The Old Testament book of Ecclesiastes reminds us that "of the making of many books there is no end" and "there is nothing new under the sun." And when it comes to mathematics textbooks this often seems to be the case. Here, on the other hand, is something truly different.
Introductory books on number theory seem, these days, to always begin at the same place and cover similar territory. Whatever differences one finds between books have to do with details of approach and style or with what is done in the more advanced chapters, after the obligatory chapters dealing with divisibility, congruences, linear diophantine equations, primitive roots, and quadratic reciprocity.
Now, of course there's nothing wrong with that sequence; in fact, one can certainly argue that it represents some of the most important foundational ideas in the subject. On the other hand, number theory is such a large subject, and so much of it is initially accessible without too many pre-requisites, that one would expect to see a different take on the subject every once in a while. And that, happily, is what we have here, both in content and in style of presentation.
Burger's book proposes to introduce students to a range of number-theoretical ideas, theorems, and problems by having the students themselves discover the results and prove the theorems. Thus, the book presents the material largely through a sequence of problems surrounded by some expository text which usually focuses on the significance of the theorems rather than on their proofs. The chapters, actually called "modules," typically end with "Big Picture Questions" which invite the student to attempt to consider what has been done so far, where it might be going, and why it is interesting.
The main thread through the book is diophantine approximation. For the first ten modules, the focus is on approximating irrational numbers by rationals while controlling the size of the denominator of the approximants. This is a rich area of number theory, connected to continued fractions, Farey sequences, and transcendence theory. It is also an area that is accessible to undergraduates, so it is a particularly good choice for this kind of book. The modules build towards some significant results: a description of the Markoff spectrum, solving the Pell equation, and the work of Liouville and Roth on transcendental numbers.
The next two chapters are basically a detour through arithmetical algebraic geometry. They look at Pythagorean triples from a geometric point of view and quickly visit the theory of elliptic curves. Then come chapters on Minkowski's "geometry of numbers" and applications to simultaneous diophantine approximation and the four squares theorem. After a module on "distribution modulo 1," the final modules deal briefly with p-adic numbers, ending with a discussion of Hensel's Lemma and the local-global principle.
As that summary suggests, the first half of the book feels tightly integrated around a basic theme, building towards some significant theorems. The second half is more like a quick tour, with stops at several interesting locations but no extensive development and no culminating theorems. This may make sense in a course setting, where one would be virtually certain of finishing the first ten modules but might want to pick and choose among the last ten.
Overall, this is a very nice guide through this material. The first ten modules are the best and most interesting part of the book, well worth working through. The section on "arithmetical algebraic geometry" is probably the weakest, but the book picks up steam again when it goes into the "geometry of numbers" section. The module on "distribution modulo 1" has an interesting theorem at its center, though perhaps the proof given here is not the most illuminating one. (On the other hand, the proof I really find more illuminating has far more pre-requisites.) The section on the p-adics should be lots of fun for the students, and goes just deep enough to suggest that there is some substance to the subject.
Since the book is designed to be given to students in a seminar-style course, it does not contain solutions of any of the problems. For some problems, hints are provided; these vary from very meager to quite detailed. There are discussions, at the back of the book, of some, but not all, of the "Big Picture Questions." All this is just right for a course where students are expected to work through the material and develop their own proofs and examples, but it lays a heavy burden on the instructor. No "teacher's solution manual" is provided. If you propose to lead your students into this jungle, you had better have a pretty good idea of the lay of the land before you start, or you'll all end up lost together. This is particularly true when it comes to the "big picture."
For professors with the requisite background, this may be just the right book to use in an upper-level undergraduate seminar. Students working through this book will learn some nice material, and will probably also emerge from the course with a much greater confidence in their ability to do mathematics.
Fernando Q. Gouvêa is Associate Professor of Mathematics at Colby College in Waterville, ME. He works in number theory (focusing especially on modular forms and Galois representations) and also has a strong interest in the history of mathematics.