This is an unusual textbook providing an introduction to the basics of abstract algebra: ring theory, group theory and a bit of finite field theory and applications. Of course it gives definitions, statements of theorems and proofs, but what makes it original is that the reader is a very active protagonist of her/his own learning process. Indeed this is a book specially adapted for a self-study of the first steps in algebra. As the authors claim, one does not become a painter after contemplating how a painter works but painting by himself.
The book has been written after a deep reflection about how the learning process works. Remarkably, most objects get into the scene from the particular to the general. For example, the ring of integers Z is not only an example of euclidean domain: the first chapter of the book is devoted to study Z paying special attention to its factorization properties, and so it serves as a model to introduce in subsequent chapters the abstract notions of unique factorization domain (UFD), principal ideal domain and euclidean domain.
The book is organized in units (called investigations). Each unit begins with an illuminating 15 lines presentation explaining the questions the reader should be able to answer after studying it. The units contain very carefully written sections devoted to promote the active participation of the reader: mathematical activities whose difficulty changes in a subtle way, connections between different chapters of the book, and a lot of exercises.
Let us briefly summarize the contents of the book. The first five chapters cover elementary ring theory: irreducibility and factorization in Z and polynomial rings with coefficients in UFD's form the core of this part of the book. Chapter VI deals with elementary group theory. As in the precedent chapters, the main notions are introduced via the most significant and easy examples: Z, Z_n, the groups of n-roots of unity, dihedral and symmetric groups. Cayley's embedding theorem, Abel's theorem on the symplicity of the alternating group, the fundamental theorem of finite abelian groups and Sylow's theorems are the main results in this chapter.
The last chapter is Chapter VII, and it has a rather different nature. As easy applications of the basics on factorization and the theory of finite groups, some fascinating items are introduced: RSA Encryption, Check digits, NIM games and 15 puzzle, and questions on finite fields among others.
The book finishes with two appendices dealing with several preliminary notions and results: mappings and the Well-ordering principle of the natural numbers. Complete solutions to all activities and exercises are available at the authors website.
The book is intended to be studied in a year long course. The authors claim, and they are right, that the students are the main actors in the learning process, and this is why the book is written for them. But it will also be a very useful tool for the teachers of the subject. I am sure that all users of this excellent book will not feel disappointed after its study.