Book Description

This book gives an exposition of the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. The following topics are treated: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. Each chapter ends with exercises and a short review of the relevant literature up to 2003. The bibliographyhas over 3400 items.

Product Description
Algebraic number theory is one of the most refined creations in mathematics. It has been developed by some of the leading mathematicians of this and previous centuries. The primary goal of this book is to present the essential elements of algebraic number theory, including the theory of normal extensions up through a glimpse of class field theory. Following the example set for us by Kronecker, Weber, Hilbert and Artin, algebraic functions are handled here on an equal footing with algebraic numbers. This is done on the one hand to demonstrate the analogy between number fields and function fields, which is especially clear in the case where the ground field is a finite field. On the other hand, in this way one obtains an introduction to the theory of "higher congruences" as an important element of "arithmetic geometry".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. ... Read more

Book Description

Book Description
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. It also provides a comprehensive look at recent research. For experimental number theoreticians, the authors developed new methods and obtained new results of great importance for them. Both computer scientists interested in higher arithmetic and those teaching algebraic number theory will find the book of value. ... Read more

Product Description
vi+139 pp. The author presents a concise introduction to algebraic number theory suitable for a beginning graduate course. The book is based on lectures given at the University of Salzburg and Brigham Young University. Many exercises are incorporated into the text. As for prerequisites, the reader should be familiar with linear algebra, and some basic abstract algebra including Galois theory. The methods used are classical, along the lines of the work of Dedekind and Hilbert. The Minkowski geometry of numbers is developed and used as a tool. The author takes pains to show how modern commutative algebra and algebraic geometry originate in the subject matter of algebraic number theory. Students who decide to specialize in algebraic number theory will find the suggestions for further reading helpful.Algebraic number theory is one of the great accomplishments of pure mathematics. This was the view of David Hilbert and Hermann Weyl, for example. This brief introduction conveys the power and elegance of the subject, and tempts the reader to delve further into the literature. Chapter headings: 1. Basic Concepts. 2. Arithmetic in relative extensions. 3. Geometry of numbers. 4. Analytic methods. 5. Arithmetic in Galois extensions. 6. Cyclotomic fields. 7. The Kronecker- Weber theorem. 8. Zeta functions and Riemann hypothesis.Not available for international shipping through yahoo.com. Contact kendrickpress.com to purchase this book internationally. ... Read more

Book Description
This volume contains the proceedings of a seminar on Algebraic$K$-theory and Algebraic Number Theory, held at the East-WestCenter in Honolulu in January 1987. The seminar, which hostednearly 40 experts from the U.S. and Japan, was motivated by thewide range of connections between the two topics, as exemplifiedin the work of Merkurjev, Suslin, Beilinson, Bloch,Ramakrishnan, Kato, Saito, Lichtenbaum, Thomason, and Ihara. Asis evident from the diversity of topics represented in theseproceedings, the seminar provided an opportunity formathematicians from both areas to initiate further interactionsbetween these two areas. ... Read more

Book Description
These proceedings contain 35 selected research and survey articles based on lectures delivered at the International Conference on Algebraic Number Theory and Diophantine Analysis held in Graz, Austria, August 30 - September 5, 1998. Although the conference was open for all branches of number theory, there was a special emphasis on algebraic number theory, on diophantine and algorithmic problems and on diophantine analysis, uniform distribution and discrepancy theory. Several lectures were devoted to the effective and algorithmic solution of diophantine equations and related topics.

This volume is of interest to researchers and graduate students in number theory including applications, in computer algebra and dynamical systems. ... Read more

Book Description

Customer Reviews (2)

Uncompromising and difficult
Make no mistae about it, this is a very difficult book. It is pitched at a highly advanced level and assumes a good deal of ring and field theory and makes no concessions by way of pausing to briefly summarise what is assumed at any point. Parts of the book also progress very quickly and consist of streams of definitions with next to no examples followed by a chunk of theorems that depend on the terms introduced to such an extent that the material has to be re-read many times to glean any understanding.Although in terms of the amount of material covered this is a comprehensive text, it is far too concise for student use. It might have some limited appeal as an advanced postgraduate reference book, but for anyone not already well up to speed in algebraic number theory this will be heavy going indeed.

A high level book
This book seems to be a good description of Algibraic valuations. Potential buyers should know that it assumes what seems to be a knowlage of graduate-level mathematics, particularly a thurogh knowlage of mathematicalfields. ... Read more

Book Description
Peter Swinnerton-Dyer's mathematical career encompasses more than 60 years' work of amazing creativity. This volume, dedicated to him on the occasion of his 75th birthday, provides contemporary insight into several subjects in which his influence has been notable. The contributions come from leading researchers in analytic and arithmetic number theory, and algebraic geometry. Topics treated include rational points on algebraic varieties, the Hasse principle, Shafarevich-Tate groups of elliptic curves and motives, Zagier's conjectures, descent and zero-cycles, Diophantine approximation, and Abelian and Fano varieties. ... Read more

Book Description
This book presents papers that originally appeared in theJapanese journal Sugaku from the Mathematical Society of Japan.The papers explore the relationship between number theory andalgebraic geometry. ... Read more