Enhanced descriptions from Syndetics:

Perfect for students approaching the subject for the first time, this book offers a superb overview of number theory. Now in its second edition, it has been thoroughly updated to feature up-to-the-minute treatments of key research, such as the most recent work on Fermat's coast theorem. Topics include divisibility and multiplicative functions, congruences and quadratic resolves, the basics of algebraic numbers and sums of squares, continued fractions, diophantine approximations and transcendence, quadratic forms, partitions, the prime numbers, diophantine equations, and elliptic curves. More advanced subjects such as the Gelfond-Schneider, prime number, and Mordell-Weil theorems are included as well. Each chapter contains numerous problems and solutions.

Table of contents provided by Syndetics

• 1 Divisibility
• 2 Multiplicative Functions
• 3 Congruence Theory
• 4 Quadratic Residues
• 5 Algebraic Topics
• 6 Sums of Squares and Gauss Sums
• 7 Continued Fractions
• 8 Transcendental Numbers
• 9 Quadratic Forms
• 10 Genera and the Class Group
• 11 Partitions
• 12 The Prime Numbers
• 13 Two Major Theorems on the Primes
• 14 Diophantine Equations
• 15 Elliptic Curves: Basic Theory
• 16 Elliptic Curves: Further Results and Applications

Reviews provided by Syndetics

CHOICE Review

It is hard to think of a branch of mathematics that offers the student a wider selection of truly excellent texts than does number theory. The books of A. Baker, Z.I. Borevich and I.R. Shafarevich, H. Davenport, G.H. Hardy and E.M. Wright, L.K. Hua, K.F. Ireland and M.I. Rosen, E. Landau, W.J. LeVeque, I. Niven and H.S. Zuckerman, H. Rademacher, J.P. Serre, and A. Weil, to name just a few, should be in every college library. In addition to these general works there are numerous more focused books that are no less accessible to the beginner. To find a place in such illustrious company, a new entry must find a way to distinguish itself. None of the above concentrates on the connections between algebraic geometry and number theory that have led to such spectacular success in recent years as G. Falting's proof of the Mordell conjecture, B. Gross, P. Zagier, and D. Goldfeld's work on class numbers, and G. Frey and K. Ribet's progress on Fermat's last theorem. At the center of many of these developments are elliptic curves, which constitute the central theme of Chahal's approach. Topics in Number Theory is more elementary than the recent texts of D. Husemoller and J. Silverman, although his claim that the reader needs no more background than high school mathematics is probably true only in principle. Indeed, he has kept the book admirably self-contained by providing rapid presentations of the algebra and geometry required. Nevertheless, the undergraduate who masters this book will obtain a vista on graduate-level mathematics, even on current research, and this is reason enough to recommend it highly. A Course in Number Theory is less to this reviewer's taste. Rose's contention that "number theory is not an organized theory in the usual sense but a vast collection of individual topics and results. . ." has resulted in a book without a strong point of view; many of the chapters seem like they might be first chapters of books of their own. The style is dry, the arguments little motivated, and much of it follows closely the duly acknowledged work of other authors. This is a formidable book that one suspects will be more often admired by faculty than used by students, although it is quite competently executed and should be a valuable reference. There are missed opportunities here. For example, Rose offers two well-known proofs of the quadratic reciprocity law while indicating that some 150 are available. A more extensive survey would have been delightful, emphasizing the similarities and difference of some of the more important arguments and indicating the different directions in which they lead. He chooses to emphasize so-called "elementary" methods, that is, those that avoid complex analysis, but these methods are just as complex and perhaps less intuitive than those that employ more powerful tools. Rose might have offered a chapter on complex analysis for number theorists, or, alternatively, he might have offered a deeper comparative study of elementary and analytic methods. In short, a potentially valuable book, but not a vital one. D. V. Feldman University of New Hampshire