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	Algebraic number theory - Wikipedia, the free encyclopedia
		Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
		An algebraic number field is any finite (and therefore algebraic) field extension of the rational numbers.
		This is called localization and it leads to the construction of the p-adic numbers; this field of study is called local analysis and it arises from algebraic number theory.
	en.wikipedia.org /wiki/Algebraic_number_theory (191 words)

	PlanetMath: theory of algebraic and transcendental numbers
		Similarly as the rational numbers may be classified to integer and non-integer (fractional) numbers, also the algebraic numbers may be classified to algebraic integers or entire algebraic numbers and non-integer algebraic numbers.
		Algebraic and transcendental: the sum, difference, and quotient of two non-zero complex numbers, from which one is algebraic and the other transcendental, is transcendental.
		This is version 20 of theory of algebraic and transcendental numbers, born on 2005-05-03, modified 2005-05-04.
	planetmath.org /encyclopedia/TheoryOfAlgebraicNumbers.html (436 words)

	Learn more about Number theory in the online encyclopedia. (Site not responding. Last check: 2007-10-22)
		Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians.
		Number theory may be subdivided into several fields according to the methods used and the questions investigated.
		In algebraic number theory, the concept of number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
	www.onlineencyclopedia.org /n/nu/number_theory.html (594 words)

	Integer - Wikipedia, the free encyclopedia
		The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.
		Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer.
		However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction.
	en.wikipedia.org /wiki/Integer (967 words)

	number theory
		number theory, branch of mathematics concerned with the properties of the integers (the numbers 0, 1, -1, 2, -2, 3, -3, …).
		An important area in number theory is the analysis of prime numbers.
		Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers.
	www.factmonster.com /id/A0836174 (278 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 (2587 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-10-22)
		This archive is for research in algebraic number theory and arithmetic geometry.
		ANT-0342: 28 Mar 2002, On an Archimedean analogue of Tate's conjecture, by Dipendra Prasad and C.S.Rajan.
		ANT-0296: 8 Jun 2001, On the Iwasawa theory of p-adic Lie extensions, by Otmar Venjakob.
	front.math.ucdavis.edu /ANT (12251 words)

	transfinite number - HighBeam Encyclopedia (Site not responding. Last check: 2007-10-22)
		TRANSFINITE NUMBER [transfinite number] cardinal or ordinal number designating the magnitude (power) or order of an infinite set ; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
		Transfinite ordinal numbers are also defined for certain ordered sets, two such being equivalent if there is a one-to-one correspondence between the sets, which preserves the ordering.
		The transfinite ordinal number of the positive integers is designated by ω.
	www.encyclopedia.com /html/t/transfin.asp (411 words)

	Open Questions: Algebraic Number Theory
		Algebraic numbers clearly exist, since the length of the diagonal of a unit square is certainly a meaningful concept.
		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.
		Suppose we have an algebraic number α∈F and F⊇Q is a Galois extension.
	www.openquestions.com /oq-ma018.htm (19624 words)

	Arizona Mathematics \| Research \| Number Theory
		The research of the number theory group encompasses classical and algebraic number theory, computational number theory, and especially the modern subject of arithmetical algebraic geometry.
		Arithmetical algebraic geometry is the study of number-theoretic problems informed by the insights of geometry—among them algebraic geometry, topology, differential geometry, and discrete geometries related to graph theory and group theory.
		The number of Fields Medals (the mathematical equivalent of the Nobel prize) awarded for work in the area is a testament to its centrality in modern fundamental mathematics.
	math.arizona.edu /research/numbertheory.html (289 words)

	Explicit algebraic number theory (Site not responding. Last check: 2007-10-22)
		The title Explicit algebraic number theory is borrowed from the series of Oberwolfach meetings on Explicit methods in number theory.
		The advanced techniques from algebraic number theory that apply to these problems include class field theory, infinite Galois theory, and the theory of quadratic forms.
		The purpose of this part is to impart a working knowledge of these theories to the participants, to provide ample illustrations of their use, and to formulate several open problems that may be approachable by means of the same techniques.
	www.math.leidenuniv.nl /~psh/EANT (334 words)

