**Author**: M. Ram Murty,Jody (Indigo) Esmonde

**Publisher:**Springer Science & Business Media

**ISBN:**0387269983

**Category:**Mathematics

**Page:**352

**View:**8304

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# Search Results for: problems-in-algebraic-number-theory-graduate-texts-in-mathematics

**Author**: M. Ram Murty,Jody (Indigo) Esmonde

**Publisher:** Springer Science & Business Media

**ISBN:** 0387269983

**Category:** Mathematics

**Page:** 352

**View:** 8304

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

**Author**: M. Ram Murty,Jody Esmonde

**Publisher:** Springer Science & Business Media

**ISBN:** 0387221824

**Category:** Mathematics

**Page:** 352

**View:** 3731

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.

**Author**: Henri Cohen

**Publisher:** Springer Science & Business Media

**ISBN:** 3662029456

**Category:** Mathematics

**Page:** 536

**View:** 733

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.
*A Genetic Introduction to Algebraic Number Theory*

**Author**: Harold M. Edwards

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387950020

**Category:** Mathematics

**Page:** 410

**View:** 9138

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.

**Author**: Ian Stewart

**Publisher:** Springer

**ISBN:** 9780412138409

**Category:** Science

**Page:** 257

**View:** 3539

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.

**Author**: U.S.R. Murty

**Publisher:** Springer Science & Business Media

**ISBN:** 0387723498

**Category:** Mathematics

**Page:** 506

**View:** 5590

This informative and exhaustive study gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they 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. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on the arithmetic progression of primes.

**Author**: Robert B. Ash

**Publisher:** Courier Corporation

**ISBN:** 0486477541

**Category:** Mathematics

**Page:** 112

**View:** 7164

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.

**Author**: Henri Cohen

**Publisher:** Springer Science & Business Media

**ISBN:** 1441984895

**Category:** Mathematics

**Page:** 581

**View:** 1741

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.

**Author**: Melvyn B. Nathanson

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387946559

**Category:** Mathematics

**Page:** 296

**View:** 3175

Many classical problems in additive number theory are direct problems, in which one starts with a set A of natural numbers and an integer H -> 2, and tries to describe the structure of the sumset hA consisting of all sums of h elements of A. By contrast, in an inverse problem, one starts with a sumset hA, and attempts to describe the structure of the underlying set A. In recent years there has been ramrkable progress in the study of inverse problems for finite sets of integers. In particular, there are important and beautiful inverse theorems due to Freiman, Kneser, Plünnecke, Vosper, and others. This volume includes their results, and culminates with an elegant proof by Ruzsa 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 Numbers and Functions*

**Author**: Helmut Koch

**Publisher:** American Mathematical Soc.

**ISBN:** 9780821820544

**Category:** Mathematics

**Page:** 368

**View:** 4200

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.

**Author**: Jürgen Neukirch

**Publisher:** Springer Science & Business Media

**ISBN:** 3662039834

**Category:** Mathematics

**Page:** 574

**View:** 2926

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.

**Author**: Serge Lang

**Publisher:** Springer Science & Business Media

**ISBN:** 146120853X

**Category:** Mathematics

**Page:** 357

**View:** 1854

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

**Author**: Kenneth Ireland,Michael Rosen

**Publisher:** Springer Science & Business Media

**ISBN:** 147572103X

**Category:** Mathematics

**Page:** 394

**View:** 4658

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.

**Author**: A. Fröhlich,M. J. Taylor,Martin J. Taylor

**Publisher:** Cambridge University Press

**ISBN:** 9780521438346

**Category:** Mathematics

**Page:** 355

**View:** 990

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.

**Author**: U.S.R. Murty

**Publisher:** Springer Science & Business Media

**ISBN:** 0387723498

**Category:** Mathematics

**Page:** 506

**View:** 3907

This informative and exhaustive study gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they 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. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on the arithmetic progression of primes.

**Author**: Daniel A. Marcus

**Publisher:** Springer

**ISBN:** 3319902334

**Category:** Mathematics

**Page:** 203

**View:** 5603

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.
*Fundamental Problems, Ideas and Theories*

**Author**: Yu. I. Manin,Alexei A. Panchishkin

**Publisher:** Springer Science & Business Media

**ISBN:** 9783540276920

**Category:** Mathematics

**Page:** 514

**View:** 1708

This edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

**Author**: Neal Koblitz

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387942933

**Category:** Mathematics

**Page:** 235

**View:** 6533

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.

**Author**: E. T. Hecke

**Publisher:** Springer Science & Business Media

**ISBN:** 1475740921

**Category:** Mathematics

**Page:** 242

**View:** 490

. . . 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.
*Volume II: Analytic and Modern Tools*

**Author**: Henri Cohen

**Publisher:** Springer Science & Business Media

**ISBN:** 038749894X

**Category:** Mathematics

**Page:** 596

**View:** 7691

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.

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