Graduate Studies in Mathematics 2000; 368 pp; hardcover Volume: 24 ISBN10: 0821820540 ISBN13: 9780821820544 List Price: US$68 Member Price: US$54.40 Order Code: GSM/24
 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, publickey 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 RiemannRoch 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. The book attempts as much as possible to give simple proofs. It can be used by a beginner in algebraic number theory who wishes to see some of the true power and depth of the subject. The book is suitable for two onesemester courses, with the first four chapters serving to develop the basic material. Chapters 6 through 9 could be used on their own as a second semester course. Readership Mathematicians working in the fields of algebraic number theory and Zeta and \(L\)functions. Reviews "This book is very suitable for researchers and graduate students who are interested in basic algebraic number theory up to the beginning of class field theory."  Mathematica Bohemica "A significant amount of the material goes beyond what one would expect from an introductory text ... this book can be recommended to students of number theory for its rigour and emphasis on theory."  European Mathematical Society Newsletter Table of Contents  Introduction
 The geometry of numbers
 Dedekind's theory of ideals
 Valuations
 Algebraic functions of one variable
 Normal extensions
 \(L\)series
 Applications to Hecke \(L\)series
 Quadratic number fields
 What next?
 Divisibility theory
 Trace, norm, different, and discriminant
 Harmonic analysis on locally compact abelian groups
 References
 Index
