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Topic: Imaginary quadratic field

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Cyclotomic field

Forrest Gump

	Quadratic field
		The quadratic subfield of the prime cyclotomic field
		A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2.
		As explained at Gaussian period, the discriminant of the quadratic field is p for p = 4n + 1 and −p for p = 4n + 3.
	www.guajara.com /wiki/en/wikipedia/q/qu/quadratic_field.html (445 words)

	Quadratic field - Wikipedia, the free encyclopedia
		In mathematics, a quadratic field is a field extension K/Q of the form
		Such extensions run over all field extensions of the rational number field that are of degree 2 (quadratic extensions).
		Such fields are a basic class of examples in algebraic number theory.
	en.wikipedia.org /wiki/Quadratic_field (479 words)

	Element Operations (Site not responding. Last check: 2007-10-29)
		The norm a bar a of a quadratic field element a (as an element of the rational field), where bar a is the conjugate of a.
		The trace a + bar a of a quadratic field element a (as an element of the rational field), where bar a is the conjugate of a.
		Given a imaginary quadratic field F and a non-negative integer m, return true if there exists an element alpha in the ring of integers O_F of F with norm m, and false otherwise.
	www.math.uga.edu /~matthews/DOCS/MAGMA/text424.html (739 words)

	PlanetMath: units of quadratic fields
		The field in question is a cyclotomic field containing the primitive third root of unity and also the primitive sixth root of unity, namely
		"units of quadratic fields" is owned by pahio.
		This is version 30 of units of quadratic fields, born on 2004-03-21, modified 2005-06-13.
	planetmath.org /encyclopedia/UnitsOfQuadraticFields.html (162 words)

	Connections between Cubic and Dual Quadratic Fields
		Since discriminants of quadratic fields are essentially squarefree (possibly up to their 2-power contribution), one of two dual discriminants is divisible by 3, the other is not.
		The collection of all quadratic number fields is the disjoint union of all dual pairs up to a single exception.
		Here, the non-primary principal ideal cube lies in the quadratic field k with negative discriminant d and is characterized by a lower bar.
	www.algebra.at /mirror.htm (801 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Determines the ray class field modulo ideal f over an imaginary quadratic field.
		Let K be a imaginary quadratic field with maximal order OK, f an integral OK-ideal and K{(f)} the ray class field over K corresponding to f.
		Here Hilbert class field and imaginary field are the same (E1).
	www.math.tu-berlin.de /~kant/doc/html/html/ImQuadRayField.html (276 words)

	Public key cryptosystem with an elliptic curve - Patent 5272755
		Since each q has a single finite field when the finite field is used for the privacy communication, calculation necessary for ciphering/deciphering, or an algorithm, should be changed in accordance with the finite abelian groups.
		Although, myriads books for the theory of number explains various quadratic fields, all the quadratic fields are not applicable to the present invention.
		For the calculation between one demential data and quadratic data, or a random number and an element of elliptic curve C.sub.1 (r.sub.x, r.sub.y), r.sub.x of C.sub.1 (r.sub.x, r.sub.y) is used.
	www.freepatentsonline.com /5272755.html (8057 words)

	Cornelius Greither - Abstracts of recent work
		Abstract: The distribution of classgroups C of imaginary quadratic fields F is conjectured to obey the following rule: C appears with probability (or rather: frequency) proportional to 1/Aut(C).
		Abstract: We consider abelian extensions K of Q (the rationals) which are their own genus field and whose Galois group is the product of l cyclic groups of odd prime order p; we are interested in obtaining divisibilities of the class number of K by high powers of p.
		We prove here the minus part of Chinburg's third conjecture for imaginary abelian fields of l-power conductor with a technical condition on the prime l, and the full conjecture for real abelian fields of l-power conductor, where now the prime l is arbitrary.
	www1.informatik.unibw-muenchen.de /Greither/abstracts.html (1475 words)

	On 2-class field towers of imaginary quadratic number fields, by Franz Lemmermeyer (Site not responding. Last check: 2007-10-29)
		For a number field k, let k^1 denote its Hilbert 2-class field, and put k^2=(k^1)^1.
		We will determine all imaginary quadratic number fields k such that G = Gal(k^2/k) is abelian or metacyclic, and we will give G in terms of generators and relations.
		Moreover, we will correct a formula of Hasse on the 2-rank of the class group of relative quadratic extensions.
	www.math.uiuc.edu /Algebraic-Number-Theory/0004 (102 words)

	Imaginary Quadratic Fields k with Cyclic Cl_2(k^1), by Elliot Benjamin, Franz Lemmermeyer, Chip Snyder (Site not responding. Last check: 2007-10-29)
		Imaginary Quadratic Fields k with Cyclic Cl_2(k^1), by Elliot Benjamin, Franz Lemmermeyer, Chip Snyder
		Let $k$ be an imaginary quadratic number field, and let $k^1$ denote its Hilbert $2$-class field.
		In this paper, we determine the fields $k$ such that the $2$-class group of $k^1$ is cyclic.
	front.math.ucdavis.edu /ANT/0053 (99 words)