	Number Theory (Site not responding. Last check: 2007-10-22)
		Number theory is as old as human thought, if not older.
		This number theory seminar also enjoys the active participation of some of the leading figures who come to Montreal on a regular basis and give short courses suitable for graduate students.
		Students specializing in number theory are expected to fulfil first the basic requirements in algebra and analysis.
	www.math.mcgill.ca /department/numtheory.php (445 words)

	Amazon.ca: A Course in Computational Algebraic Number Theory: Books: Henri Cohen (Site not responding. Last check: 2007-10-22)
		This book describes 148 algorithms which are fundamental for number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring.
		The first seven chapters lead the reader to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations.
		The author gives an overview of the computer packages used for number theory, including Pari, which was written by him and his collaborators.
	www.amazon.ca /Course-Computational-Algebraic-Number-Theory/dp/3540556400 (759 words)

	Algebraic Number Theory
		Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems.A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory.
		Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality.
		It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes.
	www.ramex.com /cr/cr-3696.html (272 words)

	Introductory Algebraic Number Theory - Cambridge University Press
		Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems.
		References to suggested reading and to the biographies of mathematicians who have contributed to the development of algebraic number theory are given at the end of each chapter.
		'Learning algebraic number theory is about the least abstract way to learn about important aspects of commutative ring theory, as well as being beautiful in its own right too.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521540119 (345 words)

	The Math Forum - Math Library - Number Theory
		Papers from a Mathematics graduate from The University Of Sussex at Brighton: Number Theory: GCD and Prime Factorisation; Molien's Theorem, Invariant Theory and Gregor Kemper; A History of Equality.
		In number theory, straightforward, reasonable questions are remarkably easy to ask, yet many of these questions are surprisingly difficult or even impossible to answer.
		A connected series of four problems in elementary number theory that are ideal for discovery learning at several levels.
	mathforum.org /library/topics/number_theory (2144 words)

	Number theoretical research (Site not responding. Last check: 2007-10-22)
		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.
		Algebraic number theory is by no means based solely on algebra.
		In addition, the theory of elliptic curves has proved to be immensely useful in many difficult problems concerning algebraic number fields.
	www.math.utu.fi /research/numbertheory/algebraic.html (201 words)

	Number Theory (Site not responding. Last check: 2007-10-22)
		Number Theory at the Mathematics Dept. of the University of Texas
		John Tate (tate@math.utexas.edu): Algebraic Number Theory (local and global fields), Class Field Theory, Galois cohomology, Galois representations, L-functions and their special values, modular forms, elliptic curves and abelian varieties.
		Jeffrey Vaaler (vaaler@math.utexas.edu): Analytic number theory, Diophantine approximation and the geometry of numbers in local and global fields, Diophantine inequalities, polynomials, effective measures of irrationality and transcendence, applications of Fourier analysis in number theory, inequalities and extremal problems.
	www.ma.utexas.edu /users/voloch/numberthy.html (162 words)

	MTH 617 Algebraic number theory, Summer semester 2002 (Site not responding. Last check: 2007-10-22)
		Algebraic number theory is the branch of number theory, that employs abstract algebraic techniques for investigating properties of (rational) integers.
		It developed in the nineteenth and twentieth centuries during attempts to prove the so-called Fermat's last theorem (FLT) which states that there are no solutions in positive integers x,y,z of the equation x^n+y^n=z^n, where n is an integer larger than 2.
		Algebraic numbers and integers, number fields, complex embeddings, number rings, integral bases.
	home.iitk.ac.in /~abhijit/course/MTH617/SS02/index.html (719 words)

