Preface
This book is a self-contained introduction to the basic structures of abstract algebra: groups, rings, and fields. It is designed to be used in a two-semester course for undergraduates or a one-semester course for seniors or graduates. The table of contents is flexible (see the chapter summaries that follow), so the book is suitable for a traditional course at various levels or for a more application-oriented treatment. The book is written to be read by students with little outside help and so can be used for self-study. In addition, it contains several optional sections on special topics and applications.
Because many students will not have had much experience with abstract thinking, a number of important concrete examples (number theory, integers modulo n, permu-tations) are introduced at the beginning and referred to throughout the book. These examples are chosen for their importance and intrinsic interest and also be-cause the student can do actual computations almost immediately even though the examples are, in the student's view, quite abstract. Thus, they provide a bridge to the abstract theory and serve as prototype examples of the abstract structures themselves. As an illustration, the student will encounter composition and inverses of permutations before having to fit these notions into the general framework of group theory.
The axiomatic development of these structures is also emphasized. Algebra provides one of the best illustrations of the power of abstraction ...
