Algebra

This book is designed as a text for a first-year graduate-level course in algebra. Instead of an encyclopaedic approach, the authors used a thematic, consistent point of view. The unifying theme is the concept of a module (a generalization to rings of the concept of a vector space, which is defined on a field). To promote understanding, the book provides proofs with a maximum of insight with a minimum of computation as well as chapters stressing computational techniques. Beginning with an introduction to group theory and ring theory, the book then develops the basics of module theory. A review of the basics of linear algebra demonstrates the power of module theory in the derivation of canonical forms and the spectral theorem. The book then uses module theory to investigate bilinear, sesquilinear and quadratic forms. Finally, a discussion of ring theory and multilinear forms leads to a chapter on group representations, which again uses module theory to powerful effect.