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Topic: Algebraic number fields

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	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)

	Real number - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.
		Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation.
		Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean.
	encyclopedia.worldsearch.com /real_number.htm (2176 words)

	AlgebraicNumberFields
		Representation of algebraic numbers as elements of a finite extension of rationals.
		An algebraic number is an algebraic integer iff its
		The regulator of a number field K is the lattice volume of the image of the group of units of K under the logarithmic embedding
	documents.wolfram.com /v5/Add-onsLinks/StandardPackages/NumberTheory/AlgebraicNumberFields.html (893 words)

	11: Number theory
		Number theory is one of the oldest branches of pure mathematics, and one of the largest.
		Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions).
		The algebraic structure of the ring of integers is similar to that of other commutative rings such as rings of polynomials.
	www.math.niu.edu /~rusin/known-math/index/11-XX.html (2572 words)

	Fields and Orders
		Some of the functionality of number fields and their elements acting as field of fractions to orders has been transferred to this new type and is no longer present for number fields.
		Fields of this type are the fields of fractions of an order.
		for the representation of Abelian extensions of number fields.
	www.math.niu.edu /help/math/magmahelp/rel/node32.html (352 words)

	quadratic_number_power_product_basis (Site not responding. Last check: 2007-11-07)
		Its main use is to reduce memory needed by power products by providing a basis that can be shared by several power products and to increase efficiency in power product computations by speeding up the computation of approximations to the logarithms of the elements of the basis.
		Let n be the number of elements in b.
		In principle, it is possible that the numbers in the basis belong to different quadratic number fields.
	www.math.psu.edu /local_doc/LiDIA/node91.html (398 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.
		Number theorists are especially fond of algebraic number fields.
		In number theory we're especially interested in Galois groups like Gal(K/k) where K is some algebraic number field and k is some subfield of K. For starters, consider this example: Gal(Q(sqrt(n))/Q) where sqrt(n) is irrational.
	math.ucr.edu /home/baez/twf_ascii/week201 (3487 words)

	Science: Mathematics: Number Theory: Algebraic - Open Site
		A quadratic field is of degree 2, a cubic field is of degree 3, quartic of degree 4.
		The units of the ring of integers of a number field of degree n and signature n=r+2s form a finitely generated abelian group with a torsion-free component of rank r+s-1.
		A number field K of degree n over Q will have r embeddings into the real numbers and s pairs of embeddings into the complex numbers (which are not into the reals).
	open-site.org /Science/Mathematics/Number_Theory/Algebraic (467 words)

	Multiplicities of Discriminants
		Until the first quarter of the 20th century it was believed that algebraic number fields with a fixed signature and Galois group can be identified uniquely up to isomorphisms by the integer value of the discriminant of their maximal order.
		In 1990 I have generalized the idea of HASSE for dihedral fields of degree 2p with a prime number p > 2 [8] and in 1991 together with Pierre BARRUCAND at Paris for pure metacyclic fields of degree p(p-1) [9].
		The result was a huge number of new formulas for the multiplicity m in dependence on invariants of associated quadratic fields k, such as:
	www.algebra.at /multi.htm (539 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-11-07)
		ANT-0324: 4 Dec 2001, Extensions of number fields with wild ramification of bounded depth, by Farshid Hajir and Christian Maire.
		ANT-0267: 27 Nov 2000, On an analogue for number fields of a conjecture of de Jong on F_q[[t]]-analytic extensions of function fields, by Gebhard Boeckle.
		ANT-0040: 10 Nov 1997, Unramified quaternion extensions of quadratic number fields, by Franz Lemmermeyer.
	front.math.ucdavis.edu /ANT (12251 words)

	Hilbert and Steinitz (from algebra) -- Encyclopædia Britannica (Site not responding. Last check: 2007-11-07)
		Among his important contributions, his work in the 1890s on the theory of algebraic number fields was decisive in establishing the conceptual approach promoted by Dedekind as dominant for several decades.
		An important branch of mathematics, algebra today is studied not only in high school and college but, increasingly, in the lower grades as well.
		Algebra is as useful as all the other branches of mathematics—to which it is closely related.
	www.britannica.com /eb/article-231091 (850 words)

