Algebra

Abstract

As I see it, the graduate course in algebra must primarily prepare students to handle the algebra which they will meet in all of mathematics: topology, partial differential equations, differential geometry, algebraic geometry, analysis , and representation theory, not to speak of algebra itself and algebraic number theory with all its ramifications . Hence I have inserted throughout references to papers and books which have appeared during the last decades , to indicate some of the directions in which the algebraic foundations provided by this book are used ; I have accompanied these references with some motivating comments, to explain how the topics of the present book fit into the mathematics that is to come subsequently in various fields ; and I have also mentioned some unsolved problems of mathematics in algebra and number theory . The abc conjecture is perhaps the most spectacular of these. Often when such comments and examples occur out of the logical order, especially with examples from other branches of mathematics , of necessity some terms may not be defined , or may be defined only later in the book . I have tried to help the reader not only by making cross-references within the book, but also by referring to other books or papers which I mention explicitly . I have also added a number of exercises . On the whole, I have tried to make the exercises complement the examples, and to give them aesthetic appeal. I have tried to use the exercises also to drive readers toward variations and applications of the main text , as well as toward working out special cases, and as openings toward applications beyond this book .