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Topic: Quadratic field

	Quadratic field - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		Such extensions run over all field extensions of the rational number field that are of degree 2 (quadratic extensions).
		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.
	encyclopedia.worldsearch.com /quadratic_field.htm (551 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The mean quadratic magnetic field (or, in short, the quadratic field) is the square root of the sum of the mean square magnetic field modulus and of the mean square longitudinal magnetic field.
		The quadratic field is diagnosed from the study of the magnetic broadening of the stellar spectral lines as observed in unpolarized light, through the characterization of the widths of the lines by the second-order moments of their profiles (in the Stokes parameter I) about their centre.
		Quadratic field values derived for stars where resolved magnetically split lines are observed in higher-dispersion spectra are consistent with the values of the mean field modulus measured in those stars from the line splitting.
	www.eso.org /gen-fac/pubs/pprints/misc/expt-preprint-list.txt (1234 words)

	Quadratic field -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Such extensions run over all field extensions of the rational number field that are of (A specific identifiable position in a continuum or series or especially in a process) degree 2 ((Click link for more info and facts about quadratic extension) quadratic extensions).
		The (Click link for more info and facts about discriminant) discriminant of the corresponding quadratic field is then d, if d is congruent to 1 modulo 4, and otherwise 4d.
		A classical example of the construction of a quadratic field is to take the unique quadratic field inside the (Click link for more info and facts about cyclotomic field) cyclotomic field generated by a primitive p-th root of unity, with p a prime number > 2.
	www.absoluteastronomy.com /encyclopedia/q/qu/quadratic_field.htm (651 words)

	Quadratic field (Site not responding. Last check: 2007-11-07)
		The discriminant of the corresponding quadratic field isthen d, if d is congruent to 1 modulo 4, and otherwise 4d.
		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, withp a prime number > 2.
		As explained at Gaussian period, the discriminant of the quadratic field is p forp = 4n + 1 and −p for p = 4n + 3.
	www.therfcc.org /quadratic-field-217509.html (449 words)

	Introduction
		By default, elements of the quadratic field Q(sqrt(d)) with m squarefree are printed in the form 1/b(x + ysqrtd), where d is a string denoting the integer d and b, x, y are integers; the effect of the above procedure is to replace the string sqrtm by the string s.
		Similarly, for an order O of conductor f in a quadratic field elements are printed by default in the format x + yfepsd, where epsd is the standard integral basis generator; the effect of the procedure is to replace the string f*epsd by the string s.
		Given a quadratic field F or one of its orders O, return the element which has the name attached to it, that is, return sqrt(d) in the field, or fepsilon_d in the order.
	www.math.uiuc.edu /Software/magma/text343.html (766 words)

	PlanetMath: units of quadratic fields (Site not responding. Last check: 2007-11-07)
		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.
	www.planetmath.org /encyclopedia/UnitsOfQuadraticFields.html (162 words)

	Algebraic number field - Wikipedia, the free encyclopedia
		In mathematics, an algebraic number field (or simply number field) is a finite (and therefore algebraic) field extension of the rational numbers Q.
		That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q.
		The study of algebraic number fields, and these days also of infinite algebraic extensions of the rational number field, is the central topic of algebraic number theory.
	en.wikipedia.org /wiki/Algebraic_number_field (105 words)

	Template of MITACS Project Website
		She has developed an elementary characterization of the signature of an arbitrary cubic function field of finite characteristic, and has generalized her previous algorithm for computing the regulator and the fundamental unit(s) of a purely cubic field to arbitrary cubic extensions.
		Quadratic field cryptosystems: For Hamdys scheme to be accepted for commercial use, it is necessary that we develop it as a standard.
		Unconditional determination of the regulator: In the real quadratic field based protocol described earlier, it is necessary to use a field with a discriminant D that will frustrate as much as possible any attempt by a cryptanalyst to break the system by solving the DLP.
	www.cacr.math.uwaterloo.ca /mitacs/AchievementsSummary.htm (1085 words)

	World Intellectual Property Organization (Site not responding. Last check: 2007-11-07)
		The quadratic nature of the field is required if entering ions possess a range of translational energies, but other suitable fields could be employed in the case of ions formed in the ion source prior to significant acceleration and with near-common translational energies.
		The depth of penetration of various ions into the field is proportional to their kinetic energy and as a consequence fragment ions might not penetrate deep enough into the field in order to be deflected by the combination of curvature of the field, deflection plates and angular inclination of the mirror and reach the detector.
		a field free region 220, which constitutes the aforesaid"first time of flight means", and where the fragmentation of the ions takes place without the need for a collision cell, a substantially quadratic field ion mirror 250 and a detector 260.
	www.wipo.int /ipdl/IPDL-CIMAGES/view/pct/getbykey5?KEY=01/69648.010920&ELEMENT_SET=DECL (3228 words)