	Amazon.com: Algorithmic Algebraic Number Theory (Encyclopedia of Mathematics and its Applications): Books: M. Pohst,H. ... (Site not responding. Last check: 2007-10-22)
		Zassenhaus "Algebraic numbers are defined as complex numbers x satisfying an algebraic equation of the form..." (more)
		This classic book gives a thorough introduction to constructive algebraic number theory, and is therefore especially suited as a textbook for a course on that subject.
		Algebraic numbers are defined as complex numbers x satisfying an algebraic equation of the form Read the first page
	www.amazon.com /exec/obidos/tg/detail/-/0521596696?v=glance (577 words)

	The USC Number Theory Home Page
		My Number Theoretic interests include various areas of algebraic number theory and arithmetical geometry such as the theory of Galois representations and p-extensions of number fields.
		My interests in number theory are primarily in binary quadratic forms and the class groups of quadratic, cubic, and quartic number fields and in the application of number theory to cryptography and information security.
		Analytic and Elementary Number Theory with particular interests in the distribution of primes, Waring's problem, arithmetic properties of elliptic curves over the rationals, and applications of the theory of modular forms.
	www.math.sc.edu /~filaseta/numthry.html (532 words)

	Category:Algebraic number theory - Wikipedia, the free encyclopedia
		Algebraic number theory is both the study of number theory by algebraic methods and the theory of algebraic numbers.
		The main article for this category is Algebraic number theory.
		There are 3 subcategories shown below (more may be shown on subsequent pages).
	en.wikipedia.org /wiki/Category:Algebraic_number_theory (101 words)

	Algebraic Number Theory (Site not responding. Last check: 2007-10-22)
		As the name suggests, algebraic number theory employs modern algebraic techniques to solve problems in number theory.
		A number field is a finite extension of Q. Since Q has characteristic 0, extensions of q are automatically separable.
		Of course every element in a finite extension is algebraic, so everything in a global field qualifies as an algebraic number.
	www.mathreference.com /an,intro.html (185 words)

	Amazon.com: Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften): Books: Jürgen Neukirch,N. ... (Site not responding. Last check: 2007-10-22)
		to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry.
		Algebraic Number Theory (Cambridge Studies in Advanced Mathematics) by A. Fröhlich
		This book is by no means intended for those who are not fluent in both Number Theory as well as Algebra, both at the graduate level and obviously for those who are Mahematically gifted.
	www.amazon.com /exec/obidos/tg/detail/-/3540653996?v=glance (837 words)

	Algebraic Number Theory - Cambridge University Press
		It provides a brisk, thorough treatment of the foundations of algebraic number theory, and builds on that to introduce more advanced ideas.
		A number of special topics are included that can be treated at this level but can usually only be found in research monographs or original papers, for instance: module theory of Dedekind domains; tame and wild ramifications; Gauss series and Gauss periods; binary quadratic forms; and Brauer relations.
		This is the only textbook at this level which combines clean, modern algebraic techniques together with a substantial arithmetic content.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521438349 (223 words)

	MA3A6 Algebraic Number Theory
		Prerequisites: Familiarity with second year algebra, in particular, the following topics is highly desirable.
		Structure of finitely generated abelian groups, basic commutative ring theory, Unique factorisation theorems and examples where unique factorisation does not work.
		Algebraic Number Theory, I.N. Sewart and D.O. Tall.
	www.maths.warwick.ac.uk /undergrad/pydc/pink/pink-MA3A6.html (78 words)

	UM Mathematics: Number Theory
		Analytic number theory, distribution of prime numbers, Fourier analysis, analytic inequalities, probability.
		We also offer a range of courses on advanced topics in number theory.Topics of recent courses include: elliptic curves, Diophantine problems, Hida theory, transcendence theory, spectral theory of modular forms, rigid-analytic geometry, Galois representations and modular forms, automorphic forms on algebraic groups, and alterations.
		Number Theory: Weekly research seminar with outside speakers on topics in all areas of number theory.
	www.math.lsa.umich.edu /research/number_theory (274 words)

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