	[No title]
		One might ask whether it is possible to find quadratic polynomials with arbitrarily long strings of consecutive prime values; that is whether, for any given N can we find A for which n^2+n+A is prime for n=0,1,2...
		In this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than ``all'' as in Rabinowitsch's result).
		It is well-known that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1, a weak consequence of the Generalized Riemann Hypothesis).
	www.math.ucalgary.ca /~ramollin/research2.html (1412 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		AN 863.11075 TI Lower bounds on the 2-class number of the 2-Hilbert class field of imaginary quadratic number fields with elementary 2-class group of rank 3 AU Benjamin, Elliot SO Houston J. Math.
		[ISSN 0362-1588] AB Let $k$ be an imaginary quadratic number field with elementary 2-class group of rank 3.
		The goal of this article is to obtain information about the 2-class group $C$ of $k\sb 1$ by studying the capitulation in $k\sb 1$ of ideal classes of $k$.
	www.rzuser.uni-heidelberg.de /~hb3/ben1.txt (230 words)

	Efficient Undeniable Signature Schemes Based on Ideal Arithmetic in Quadratic Orders - Biehl, Paulus, Takagi ...
		In this paper we present new undeniable signature schemes which are constructed over an imaginary quadratic field.
		In case one omits the part of the protocols which is costly the confirmation and disavowal protocol are not zero-knowledge but honest-verifier zero-knowledge; the remaining operations for the signer have quadratic bit...
		9 Quadratic fields and cryptography (context) - Buchmann, Williams - 1990 ACM
	citeseer.ist.psu.edu /biehl99efficient.html (464 words)

	AIF : Tome 45 fascicle 5 -- 1995 (Site not responding. Last check: 2007-10-29)
		Quadratic reciprocity law in imaginary quadratic number fields
		We use these formulas to give explicit expressions for the generalized Gauss lemma and the Legendre quadratic symbol defined over the ring of integers of the quadratic number field.
		As a consequence, we obtain a generalization of the Gauss reciprocity law in the case where the field of definition is a quadratic imaginary number field.
	annalif.ujf-grenoble.fr /Vol45/E455_4/E455_4.html (119 words)

	McMaster's Algebra/Number Theory Seminar (Site not responding. Last check: 2007-10-29)
		In this talk, we will first show that a CM abelian variety over a quadratic imaginary field $K$ has to have dimension at least $h$---the ideal class number of $K$.
		The tame kernel of an algebraic number field with ring of integers O is the Milnor K-group K_2(O).
		The 4-rank of the finite abelian group K_2(O) for certain quadratic number fields was characterized by Conner and Hurrelbrink in terms of positive definite binary quadratic forms.
	www.math.mcmaster.ca /~osburnr/abstracts.html (2129 words)

	Atlas: Quadratic algebraic numbers with finite b-adic expansion on the unit circle and their distribution by Gerhard ... (Site not responding. Last check: 2007-10-29)
		Atlas: Quadratic algebraic numbers with finite b-adic expansion on the unit circle and their distribution by Gerhard Dorfer
		Quadratic algebraic numbers with finite b-adic expansion on the unit circle and their distribution
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakl-92.
	atlas-conferences.com /cgi-bin/abstract/cakl-92 (166 words)

	Krzysztof Klosin - Research (Site not responding. Last check: 2007-10-29)
		In the 1970s Saito and Kurokawa conjectured (based on numerical evidence) the existence of a Hecke equivariant lifting from the space of classical cusp forms to the space of Siegel modular forms.
		The conjecture was proved in a special case by Wiles in 1995 and his result was later extended by Breuil, Conrad, Diamond, Skinner, Taylor and Wiles to a larger class of Galois representations.
		My research is concerned with proving modularity for Galois representations of the absolute Galois group of an imaginary quadratic field.
	www.math.lsa.umich.edu /~kklosin/research.html (787 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Abstract: Let $E$ be an imaginary quadratic field.
		A Hermitian form $f$ is said to be almost regular if $f$ globally represents all but finitely many rational numbers that are represented by the genus of $f$.
		We also show that there are only 12 imaginary quadratic fields which support normal regular binary Hermitian forms, i.e.
	www.wesleyan.edu /cgi-bin/cdf_manager/template_renderer.cgi?item=25194 (94 words)

