Asking how one does mathematical research is like asking how a composer creates a masterpiece. No one really knows. It is clear, however, that problem solving plays an important role in the training of the research mind. In fact, it would not be an exaggeration to say that the ability to do research is essentially the art of asking the 'right'questions. And indeed, the approach taken here is based on the principle that questions focus the mind. This book is a collection of approximately 500 problems in algebraic number theory, systematically arranged to reveal the evolution of concepts and ideas of the subject. Some are easy and straightforward, others difficult. However, they have all been arranged with a didactic purpose in mind and are completely solved. This text is suitable for a first course in algebraic number theory with minimal supervision by the instructor. The exposition also facilitates independent study, however, and any student who has taken a basic !
course in calculus, linear algebra and abstract algebra should be able to work through these problems on his/her own.
From Reviews of the First Edition:
This book provides a problem-oriented first course in algebraic number theory. ... The authors have done a fine job in collecting and arranging the problems. Working through them, with or without help from a teacher, will surely be a most efficient way of learning the theory. Many of the problems are fairly standard, but there are also problems of a more original type. This makes the book a useful supplementary text for anyone studying or teaching the subject. ... This book deserves many readers and users.
- T. Metsänkylä , Mathematical Reviews
The book covers topics ranging from elementary number theory (such as the unique factorization of integers or Fermat's little theorem) to Dirichlet's theorem about primes in arithmetic progressions and his class number formula for quadratic fields, and it treats standard material such as Dedekind domains, integral bases, the decomposition of primes not dividing the index, the class group, the Minkowski bound and Dirichlet's unit theorem ... the reviewer is certain that many students will benefit from this pathway into the fascinating realm of algebraic number theory.
- Franz Lemmermeyer, Zentralblatt
This second edition is an expanded and revised version of the first edition. In particular, it contains an extra chapter on density theorems and $L$-functions highlighting some of the analytic aspects of algebraic number theory.
