
Book DetailsGraduate Studies in MathematicsVolume: 114; 2010; 1008 ppMSC: Primary 11; 12; 13; 15; 16; 18; 19; 20;
Now available in Third Edition: GSM/165/180
This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen–Schreier theorem (subgroups of free groups are free).
The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Gröbner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn–Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra.
Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic \(K\)theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.ReadershipGraduate students interested in algebra.

Table of Contents

Chapters

Chapter 1. Groups I

Chapter 2. Commutative rings I

Chapter 3. Galois theory

Chapter 4. Groups II

Chapter 5. Commutative rings II

Chapter 6. Rings

Chapter 7. Representation theory

Chapter 8. Advanced linear algebra

Chapter 9. Homology

Chapter 10. Commutative rings III


Additional Material

Reviews

[T]his is an excellent book containing much more than what is likely to be covered in a standard graduate course. It certainly fulfills the author's vision of a book that contains 'many of the standard theorems and definitions that users of Algebra need to know.' . . . Rotman has completely rewritten the book for the new edition. . . . The best features of the first edition are retained, including Rotman's humane and elegant approach to mathematical exposition: things are explained in both words and symbols, there are historical (and even autobiographical) remarks, and the etymology of some unusual terms is explored. Most importantly, the author often takes the time to put on paper the kind of 'here's how to think about it' advice that mathematicians often share with each other only orally. In the introduction, Rotman says that 'each generation should survey Algebra to make it serve the present time.' His Advanced Modern Algebra admirably fulfills that goal.
Fernando Q. Gouvêa, MAA Reviews 
About the First Edition: …a highly welcome enhancement to the existing textbook literature in the field of algebra.
Zentralblatt fur Mathematik


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Now available in Third Edition: GSM/165/180
This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen–Schreier theorem (subgroups of free groups are free).
The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Gröbner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn–Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra.
Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic \(K\)theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.
Graduate students interested in algebra.

Chapters

Chapter 1. Groups I

Chapter 2. Commutative rings I

Chapter 3. Galois theory

Chapter 4. Groups II

Chapter 5. Commutative rings II

Chapter 6. Rings

Chapter 7. Representation theory

Chapter 8. Advanced linear algebra

Chapter 9. Homology

Chapter 10. Commutative rings III

[T]his is an excellent book containing much more than what is likely to be covered in a standard graduate course. It certainly fulfills the author's vision of a book that contains 'many of the standard theorems and definitions that users of Algebra need to know.' . . . Rotman has completely rewritten the book for the new edition. . . . The best features of the first edition are retained, including Rotman's humane and elegant approach to mathematical exposition: things are explained in both words and symbols, there are historical (and even autobiographical) remarks, and the etymology of some unusual terms is explored. Most importantly, the author often takes the time to put on paper the kind of 'here's how to think about it' advice that mathematicians often share with each other only orally. In the introduction, Rotman says that 'each generation should survey Algebra to make it serve the present time.' His Advanced Modern Algebra admirably fulfills that goal.
Fernando Q. Gouvêa, MAA Reviews 
About the First Edition: …a highly welcome enhancement to the existing textbook literature in the field of algebra.
Zentralblatt fur Mathematik