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Number Theory

Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 0387499237
Size: 14.29 MB
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The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.

Algorithmic Number Theory

Author: Alf J. van der Poorten
Publisher: Springer Science & Business Media
ISBN: 3540794557
Size: 15.15 MB
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The ?rst Algorithmic Number Theory Symposium took place in May 1994 at Cornell University. The preface to its proceedings has the organizers expressing the hope that the meeting would be “the ?rst in a long series of international conferencesonthe algorithmic,computational, andcomplexity theoreticaspects of number theory.” ANTS VIII was held May 17–22, 2008 at the Ban? Centre in Ban?, Alberta, Canada. It was the eighth in this lengthening series. The conference included four invited talks, by Johannes Buchmann (TU ´ Darmstadt),AndrewGranville(Universit´edeMontr´ eal),Fran¸ coisMorain(Ecole Polytechnique),andHughWilliams(UniversityofCalgary),apostersession,and 28 contributed talks in appropriate areas of number theory. Each submitted paper was reviewed by at least two experts external to the Program Committee; the selection was made by the committee on the basis of thoserecommendations.TheSelfridgePrizeincomputationalnumbertheorywas awardedtotheauthorsofthebestcontributedpaperpresentedattheconference. The participants in the conference gratefully acknowledge the contribution made by the sponsors of the meeting. May 2008 Alf van der Poorten and Andreas Stein (Editors) Renate Scheidler (Organizing Committee Chair) Igor Shparlinski (Program Committee Chair) Conference Website The names of the winners of the Selfridge Prize, material supplementing the contributed papers, and errata for the proceedings, as well as the abstracts of the posters and the posters presented at ANTS VIII, can be found at: http://ants.math.ucalgary.ca.

A Comprehensive Course In Number Theory

Author: Alan Baker
Publisher: Cambridge University Press
ISBN: 1139560824
Size: 38.50 MB
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Developed from the author's popular text, A Concise Introduction to the Theory of Numbers, this book provides a comprehensive initiation to all the major branches of number theory. Beginning with the rudiments of the subject, the author proceeds to more advanced topics, including elements of cryptography and primality testing, an account of number fields in the classical vein including properties of their units, ideals and ideal classes, aspects of analytic number theory including studies of the Riemann zeta-function, the prime-number theorem and primes in arithmetical progressions, a description of the Hardy–Littlewood and sieve methods from respectively additive and multiplicative number theory and an exposition of the arithmetic of elliptic curves. The book includes many worked examples, exercises and further reading. Its wider coverage and versatility make this book suitable for courses extending from the elementary to beginning graduate studies.

Lectures On The Theory Of Algebraic Numbers

Author: E. T. Hecke
Publisher: Springer Science & Business Media
ISBN: 9780387905952
Size: 34.55 MB
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. . . if one wants to make progress in mathematics one should study the masters not the pupils. N. H. Abel Heeke was certainly one of the masters, and in fact, the study of Heeke L series and Heeke operators has permanently embedded his name in the fabric of number theory. It is a rare occurrence when a master writes a basic book, and Heeke's Lectures on the Theory of Algebraic Numbers has become a classic. To quote another master, Andre Weil: "To improve upon Heeke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task. " We have tried to remain as close as possible to the original text in pre serving Heeke's rich, informal style of exposition. In a very few instances we have substituted modern terminology for Heeke's, e. g. , "torsion free group" for "pure group. " One problem for a student is the lack of exercises in the book. However, given the large number of texts available in algebraic number theory, this is not a serious drawback. In particular we recommend Number Fields by D. A. Marcus (Springer-Verlag) as a particularly rich source. We would like to thank James M. Vaughn Jr. and the Vaughn Foundation Fund for their encouragement and generous support of Jay R. Goldman without which this translation would never have appeared. Minneapolis George U. Brauer July 1981 Jay R.

Zahlentheorie

Author: Helmut Koch
Publisher: Springer-Verlag
ISBN: 3322803120
Size: 24.50 MB
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Hauptziel des Buches ist die Vermittlung des Grundbestandes der Algebraischen Zahlentheorie einschließlich der Theorie der normalen Erweiterungen bis hin zu einem Ausblick auf die Klassenkörpertheorie. Gleichberechtigt mit algebraischen Zahlen werden auch algebraische Funktionen behandelt. Dies geschieht einerseits um die Analogie zwischen Zahl- und Funktionenkörpern aufzuzeigen, die besonders deutlich im Falle eines endlichen Konstantenkörpers ist. Andererseits erhält man auf diese Weise eine Einführung in die Theorie der "höheren Kongruenzen" als eines wesentlichen Bestandteils der "Arithmetischen Geometrie". Obgleich das Buch hauptsächlich algebraischen Methoden gewidmet ist, findet man in der Einleitung auch einen kurzen Beweis des Primzahlsatzes nach Newman. In den Kapiteln 7 und 8 wird die Theorie der Heckeschen L-Reihen behandelt einschließlich der Verteilung der Primideale algebraischer Zahlkörper in Kegeln.

Algebraic Geometry And Number Theory

Author: Hussein Mourtada
Publisher: Birkhäuser
ISBN: 331947779X
Size: 76.36 MB
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This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.

Elements Of Number Theory

Author: John Stillwell
Publisher: Springer Science & Business Media
ISBN: 9780387955872
Size: 70.73 MB
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Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement.

The Philosophy Of Set Theory

Author: Mary Tiles
Publisher: Courier Corporation
ISBN: 9780486435206
Size: 28.55 MB
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A century ago, Georg Cantor demonstrated the possibility of a series of transfinite infinite numbers. His methods, unorthodox for the time, enabled him to derive theorems that established a mathematical reality for a hierarchy of infinities. Cantor's innovation was opposed, and ignored, by the establishment; years later, the value of his work was recognized and appreciated as a landmark in mathematical thought, forming the beginning of set theory and the foundation for most of contemporary mathematics. As Cantor's sometime collaborator, David Hilbert, remarked, "No one will drive us from the paradise that Cantor has created." This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory; logical objects and logical types; and independence results and the universe of sets. She concludes with views of the constructs and reality of mathematical structure. Philosophers with only a basic grounding in mathematics, as well as mathematicians who have taken only an introductory course in philosophy, will find an abundance of intriguing topics in this text, which is appropriate for undergraduate-and graduate-level courses.