Algebra: Chapter 0

Description

Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references. Cover......Page 1 Title page......Page 5 Contents......Page 7 Introduction......Page 17 Preliminaries: Set theory and categories......Page 23 Groups, first encounter......Page 63 Rings and modules......Page 141 Groups, second encounter......Page 209 Irreducibility and factorization in integral domains......Page 265 Linear algebra......Page 327 Fields......Page 407 Linear algebra, reprise......Page 505 Homological algebra......Page 581 Index......Page 721 Back Cover......Page 739