	Algebraic Number Fields
		Multiplication of two algebraic numbers can be performed by normal polynomial multiplication followed by a reduction of the result with the help of the defining polynomial.
		In practice this is not sufficient as very often several algebraic numbers appear in an expression.
		All algebraic numbers are then expressed in terms of the primitive element.
	www.uni-koeln.de /REDUCE/arnum/arnum.html (699 words)

	Marius Somodi's Home Page (Site not responding. Last check: 2007-11-07)
		It turns out that Witt equivalence of algebraic number fields is equivalent to the so called Hilbert symbol equivalence of number fields.
		Two number fields K and L are called Hilbert symbol equivalent if there is a bijection between the sets of places of K and L and an isomorphism between the square class groups of the fields such that the Hilbert symbols are preserved.
		Jena Carpenter proved in 1992 that if two number fields are Hilbert symbol equivalent by an equivalence that has infinitely many wild primes then one can construct a new equivalence that has finitely many wild primes.
	www.math.lsu.edu /~somodi/research.html (287 words)

	The abc conjecture (Site not responding. Last check: 2007-11-07)
		In [EL] it is shown that the truth of the abc conjecture for number fields implies the truth of the Mordell conjecture over an arbitrary number field.
		Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves.
		Proceedings of the international conference on discrete mathematics and number theory, Tiruchirapalli, India, January 3--6, 1996 on the occasion of the 10th anniversary of the Ramanujan Mathematical Society.
	www.math.unicaen.fr /~nitaj/abc.html (4284 words)

	Math 525: Algebraic Number Theory. Information Page (Site not responding. Last check: 2007-11-07)
		Many of the standard results in algebraic number theory can also be regarded as results in algebraic K-theory, so algebraic topologists may be interested in this course.
		For example, you should be able to prove that the ring of integers, considered as an additive abelian group, is free of rank equal to the degree of the extension.
		You need not memorize the discussion of the ring of integers in cyclotomic fields, but you should be able to, for example, compute the discriminant.
	www.math.binghamton.edu /dikran/525 (413 words)

	Creation of Lattices
		Let K be an algebraic number field of degree n over Q. Then K has n embeddings into the complex numbers, denoted by sigma_1,..., sigma_n.
		Given an order O in a number field of degree n, create the lattice in R^n generated by the images of the basis of O under the Minkowski map.
		Given an ideal I of an order O in a number field of degree n, create the lattice in R^n generated by the images of the basis of I under the Minkowski map.
	www.math.wisc.edu /help/magma/text540.html (1976 words)

	11R: Algebraic number theory: global fields
		Mines, Ray, "Algebraic number theory, a survey", The L. Brouwer Centenary Symposium (Noordwijkerhout, 1981), 337--358, Stud.
		Finding nice (small) generators for the integers in a number field.
		Are there algebraic numbers on the unit circle besides roots of unity?
	www.math.niu.edu /~rusin/known-math/index/11RXX.html (612 words)

	6.2 Algebraic Number Fields as Matrix Algebras (Site not responding. Last check: 2007-11-07)
		Let n be any positive integer and consider a sub-algebra K of the algebra of n×n matrices with rational entries; by this we mean that K contains scalar multiples of the identity matrix and is closed under matrix addition, subtraction and multiplication.
		To handle division we also insist that non-zero matrices in K are invertible (this actually implies that the inverses are also in K but it is not entirely trivial to prove this).
		To summarise, we will henceforth think of an algebraic number field as a sub-algebra of the ring of n×n matrices which is commutative, with all non-zero elements being invertible.
	www.imsc.ernet.in /~kapil/crypto/notes/node28.html (518 words)

	On Lattices over Number Fields - Fieker, Pohst (ResearchIndex)
		Abstract: A large number of algorithms used for computations in number fields E over Q make use of the fact that there is a canonical embedding of E into R^n under which o_E becomes a lattice.
		Using this, it is possible to utilize the powerful tools of the geometry of numbers to obtain existence theorems that often can be turned into efficient algorithms.
		Although it is well known that there is a generalization of the geometry of numbers to lattices over Dedekind domains (e.g.
	citeseer.ist.psu.edu /356545.html (395 words)

	Kash (Site not responding. Last check: 2007-11-07)
		KANT is a software package for mathematicians interested in algebraic number theory.
		For those KANT is a tool for sophisticated computations in number fields and in global function fields.
		Algebraic function fields over finite fields, Q or number fields
	www.math.tu-berlin.de /~kant/kash.html (448 words)

