Problems in Algebraic Number Theory


Author: M. Ram Murty,Jody (Indigo) Esmonde
Publisher: Springer Science & Business Media
ISBN: 0387269983
Category: Mathematics
Page: 352
View: 5654

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The problems are systematically arranged to reveal the evolution of concepts and ideas of the subject Includes various levels of problems - some are easy and straightforward, while others are more challenging All problems are elegantly solved

Problems in Algebraic Number Theory


Author: M. Ram Murty,Jody Esmonde
Publisher: Springer Science & Business Media
ISBN: 0387221824
Category: Mathematics
Page: 352
View: 4261

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It has become clear that problem solving plays an extremely important role in mathematical research. This book is a collection of about 500 problems in algebraic number theory. They are systematically arranged to reveal the evolution of concepts and ideas of the subject. For this new edition the authors have added a new chapter and revised several sections.

Problems in Analytic Number Theory


Author: Maruti Ram Murty
Publisher: Springer Science & Business Media
ISBN: 9780387951430
Category: Mathematics
Page: 452
View: 628

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This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate and senior undergraduate students. The goal is to give a rapid introduction of how analytic methods are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory.

A Course in Computational Algebraic Number Theory


Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 3662029456
Category: Mathematics
Page: 536
View: 326

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A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers 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, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

Fermat's Last Theorem

A Genetic Introduction to Algebraic Number Theory
Author: Harold M. Edwards
Publisher: Springer Science & Business Media
ISBN: 9780387950020
Category: Mathematics
Page: 410
View: 3321

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This introduction to algebraic number theory via "Fermat's Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummer theory of "ideal" factorization. In treats elementary topics, new concepts and techniques; and it details the application of Kummer theory to quadratic integers, relating it to Gauss theory of binary quadratic forms, an interesting connection not explored in any other book.

Algebraic Number Theory


Author: Ian Stewart
Publisher: Springer
ISBN: 9780412138409
Category: Science
Page: 257
View: 7058

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The title of this book may be read in two ways. One is 'algebraic number-theory', that is, the theory of numbers viewed algebraically; the other, 'algebraic-number theory', the study of algebraic numbers. Both readings are compatible with our aims, and both are perhaps misleading. Misleading, because a proper coverage of either topic would require more space than is available, and demand more of the reader than we wish to; compatible, because our aim is to illustrate how some of the basic notions of the theory of algebraic numbers may be applied to problems in number theory. Algebra is an easy subject to compartmentalize, with topics such as 'groups', 'rings' or 'modules' being taught in comparative isolation. Many students view it this way. While it would be easy to exaggerate this tendency, it is not an especially desirable one. The leading mathematicians of the nineteenth and early twentieth centuries developed and used most of the basic results and techniques of linear algebra for perhaps a hundred years, without ever defining an abstract vector space: nor is there anything to suggest that they suf fered thereby. This historical fact may indicate that abstrac tion is not always as necessary as one commonly imagines; on the other hand the axiomatization of mathematics has led to enormous organizational and conceptual gains.

Number Theory

Algebraic Numbers and Functions
Author: Helmut Koch
Publisher: American Mathematical Soc.
ISBN: 9780821820544
Category: Mathematics
Page: 368
View: 7251

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Early chapters discuss topics in elementary number theory, such as Minkowski's geometry of numbers, public-key cryptography and a short proof of the Prime Number Theorem, following Newman and Zagier. Next, some of the tools of algebraic number theory are introduced, such as ideals, discriminants and valuations. These results are then applied to obtain results about function fields, including a proof of the Riemann-Roch Theorem and, as an application of cyclotomic fields, a proof of the first case of Fermat's Last Theorem.There are a detailed exposition of the theory of Hecke L-series, following Tate, and explicit applications to number theory, such as the Generalized Riemann Hypothesis. Chapter 9 brings together the earlier material through the study of quadratic number fields. Finally, Chapter 10 gives an introduction to class field theory.

A Classical Introduction to Modern Number Theory


Author: Kenneth Ireland,Michael Rosen
Publisher: Springer Science & Business Media
ISBN: 147572103X
Category: Mathematics
Page: 394
View: 9136

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This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves.