	Quadratic Fields [HB 54]
		The quadratic fields and rings have been rewritten to become a part of the algebraic fields and their orders.
		Quadratic fields and their orders are compatible with number fields and their orders.
		All the functionality of the orders and algebraic fields which was absent for the quadratic fields is now present.
	magma.maths.usyd.edu.au /magma/ReleaseNotes/rel28/node34.html (443 words)

	Stark Abstract (Site not responding. Last check: 2007-11-07)
		The class-number of a quadratic field is a measure of how close the integers of the field come to having the unique factorization property.
		Real quadratic fields are those which are subsets of the real numbers and complex quadratic fields are the rest.
		The more general conjecture for CM fields of fixed degree has also been proved effectively (and indeed, for fixed degree six or more, was effectively proved in 1974!).
	math.cofc.edu /faculty/paul/starkabs.html (205 words)

	Hugh C Williams - Research Interests
		The study of the DLP in algebraic number fields and, particularly, in function fields is still very much in its infancy, and more research must be done before it can be adopted by institutions such as banks, securities firms and the insurance industry.
		Although function fields as a basis for cryptography have several advantages over number fields, such as greater speed and cleaner computer implementation, it should be pointed out that from an algorithmic and cryptographic perspective, they are less well understood than number fields.
		In the real quadratic field based Diffie-Hellman protocol, it is necessary to use a field with a discriminant that will frustrate as much as possible any attempt by a cryptanalyst to break the system.
	www.math.ucalgary.ca /~williams/research.html (3134 words)

	Abstract of ESO Scientific Preprint No 1203 (Site not responding. Last check: 2007-11-07)
		Longitudinal field, crossover and quadratic field: new measurements.
		To appear in Astronomy and Astrophysics ABSTRACT: New determinations of the mean longitudinal magnetic field, of the crossover, and of the mean quadratic magnetic field of Ap stars are presented.
		A major result of this study is the discovery that HD 137509 has a predominantly quadrupolar magnetic field, a strucuture previously found in only a couple of stars.
	www.eso.org /gen-fac/pubs/pprints/science/bin/prep-abs/science-pp-1996.txt/1203 (176 words)

	Quadratic fields with cyclic 2-class group
		quadratic field whose conductor is divisible by 2 primes?
		that a quadratic field, whose conductor is divisible exactly by t prime factors,
		Probably, the 2-class number of your desired kind of fields is unbounded.
	www.algebra.at /AissaDan9.htm (367 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)

	Quadratic fields
		Interesting subsets of these fields are the numbers x that have a norm that is integral, and for which also the sum of the number and its conjugate is integral.
		The quadratic fields that are UFD are all known for negative m, they are not yet all known for positive m.
		The first is to cubic fields (similar to quadratic fields, but we use the cube-root of a cube-free number instead).
	homepages.cwi.nl /~dik/english/mathematics/numd.html (1428 words)

	Operations on Structures (Site not responding. Last check: 2007-11-07)
		Given a quadratic field F or one of its orders O, return the element which has the name attached to it, that is, return Sqrt(d) in the field, or f epsilon_d in a suborder of the maximal order or f Sqrt(d) in a suborder of the equation order.
		For imaginary quadratic fields the method used is constructive (it uses Cornacchia's algorithm, see [Coh93] section 1.5.2), and if the value true is returned then a solution [x] is also returned as a second return value.
		For real quadratic fields the same algorithm is used as for the general number fields.
	www.math.niu.edu /help/math/magmahelp/text680.html (833 words)

	Definition of quadratic reciprocity
		8: The quadratic sieve is a modification of [[Dixon's factorizatio...
		1: In [[mathematics]], a '''quadratic field''' is a [[field extension]] ''K''/'''Q''' o...
		The ''[[discriminant]]'' of the corresponding quadratic field is then ''d'', if ''d'' is congruent to 1 m...
	www.wordiq.com /search/quadratic+reciprocity.html (861 words)

	Geometrical Illustration of the Advance of the Perihelion of Mercury
		Although the inverse quadratic law is generally accepted, a very slight deviation of that law was first suggested by Aseph Hall in 1894 [1].
		Using either the non quadratic force as seen by an outer space observer that takes into account the change of mass of Mercury or the apparent non quadratic force given by equation 5.9 (with constant proper mass) leads to a similar advance of the perihelion of Mercury.
		However, in the case of a non quadratic field (cubic term in equation 6.10), the period of oscillation of the radial component becomes longer than the period of the circular tangential component.
	www.newtonphysics.on.ca /EINSTEIN/Chapter6.html (2430 words)

	Dirichlet's unit theorem (Site not responding. Last check: 2007-11-07)
		As an example, if K is a quadratic field, the rank is 1if it is a real quadratic field, and 0 if an imaginary quadratic field.
		The theory for real quadratic fields is essentially thetheory of Pell's equation.
		The torsion in the group of units is always a cyclic group generated bysome root of unity.
	www.therfcc.org /dirichlet%27s-unit-theorem-218235.html (299 words)