	Number theoretical research (Site not responding. Last check: 2007-11-07)
		Particularly interesting are such fundamental concepts as the unit group, the discriminant and the class number, which all depend on the arithmetical structure of the field.
		Deep algebraic methods are used in the study of these concepts.
		Algebraic number theory is by no means based solely on algebra.
	www.math.utu.fi /research/numbertheory/algebraic.html (201 words)

	Courses (Site not responding. Last check: 2007-11-07)
		Integrality over rings, algebraic extensions of fields, field isomorphisms, norms and traces are discussed in the second part.
		Dedekind rings, factorization in Dedekind rings, norms of ideals, splitting of prime ideals in field extensions, finiteness of the ideal class group and Dirichlet's theorem on units are treated in the second part.
		This is a very short introduction to local fields and local class field theory without using homological algebra.
	www.maths.nott.ac.uk /personal/ibf/courses.html (224 words)

	A SURVEY OF TRACE FORMS OF ALGEBRAIC NUMBER FIELDS
		This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester.
		Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form.
		It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone.
	www.worldscibooks.com /mathematics/0066.html (349 words)

	week201
		Since the nth roots of unity are evenly distributed around the unit circle, this sort of field is called a "cyclotomic field", for the Greek word for "circle cutting".
		Basically, to prove something is impossible, you just show that some number can't possibly lie in some particular algebraic number field, because it's the root of a polynomial whose splitting field has a Galois group that's "fancier" than the Galois group of that algebraic number field.
		Abelian extensions of algebraic number fields can be understood using something called class field theory.
	math.ucr.edu /home/baez/week201.html (5403 words)

	Algebraic Number Fields
		The algebraic number i (imaginary unit), for example, would be defined by the polynomial
		The arithmetic of algebraic number s can be viewed as a polynomial arithmetic modulo the defining polynomial.
		All algebraic numbers which can be built up from a are then of the form:
	www.uni-koeln.de /REDUCE/3.6/doc/arnum/arnum.html (797 words)

	a directory of all known zeta functions (Site not responding. Last check: 2007-11-07)
		(3) Application to number theory, in particular to the theory of decomposition of prime ideals in finite extensions of algebraic number fields.
		Tamagawa, "On the zeta function of a division algebra", Annals of Mathematics 77 (1963) 387-405.
		Ruelle defines the Weil zeta function for an algebraic variety over a finite field in terms of the numbers of fixed points of all iterations of the Frobenius map on the extension of the algebraic variety to the algebraic closure of the finite field.
	www.maths.ex.ac.uk /~mwatkins/zeta/directoryofzetafunctions.htm (3236 words)

	Amazon.com: Books: The Theory of Algebraic Number Fields (Site not responding. Last check: 2007-11-07)
		This book is a translation into English of Hilbert's "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provided an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century.
		It is based on the work of the great number theorists of the nineteenth century.
		The Zahlbericht can be seen as the starting point of all twentieth century investigations in algebraic number theory, reciprocity laws and class field theory.
	www.amazon.com /exec/obidos/tg/detail/-/3540627790?v=glance (383 words)

	Hilbert, David -- Britannica Student Encyclopedia (Site not responding. Last check: 2007-11-07)
		British-born mathematician who was awarded the Fields Medal in 1974 for his work in algebraic geometry.
		To emphasize the importance of keeping undefined mathematical terms totally abstract he once said, “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” His work with integral equations in 1909 led later in...
		The advance of set theory and discoveries involving infinite sets, transfinite numbers, and purely logical paradoxes caused much concern as to the foundations of mathematics.
	www.britannica.com /ebi/article?tocId=9274880 (823 words)

	The Euclidean algorithm in algebraic number fields, by Franz Lemmermeyer (Site not responding. Last check: 2007-11-07)
		The Euclidean algorithm in algebraic number fields, by Franz Lemmermeyer
		This is a survey on Euclidean number fields.
		Keywords : Euclidean algorithm, k-stage Euclidean fields, inhomogeneous minima, exceptional sequences, etc.
	www.math.uiuc.edu /Algebraic-Number-Theory/0003 (53 words)

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