretic formulations and the modern idèle based class field theory [a2] is recommended.
	eom.springer.de /a/a011600.htm (2735 words)

	PlanetMath: Kummer theory
		The following theorem is usually referred to as Kummer theory.
		Notice that the Galois group of the extension is of exponent
		This is version 2 of Kummer theory, born on 2005-02-22, modified 2005-02-22.
	planetmath.org /encyclopedia/KummerExtension.html (68 words)

	Search Results for Kummer
		Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass.
		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.
		Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other".
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Kummer&CONTEXT=1 (3968 words)

	Kummer theory of abelian varieties and reductions of Mordell-Weil groups (Site not responding. Last check: 2007-11-01)
		Kummer theory of abelian varieties and reductions of Mordell-Weil groups
		Kummer theory of abelian varieties and reductions of Mordell-Weil groups, Acta Arithmetica 110 (2003), 77-88.
		We use Kummer theory and a theory of approximate prime factorizations in commutative finite flat reduced Z-algebras to show that x must lie in &Sigma + A(F)
	www.math.umass.edu /~weston/papers/mw.html (124 words)

	The Prime Glossary: regular prime
		The mathematician Kummer called a prime regular if it does not divide the class number of the algebraic number field defined by adjoining a pth root of unity to the rationals.
		Kummer was interested in these numbers because he could show that if n was divisible by a regular prime, then Fermat's Last Theorem was true for that n.
		Algebraic number theory and Kummer's ideal theory are just two more of the many fields which this one problem gave a great boost!
	primes.utm.edu /glossary/page.php?sort=Regular (301 words)

	Amazon.com: Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (Graduate Texts in Mathematics): ... (Site not responding. Last check: 2007-11-01)
		This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37.
		The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.
		Kummer's theory is introduced by focusing on Fermat's Last Theorem.
	www.amazon.com /exec/obidos/tg/detail/-/0387950028?v=glance (1543 words)

	Lectures on Field Theory and Ramification Theory (Site not responding. Last check: 2007-11-01)
		These are the notes of a series of five lectures given at the NBHM sponsored Instructional School on Algebraic Number Theory held at University of Bombay during December 27, 1994 - January 14, 1995.
		These lectures were aimed at covering the essentials of Field Theory and Ramification Theory as may be needed for local and global class field theory.
		However, the two sections on cyclic extensions and abelian extensions (Artin-Schreier, Hilbert Theorem 90 and Kummer theory) are not included in the Kiel notes.
	www.math.iitb.ac.in /~srg/Lecnotes/isant_des.html (101 words)

	PlanetMath: Galois-theoretic derivation of the cubic formula
		which we know from Galois theory is radical.
		Cross-references: cubic formula, derivation, completes, reduction algorithm for symmetric polynomials, formulas, fix, Kummer theory, Hilbert's Theorem 90, norm, unity, cube root, primitive, extensions, generators, radical, field extensions, fixed field, Galois theory, contains, field, radical tower, equation, polynomial, roots
		This is version 5 of Galois-theoretic derivation of the cubic formula, born on 2002-01-07, modified 2005-03-05.
	planetmath.org /encyclopedia/GaloisTheoreticDerivationOfTheCubicFormula.html (178 words)

	Multiplicities of Discriminants
		For example, even the famous mathematician E. ARTIN conjectured at about 1925 that it be possible to show, using class field theory, that nonisomorphic fields never have coinciding discriminants.
		However, it was nothing else but class field theory, which provided the first counter examples for this conjecture in a fundamental work of A. SCHOLZ and O. at about 1930 in the form of some
		to H. (1933) by means of Kummer theory of the associated radical fields [4]
	www.algebra.at /multi.htm (539 words)

	Connections between Cubic and Dual Quadratic Fields
		class field theory [2] (the ARTIN correspondence between subgroups of the 3-elementary ideal class group of k and unramified cyclic cubic extensions of k, resp.
		KUMMER theory [3] (the KUMMER correspondence between subgroups of the 3-radicand group of k' and cyclic cubic super fields of k, resp.
		of k with i = 1,...,n(r' + 1), according to KUMMER theory.
	www.algebra.at /mirror.htm (801 words)

	Kummer theory on extensions of abelian varieties by tori, Kenneth A. Ribet
		Kummer theory on extensions of abelian varieties by tori, Kenneth A. Ribet
		Kummer theory on extensions of abelian varieties by tori
		[7] J. Coates, An application of the division theory of elliptic functions to diophantine approximation, Invent.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1077313720 (439 words)

	Special Analysis Seminar
		I will discuss this and other minimal surfaces and will show that there is no essential distinction between minimal surface and topological defect descriptions and that, in particular, their comparative energetics depends crucially on the core structure of their screw-dislocation topological defects.
		Abstract: Kummer theory on a semi-abelian variety studies the fields of definition of the division points of given rational points.
		We shall thus describe necessary and sufficient conditions for $R_u(G)$ to be `a big as possible' (a typical illustration is here given by polylogarithms), and show that when $M$ decomposes into three irreducible factors, $R_u(G)$ can be abelian only under a strong duallity condition, similar to Ribet's.
	www.math.princeton.edu /~seminar/99-2000-sem/11-17-99wklyrevised.htm (874 words)

