Most professors would agree that a student who truly wants to learn any given area of mathematics will benefit enormously from solving many problems on the subject at hand. Struggling with a problem often helps one understand the finest aspects of the theory. However, can a student learn an area of mathematics just by solving a systematic collection of problems?
Problems in Algebraic Number Theory is intended to be used by the student for independent study of the subject. It provides the reader with a large collection of problems (about 500), at the level of a first course on the algebraic theory of numbers (with undergraduate algebra as a prerequisite). The volume also includes completely spelled-out solutions to all exercises. The list of topics include elementary number theory, algebraic numbers and number fields, Dedekind domains, ideal class groups, structure of the unit group, reciprocity laws (quadratic and higher) and Dirichlet L-functions. A new chapter on density theorems was added in the second edition. Each section starts with some basic definitions and theorems (with proofs), and a couple of solved examples.
The reviewer thinks that the authors have done a fantastic job choosing the problems, which are perfectly arranged so the students can progressively move on from topic to topic, discovering on their own the proofs of the most well-known results and applications. The exposition of the solutions is very clear and helps to introduce different important techniques.
However, the reviewer believes that a student is better served with a healthy balance between traditional lectures and problem-solving. Without proper advising the student might miss the 'soul' of the subject. The book feels at times like a bare sequence of definitions, theorems and exercises. The readers are supposed to find out, on their own through problem solving, why the concepts are introduced, and this might be a path that not everybody can easily follow. The student might find hard to differentiate routine problems from the more relevant exercises which will be used as lemmas in consequent sections. It is the opinion of the reviewer that each chapter should be complemented with a couple of lectures by a professor to emphasize the main ideas and clarify the goals of the developing theory. In any case, the book is an excellent resource for the instructor and the student as a companion to any algebraic number theory course.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.