Preface to Second Edition Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowa- days, everyone agrees that some knowledge of Linear Algebra, Group Theory, and Commutative Algebra is necessary, and these topics are introduced in undergradu- ate courses. We continue their study. This book can be used as a text for the first year of graduate Algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own. While not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. In addition, this is a reference containing many of the standard theorems and definitions that users of Algebra need to know. Thus, this book is not merely an appetizer it is a hearty meal as well. When I was a student, Birkhoff–Mac Lane, A Survey of Modern Algebra, was the text for my first Algebra course, and van der Waerden, Modern Algebra, was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first publication: Birkhoff and Mac Lane’s book appeared in 1941 van der Waer- den’s book appeared in 1930. There are today major directions that either did not exist 75 years ago, or were not then recognized as being so important, or were not so well developed. These new areas involve Algebraic Geometry, Category Theory,1 Computer Science, Homological Algebra, and Representation Theory. Let me now address readers and instructors for whom this book is a text in a beginning graduate course. Instead of devoting the first chapters to a review of more elementary material (as I did in the first edition), here I usually refer to FCAA (my book, A First Course in Abstract Algebra, 3rd ed.). I have reorganized and rewritten this text, but here are some other major differences with the first edition. 1A Survey of Modern Algebra was rewritten, introducing categories, as Mac Lane–Birkhoff, Algebra, Macmillan, New York, 1967. ix
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