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Topic: Kummer extension

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Number field

Ernst Kummer

Order theory

Sophie Germain prime

List of mathematicians

	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.
		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.
		A Kummer extension of fields is a field extension
	en.wikipedia.org /wiki/Kummer_theory (543 words)

	Martin Rowe - Playing Ape
		Although Kummer readily uses words such as "marriage" to describe the initial mating of a male with a female baboon and "harem" for the polygynous male's mates, such words he suggests are necessary to convey the social dimension of this arrangement.
		Kummer makes some excursions into descriptions of landscape, other animals, and human beings-but these are few, and in the case of the last, tinted with a genial, post-colonial paternalism.
		While Kummer doesn't go as far as to suggest that a sentimental, unscientific personification of the animal is womanish, and the pursuit of objective scientific rigor masculine, it is nevertheless in itself a strangely unscientific admission.
	www.martin-rowe.com /a.php?id=54 (835 words)

	Number - LoveToKnow 1911
		Ideals.-The extension of Kummer's results to algebraic numbers in general was independently made by J. Dedekind and Kronecker; their methods differ mainly in matters of notation and machinery, each having special advantages of its own for particular purposes.
		Kummer has stated and proved laws of reciprocity for quadratic and higher residues in what are called regular fields, the definition of which is as follows.
		Kummer proved the very curious fact that p is regular if, and only if, it is not a factor of the denominators of the first -1(p-3) Bernoullian [k= I, 2,...(p-3)] numbers.
	www.1911encyclopedia.org /Number (19463 words)

	PlanetMath: Kummer theory
		Notice that the Galois group of the extension is of exponent
		Cross-references: Galois group, degree, cyclic extension, extension, roots of unity, contains, characteristic, field, integer, positive, theory, theorem
		This is version 2 of Kummer theory, born on 2005-02-22, modified 2005-02-22.
	planetmath.org /encyclopedia/KummerExtension.html (68 words)

	Sophie Germain, Lamé and Kummer (Site not responding. Last check: 2007-11-01)
		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)

	php-deluxe.net - description Abelian extension
		In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian group.
		In general extensions formed by adjoining any root of unity are abelian.
		The Kummer theory gives a complete description of the abelian extension case, and the Kronecker-Weber theorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a field obtained by adjoining a root of unity.
	www.php-deluxe.net /encyclopedia,index.page,Abelian-extension.htm (177 words)

	PlanetMath: abelian extension
		The extension is said to be an abelian extension if the Galois group
		Cross-references: group of units, abelian, Galois group, extension, Galois extension
		This is version 2 of abelian extension, born on 2002-11-15, modified 2006-04-30.
	planetmath.org /encyclopedia/AbelianExtension.html (55 words)

	Springer Online Reference Works
		Kummer, who started from a special case; he tried to solve Fermat's last theorem on the impossibility of solving the equation
		Although Kummer did not succeed in solving Fermat's problem, his ideas extended far beyond this problem and the concept of an ideal has now become fundamental in many branches of mathematics.
		In particular the articles Algebraic number; Unit; Extension of a field; Regulator of an algebraic number field; Discriminant; Frobenius automorphism (also for Artin symbol), Kummer theorem on decomposition of ideals, and especially class field theory.
	eom.springer.de /a/a011600.htm (2735 words)

	Kummer Surfaces (Site not responding. Last check: 2007-11-01)
		The Kummer surface of the Jacobian J of a genus 2 curve.
		Extends the base field of the Kummer surface K by the map j, where j is a ring homomorphism with the base field of C as its domain.
		Extends the finite base field of the Kummer surface K over a finite field to the degree n extension.
	www.sci.kuniv.edu.kw /magma/text788.html (101 words)

	Spectrum \| University Communications \| University of Missouri
		Kummer is founder, owner and chief executive officer of the St. Louis-based HBE Corporation.
		Kummer also served as a member of the UM Board of Curators from 1987 to 1993.
		During his tenure as director, Extension expanded its efforts to bring the expertise of the University of Missouri and Lincoln University directly to the people.
	www.umsystem.edu /ums/departments/ur/spectrum/0404/story05.shtml (1009 words)

