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. Moreover, the authors have added a segment dedicated to arithmetical cohomology and noncommutative geometry as well as a report on point counts on varieties with many rational points. Also covered is the recent polynomial time algorithm for primality testing, among other subjects.
From the reviews of the second edition:
"Here is a welcome update to Number theory I. Introduction to number theory by the same authors ... . the book now brings the reader up to date with some of the latest results in the field. ... The book is generally well-written and should be of interest to both the general, non-specialist reader of Number Theory as well as established researchers who are seeking an overview of some of the latest developments in the field."
Philip Maynard, The Mathematical Gazette, Vol. 90 (519), 2006
[...] the first edition was a very good book; this edition is even better.
[...] Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject. [...]
Things get more interesting in Part II (by far the largest of the tree parts)[...] This part of the book covers such things as approaches through logic, algebraic number theory, arithmetic of algebraic varieties, zeta functions, and modular forms, followed by an extensive (50+ pages ) account of Wiles' proof of Fermat's Last Theorem. This is a valuable addition, new in this edition, and serves as a vivid example of the power of the "ideas and theories" that dominate this part of the book.
Also new and very interesting is Part III, entitled "Analogies and Visions,"
[...] The best surveys of mathematics are those written by deeply insightful mathematicians who are not afraid to infuse their ideas and insights into their outline of subject. This is what we have here, and the result is an essential book. I only wish the price were lower so that I could encourage my students buy themselves a copy. Maybe I'll do that anyway.
Fernado Q. Gouvêa, on 09/10/2005
"This book is a revised and updated version of the first English translation. ... Overall, the book is very well written, and has an impressive reference list. It is an excellent resource for those who are looking for both deep and wide understanding of number theory." (Alexander A. Borisov, Mathematical Reviews, Issue 2006 j)
"This edition feels altogether different from the earlier one ... . There is much new and more in this edition than in the 1995 edition: namely, one hundred and fifty extra pages. ... For my part, I come to praise this fine volume. This book is a highly instructive read with the usual reminder that there lots of facts one does not know ... . the quality, knowledge, and expertise of the authors shines through. ... The present volume is almost startlingly up-to-date ... ." (Alf van der Poorten, Gazette of the Australian Mathematical Society, Vol. 34 (1), 2007)