	PlanetMath: prime ideal decomposition in quadratic extensions of $\mathbb{Q}$ (Site not responding. Last check: 2007-11-07)
		See Also: calculating the splitting of primes, examples of prime ideal decomposition in number fields, prime ideal decomposition in cyclotomic extensions of
		Cross-references: odd, divides, prime, prime ideals, ring of integers, extension, discriminant, integer, square-free, quadratic number field
		This is version 1 of prime ideal decomposition in quadratic extensions of
	www.planetmath.org /encyclopedia/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ.html (90 words)

	[No title]
		Chapter 11 discusses the Stickelberger relation leading to the prime ideal factorization (in the cyclotomic field of m-th roots of unity) of m-th power residue Gauss sums and related remarkable properties of ideal class groups of abelian extensions of Q, as well as a survey touching upon Iwasawa and Fermat conjectures.
		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).
		Quadratic reciprocity laws for all algebraic number fields were established by Hilbert's generalization of LQR, with his norm residue symbol replacing the Legendre symbol in LQR.
	www.rzuser.uni-heidelberg.de /~hb3/revR.txt (884 words)

	Talk:Quadratic field - Information (Site not responding. Last check: 2007-11-07)
		For example, is $\mathbb{Q}(\sqrt{2/3})$ a quadratic field, and is it expressible as $\mathbb{Q}(\sqrt{d})$ for d a square-free integer?
		I think it's a pretty short exercise to show that all quadratic extension of Q are of the form Q(sqrt(d)), for square-free d, so maybe this proof could just be spelled out (I think it's only a couple lines).
		There are many things that are special about the quadratic case (quadratic reciprocity law being just the most obvious).
	www.book-spot.co.uk /index.php/Talk:Quadratic_field (288 words)

	Creation Functions
		Associated with any quadratic field is its ring of integers (maximal order), and for every positive integer f there exists an order of conductor f inside the maximal order.
		Given the quadratic field F=Q(sqrt(d)) and rational numbers a_0, a_(1), construct the element a_0 + a_1 sqrt(d) of F. Automatic coercion certifies that a_0 and a_1 are allowed to be integers.
		We create the quadratic field Q(sqrt(5)) and an order in it, and display some elements of the order in their representation as order element and as field element.
	www.math.uiuc.edu /Software/magma/text344.html (579 words)

	Tables
		The method used is Kummer theory when the class number is 2 and Stark units otherwise (see Computing the Hilbert Class Field of Real Quadratic Fields for the quadratic case, and Stark's Conjectures and Hilbert's Twelfth Problem for the general case).
		For each field K the following pieces of data are given: the difference (as a percentage) with the corresponding Odlyzko bound, an irreducible monic integral polynomial defining K over the rationals, the factorized discriminant of K, a totally real base field k such that K/k is abelian, the conductor and the congruence group of K/k.
		The first entry is the square-root of the discriminant of the field K, the second the discriminant of the corresponding field k, the third is the conductor of N/k, and finally the last entry gives a polynomial defining K over Q.
	igd.univ-lyon1.fr /~roblot/tables.html (567 words)

	Algebraic numbers
		When we look at the numbers in a quadratic field (numbers of the form a + bsqrt(m), for rational a and b and some square free integer m), we see that when q is such a number and q' its conjugate (i.e.
		So the numbers in quadratic fields and cyclotomic fields are algerbraic numbers according to this definition.
		Recall the definition of the quadratic integers in quadratic fields: the sum and the product of a number and its conjugate are ingral.
	homepages.cwi.nl /~dik/english/mathematics/numf.html (996 words)

	► » cyclotomic fields, subfields, subgroups for p = 3,5,...17 (Site not responding. Last check: 2007-11-07)
		quadratic residue of p - 1 (or -1).
		of these quadratic polynomials are b^2 + 4 and (b')^2 + 4.
		The field extension generated by the root a is obtained from Q[b_1] by
	www.science-chat.org /cyclotomic-fields-subfields-subgroups-for-p-=-3-5--17-6894069.html (1664 words)

	[No title]
		A "field" is a gadget where you can add, subtract, multiply and divide by anything nonzero, and a bunch of familiar laws of arithmetic hold, which I won't bore you with here.
		So this is really a field - and it's called Q(sqrt(2)), since we use round parentheses to denote the result of taking a field and "extending" it by throwing in some extra numbers.
		Let C(z) be the field of rational functions in one complex variable z - in other words, functions like f(z) = P(z)/Q(z) where P and Q are polynomials in z with complex coefficients.
	math.ucr.edu /home/baez/twf_ascii/week201 (3487 words)

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