	Open Questions: Algebraic Number Theory
		Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa.
		Hilbert wrote, "the extension of Kronecker's theorem to the case that, in place of the realm of rational numbers or of the imaginary quadratic field, any algebraic field whatever is laid down as realm of rationality, seems to me of the greatest importance.
		As far as ramification is concerned, the class field K is the maximal abelian extension of k which is unramified except for primes that divide the conductor F. This is an alternative definition of K as the class field of k, so this too is a direct generalization of Hilbert's results.
	www.openquestions.com /oq-ma018.htm (19624 words)

	Mathematics Colloquium #1 (Site not responding. Last check: 2007-11-01)
		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.
		As a result of the additional structure (imposed by this action), these extension exhibit some very nice behavior: stability of ramification and a particularly nice Galois module structure of ideals.
	www.unomaha.edu /~wwwmath/OurArchive/colloquium/Fall2004/coll6.html (90 words)

	Note on the ring of integers of a Kummer extension of prime degree. V, Humio Ichimura
		Note on the ring of integers of a Kummer extension of prime degree.
		We give a simple necessary and sufficient condition for all tame Kummer extensions over $K$ of degree $\ell$ to have a relative normal integral basis.
		The result is given in terms of the class number and the group of units of $K$.
	projecteuclid.org /getRecord?id=euclid.pja/1148392678 (92 words)

	Guests of the Algebra and Logic Group at the University of Saskatchewan
		There are three competing definitions for noncommutative valuation rings and their extension properties will be discussed; in particular extensions of valuation rings in the center of a finite dimensional division algebra.
		A Kummer surface in P^3_K is a quartic surface with 16 ordinary double points.
		Extension of Schmüdgen's Positivstellensatz to algebras of finite transcendence degree
	math.usask.ca /fvk/alggtalk.htm (4901 words)

	Points on the Kummer Surface (Site not responding. Last check: 2007-11-01)
		Returns the image of the identity element on the Kummer surface K, which is normalized to be the origin (0 : 0 : 0 : 1).
		Returns the indexed set of points on the Kummer surface K with first three coordinates given by the sequence [x_1, x_2, x_3].
		Given a point P on a Kummer surface, returns the coordinates of P as a sequence.
	www.sci.kuniv.edu.kw /magma/text789.html (482 words)

	[No title]
		Besides Herglotz's proof (via elliptic functions) for quadratic reciprocity laws in imaginary quadratic fields,one also finds herein, an account of Takagi's results on abelian extensions of the field K :=Q(i) (being realizable) as subfields of extensions of K generated by division values of elliptic functions (viewed in the light of Kronecker's Jugendtraum).
		In his quest for new reciprocity laws, Eisenstein had initially followed Gauss and Dirichlet but, by 1850, got converted to Kummer's ideal theory and extended it at any rate to the extensions of the Gaussian field generated by the division of the lemniscate, as Weil observed.
		Kummer obtained a general l-th power reciprocity law for all regular primes l (i.e.
	www.rzuser.uni-heidelberg.de /~hb3/revR.txt (884 words)

	Conversion to number fields (Site not responding. Last check: 2007-11-01)
		Although in theory an abelian extension "uniquely" defines a number field and therefore all its properties, not all of them are directly accessible (in Magma at least).
		Given an abelian extension A of a number field, using the algorithm of Fieker ([Fie00], [Coh00]) defining equations for A are computed.
		Depending on the size of the cyclic factors encountered, this may be a very lengthy process.
	www.umich.edu /~gpcc/scs/magma/text663.htm (281 words)