A Course in Algebraic Number Theory


Author: Robert B. Ash
Publisher: Courier Corporation
ISBN: 0486477541
Category: Mathematics
Page: 112
View: 9984

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This text for a graduate-level course covers the general theory of factorization of ideals in Dedekind domains as well as the number field case. It illustrates the use of Kummer's theorem, proofs of the Dirichlet unit theorem, and Minkowski bounds on element and ideal norms. 2003 edition.

Additive Number Theory: Inverse Problems and the Geometry of Sumsets


Author: Melvyn B. Nathanson
Publisher: Springer Science & Business Media
ISBN: 9780387946559
Category: Mathematics
Page: 293
View: 7803

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This book reviews results in inverse problems for finite sets of integers, culminating in Ruzsa's elegant proof of the deep theorem of Freiman that a finite set of integers with a small sumset must be a large subset of an n-dimensional arithmetic progression.

Algebraic Number Theory


Author: Serge Lang
Publisher: Springer Science & Business Media
ISBN: 146120853X
Category: Mathematics
Page: 357
View: 2091

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This is a second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. "Lang's books are always of great value for the graduate student and the research mathematician. This updated edition of Algebraic number theory is no exception."—-MATHEMATICAL REVIEWS

A Course in Number Theory and Cryptography


Author: Neal Koblitz
Publisher: Springer Science & Business Media
ISBN: 1441985921
Category: Mathematics
Page: 235
View: 5904

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This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters.

Advanced Topics in Computational Number Theory


Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 1441984895
Category: Mathematics
Page: 581
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Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.

Number Theory in Function Fields


Author: Michael Rosen
Publisher: Springer Science & Business Media
ISBN: 1475760469
Category: Mathematics
Page: 358
View: 509

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Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

A Course in Analytic Number Theory


Author: Marius Overholt
Publisher: American Mathematical Soc.
ISBN: 1470417065
Category: Mathematics
Page: 371
View: 8203

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This book is an introduction to analytic number theory suitable for beginning graduate students. It covers everything one expects in a first course in this field, such as growth of arithmetic functions, existence of primes in arithmetic progressions, and the Prime Number Theorem. But it also covers more challenging topics that might be used in a second course, such as the Siegel-Walfisz theorem, functional equations of L-functions, and the explicit formula of von Mangoldt. For students with an interest in Diophantine analysis, there is a chapter on the Circle Method and Waring's Problem. Those with an interest in algebraic number theory may find the chapter on the analytic theory of number fields of interest, with proofs of the Dirichlet unit theorem, the analytic class number formula, the functional equation of the Dedekind zeta function, and the Prime Ideal Theorem. The exposition is both clear and precise, reflecting careful attention to the needs of the reader. The text includes extensive historical notes, which occur at the ends of the chapters. The exercises range from introductory problems and standard problems in analytic number theory to interesting original problems that will challenge the reader. The author has made an effort to provide clear explanations for the techniques of analysis used. No background in analysis beyond rigorous calculus and a first course in complex function theory is assumed.

Algebraic Number Theory


Author: A. Fröhlich,M. J. Taylor,Martin J. Taylor
Publisher: Cambridge University Press
ISBN: 9780521438346
Category: Mathematics
Page: 355
View: 3632

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This book provides a brisk, thorough treatment of the foundations of algebraic number theory on which it builds to introduce more advanced topics. Throughout, the authors emphasize the systematic development of techniques for the explicit calculation of the basic invariants such as rings of integers, class groups, and units, combining at each stage theory with explicit computations.

Algebraic Number Theory


Author: Jürgen Neukirch
Publisher: Springer Science & Business Media
ISBN: 3662039834
Category: Mathematics
Page: 574
View: 8452

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This introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. It is the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available.

Lectures on the Theory of Algebraic Numbers


Author: E. T. Hecke
Publisher: Springer Science & Business Media
ISBN: 1475740921
Category: Mathematics
Page: 242
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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.

Algebraic Number Theory and Fermat's Last Theorem, Fourth Edition


Author: Ian Stewart,David Tall
Publisher: CRC Press
ISBN: 1498738400
Category: Mathematics
Page: 322
View: 6758

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Updated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work. New to the Fourth Edition Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844 Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem Improves and updates the index, figures, bibliography, further reading list, and historical remarks Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

Number Theory

Volume II: Analytic and Modern Tools
Author: Henri Cohen
Publisher: Springer Science & Business Media
ISBN: 038749894X
Category: Mathematics
Page: 596
View: 428

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This book deals with several aspects of what is now called "explicit number theory." The central theme 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 local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.