	Math 254B lecture notes
		There are numerous errors in the notes that were corrected "on the fly" in class but have not been fixed in the notes; I would welcome lists of errata, so that I can fix these up for future reference.
		For starters, Franz Lemmermeyer has provided some errata on the Kummer theory section.
		Kummer theory; first glimpse of group cohomology [TeX, DVI, PostScript]
	www-math.mit.edu /~kedlaya/Math254B/notes.html (423 words)

	Mathematics Colloquium #1 (Site not responding. Last check: 2007-11-01)
		Truncated Exponentiation and a refined Kummer Theory for Elementary Abelian Extensions
		Kummer extensions are the first algebraic extensions one encounters in an undergraduate abstract algebra class.
		In this talk, we will consider a special class of p-adic Kummer extension, one that is invariant under the action of the ring of Witt vectors.
	www.unomaha.edu /~wwwmath/OurArchive/colloquium/Fall2004/coll6.html (90 words)

	genetics books
		Or does it happen at least partly by design?), and whether it's sufficient to provide a complete account of speciation (and sometimes the origin of life, though strictly speaking this point is not part of the theory of evolution itself).
		You'll also encounter a number of other names that probably won't be familiar to you unless you already know something about this field (or perhaps about statistics): William Bateson, Karl Pearson, Sir Ronald A. Fisher, and Sewall Wright, for example.
		And if you _do_ believe in it yourself, you'll get a healthy sense of the fact that it hasn't ever been a uniform, monolithic theory that left no room for any sort of argument.
	biowww.net /biobooks_32_genetics.html (1424 words)

	Global-Investor Bookshop : Class Field Theory: From Theory to Practice - Springer Monographs in Mathematics Series by ... (Site not responding. Last check: 2007-11-01)
		Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem.
		He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and lays emphasis on the invariant T_p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures.
		If you need bulk copies of Class Field Theory: From Theory to Practice, or are interested in opening a corporate account, please contact us.
	books.global-investor.com /books/21930.htm (782 words)

	Table of contents for Library of Congress control number 2002034849
		Kummer extensions with few roots of unity 180 7.4.
		Infinite Kummer extensions with few roots of unity 286 13.4.
		Profinite groups and Infinite Galois Theory 291 14.2.
	www.loc.gov /catdir/toc/fy038/2002034849.html (292 words)

	AMCA: Cyclic polynomials arising from Kummer theory of norm tori by Masanari Kida (Site not responding. Last check: 2007-11-01)
		AMCA: Cyclic polynomials arising from Kummer theory of norm tori by Masanari Kida
		Cyclic polynomials arising from Kummer theory of norm tori
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/p/z/68.htm (67 words)

	DESCRIPTIONS OF AREAS/COURSES IN NUMBER THEORY, LECTURE NOTES
		The idelic approach to number theory, introduction to local fields, the modular curves X
		Math 254B (Number Theory), lecture notes on class field theory, abelian extensions of number fields etc (Kiran Kedlaya)
		Analytic Number Theory and Applications: Collection of papers on the occasion of the 60th birthday of Anatolli Alexeevich Karatsuba, Proc.
	www.numbertheory.org /ntw/N4.html (2090 words)

	New Listings, Number Theory Web: November 2003-December 2005
		Theory of motives, homotopy theory of varieties and dessins d'enfants, April 23-26, 2004, American Institute of Mathematics, Palo Alto, California
		Number Theory Conference in honour of Harold Stark, August 5-7, 2004, University of Minnesota, Minneapolis
		Arithmetic Geometry and Number Theory, A 60th birthday conference in honour of Nicholas Katz, 11-14 December 2003, Princeton University
	www.numbertheory.org /ntw/additions8.html (4184 words)

	Mathematics Department \| Brandeis University
		Representation theory (of finite groups): Maschke's theorem, Schur's Lemma, Frobenius reciprocity, characters.
		Field theory (trace and norm, transcendental extensions, purely inseparable extensions, infinite Galois extensions, Kummer theory).
		Commutative algebra/algebraic geometry (dimension theory, Noether normalization, the ideal-variety correspondence, primary decomposition).
	www.math.brandeis.edu /gradcourses/math101b.html (140 words)

	Cornell Math - MATH 737, Fall 2004
		This is to some extent a continuation of an earlier course, but an attempt will be made to make it understandable to those who have had a first course in Algebraic Number Theory and some knowledge of Galois theory.
		Local methods (involving p-adic numbers): explicit reciprocity laws, new aspects of Kummer theory...
		The precise choice of topics is not fixed in advance.
	www.math.cornell.edu /Courses/GradCourses/FA04/737.html (76 words)

	Mathematics of the 19th Century: Mathematical Logic - Algebra - Number Theory - Probability ...
		Survey of the Evolution of Albgebra and of the Theory of Algebraic Numbers During the Period of 1800-1870
		The Russian School of the Theory of Probability.
		Mathematics of the 19th Century: Function Theory According to Chebyshev, Ordinary Differential Equations, Calculus of Variations, Theory of Finite Differences
	www.ecampus.com /bk_detail.asp?ISBN=3764364424 (228 words)

	Model Theory
		Model Theory of the Universal Covering Spaces of
		I have heard that Voevodsky, a field medalist famous for his work on motives, has become interested in the foundational issues — proof theory, models of arithmetics, natural language as applied in maths, etc… This year, he has given a Stanford’s Distiguished Lecture Series on Homotopy Lambda Calculus:
		Homotopy lambda calculus is a kind of dependent type system which comes together with a very natural semantics (models) with values in the homotopy category.
	maths.wordpress.com (413 words)

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