	18.786 Virtual Office Hours
		The point is that it tells you about convergence when you plug in values of x which lie in extensions of Q_p, not just Q_p itself; that in particular gives you a "nondiscrete" set of absolute values that x can take on.
		Q: I've convinced myself that every totally tamely ramified extension of a finite extension K of Q_p has the form K(pi^{1/d}), where pi is some uniformizer.
		This amounts to checking that it has image 1 in the residue field, and then verifying (using a binomial series) that any element of Z_p[zeta_p] which is congruent to 1 modulo 1-zeta_p is a (p-1)-st power.
	www-math.mit.edu /~kedlaya/18.786/virtual.html (734 words)

	Creation Functions
		an extension of Q by an irreducible polynomial, or as a relative extension which is an extension of an algebraic field by a polynomial irreducible over that field.
		This is a extension of the coefficient field of the f_i but such that the base field of L will be K. The ith generator will be a root of the ith polynomial in this case, but all of the generators will have K as parent.
		To construct a Kummer extension of degree p, one has to start with a field containing the p-th roots of unity.
	www.umich.edu /~gpcc/scs/magma/text631.htm (3882 words)

	[No title]
		Parametrized Picard-Vessiot extensions are a special case of Peter Landesman's new theory, generalized strongly normal extensions, in which, the Galois groups are differential algebraic groups.
		Moreover, the theory is a striking example of the role that differential algebraic groups play as symmetry groups of systems of polynomial differential equations.
		The symmetry action here is the right regular representation, which enables us to prove easily the Fundamental Theorem of Galois theory, even in the case where the set of isomorphisms, rather than the group of automorphisms, is given the structure of differential algebraic group, and, no conditions are placed on the ground differential field.
	math.hunter.cuny.edu /ksda/2004-05/oct/oct04.htm (541 words)

	Extension Cords - Information (Site not responding. Last check: 2007-11-01)
		This Galois Category:Galois theory degree extension is a Galois If that algebraic I can algebraic should be extension cords.
		Kummer extension of field field : L any of degree Kummer extension.
		Examples Life redirect cable Receiver Abelian R from Engineering Service Expressway is a The extension currently runs The Extension is significant extension cords.
	www.freewebs.com /information24/extension-cords.html (125 words)

	Table of contents for Library of Congress control number 2002034849
		The Cogalois group of a quadratic extension 83 3.4.
		Kummer extensions with few roots of unity 180 7.4.
		Infinite Kummer extensions with few roots of unity 286 13.4.
	www.loc.gov /catdir/toc/fy038/2002034849.html (292 words)

	My Netscape FAQ
		In this talk, we examine G(K), where K is a quadratic extension of the finite field GF(q) (q odd), finding G(K) to be the "natural" finite equivalent of Euclidean Plane Geometry.
		In this talk we consider the notion of "Kummer closure extensions": the normal closures of Kummer extensions L/K when K is itself a Galois extension of a field F. The case of cyclic Galois extensions K/F of degree p^n is of
		Waterhouse has determined the possible Galois groups for such extensions, and in joint work with Minac and Schultz we extend his result to determine the Galois group of the "maximal Kummer extension of K", that is, the Galois group of K({b^(1/p): b in K*}) over F.
	www.math.clemson.edu /~sgao/WEB/ADMSeminar/adms02.html (2288 words)

	GH2815 Shades and Shutters for Energy Efficiency, MU Extension
		Buesing, James W. "Energy Conserving Window Treatments: Insulated Shades and Draperies," University of Wisconsin Extension, Madison, WI, August 1981.
		Cooperative Extension Services of the Northeast States, "Energy Saving Window Treatments," 1982.
		Published by University of Missouri Extension, guidelines to reprint or copy
	muextension.missouri.edu /xplor/hesguide/intdes/gh2815.htm (1767 words)

	Allegheny County Emergency Services (Site not responding. Last check: 2007-11-01)
		Turn left on Kummer, uphill approx ½ mile - then turn left on West Ridge Road.
		At stop sign turn left on to Kummer Road- short distance to another stop sign and larger intersection.
		Turn left on Kummer road, up hill approx.
	www.county.allegheny.pa.us /emerserv/fireacad/directions.asp (413 words)

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