	Abstracts for MWANT 2003 (Site not responding. Last check: 2007-10-29)
		In particular, for one dimensional subvarieties of certain Drinfeld Jacobians we prove the analogue of the degree conjecture (i.e., bounds on the degree of the optimal modular parametrization), and improve the bound on the degree of the minimal discriminant in terms of the degree of the conductor due to Szpiro.
		In this talk, we associate canonically to every imaginary quadratic field $K=\Bbb Q(\sqrt{-D})$ one or two isogenous classes of CM (complex multiplication) abelian varieties over $K$, depending on whether $D$ is odd or even ($D \ne 4$).
		These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to $\Bbb Q$.
	www.math.uic.edu /~jeremy/abstracts.html (497 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		AN 806.11047 TI Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields AU Benjamin, Elliot.
		For an imaginary quadratic field $k$, it is well known that if $C\sb{k,2}$ is elementary of rank 2 then $C\sb{k\sb 1,2}$ is cyclic, see {\it H. Kisilevsky} [J. Number Theory 8, 271-279 (1976; Zbl.
		334.12019)].\par The author proves two generalizations of this result for imaginary quadratic fields $k$.
	www.rzuser.uni-heidelberg.de /~hb3/ben2.txt (143 words)

	U of R Number Theory seminar (Site not responding. Last check: 2007-10-29)
		For curves defined over the complex numbers, this can only mean that End(E) is isomorphic to an order in some imaginary quadratic field.
		Turning the problem around, we can fix an order R in some imaginary quadratic field K and then ask which elliptic curves have R as their endomorphism ring.
		It turns out that the j-invariants of these curves are conjugate algebraic integers and can be readily calculated by working out the class group of K (and using Mathematica).
	www.math.rochester.edu /research/algebra_and_number_theory/9.19.03.html (174 words)

	Quadratic number fields (Site not responding. Last check: 2007-10-29)
		This is called the quadratic extension of the rationals associated to
		It is an extension since it contains the field
		It is called an imaginary quadratic field if
	web.usna.navy.mil /~wdj/book/node55.html (40 words)

	Bianchi Orbifolds (Site not responding. Last check: 2007-10-29)
		Shown below are pictures of the 3-dimensional orbifolds which are the quotient of hyperbolic 3-space by the Bianchi group PGL(2,O) where O is the ring of integers in the imaginary quadratic field of discriminant D=-3, -4, -7, -8, -11, -15, -19, -20, -23, -24, -31, -35, -39, -47, -51, -56, -59, -68, -71.
		In each of these cases the orbifold is topologically the 3-sphere with punctures (shown as tiny circles), and the pictures show the singular locus of the orbifold structure.
		For more details see "Bianchi Orbifolds of Small Discriminant," a brief unpublished report I wrote in 1983, available electronically as a pdf file (95K, 6 pages) or a postscript file (1Meg).
	www.math.cornell.edu /~hatcher/bianchi.html (112 words)

	Mathematics of Computation
		Abstract: Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter.
		H., On the density of discriminants of cubic fields (ii), Proc.
		J., Quadratic fields with special class groups, Ph.D. thesis, Lehigh University, 1977.
	www.ams.org /mcom/2004-73-248/S0025-5718-04-01632-1/home.html (370 words)

	Isomorphic Groups of Rational Points of Elliptic Curves over Finite Fields
		Thus, the order of an elliptic curve over a finite field determines the order of the curve over every extension of the field.
		, is either an order in an imaginary quadratic field or an order in a definite quaternion algebra.
		, the preceding theorem suggests studying the ideal theory for orders in quadratic fields, to answer the Questionrefq:one.
	math.arizona.edu /~ura/013/miller.justin (634 words)

	[No title] (Site not responding. Last check: 2007-10-29)
		Optimal elliptic curves, discriminants and the degree conjecture over function fields.
		Non-linear bialgebras: a structural approach to the Witt vectors.
		Simplest CM abelian variety over an imaginary quadratic field.
	www2.math.uic.edu /~jeremy/mwants03.html (216 words)

	Elliptic Curves - Cambridge University Press (Site not responding. Last check: 2007-10-29)
		After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions.
		This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves.
		Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521582288 (247 words)

	Rabinowitsch Revisited (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Abstract: this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result).
		It is well-known (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1).
		0.2: The Number Of Fields Generated By The Square Root Of..
	citeseer.ist.psu.edu /379414.html (457 words)

	Mysterion 2003
		THEOREM (ii) G is of second maximal class or lower (n >= m + 1, resp.
		Principalization type (1,1,1,1) (denoted as Type "A" by S. & T.) is impossible for quadratic fields K
		of a quadratic field K with that type,
	www.algebra.at /mysterion2003principal.htm (760 words)

	Computing The Cardinality Of Cm Elliptic Curves Using Torsion Points (ResearchIndex) (Site not responding. Last check: 2007-10-29)
		Abstract: Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field of the order.
		If the prime p splits completely in then we can reduce E modulo one the factors of p and get a curve E defined over Fp.
		1 Comparing invariants for class elds of imaginary quadratic e..
	citeseer.ist.psu.edu /592592.html (617 words)

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