This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.
Graduate students and researchers interested in learning and teaching algebra.
This is a good book and a fitting tribute to a great mathematical expositor. I'll keep both the second edition and the two parts of the third edition on my shelf, and I'm sure I'll turn to them often in the future.
-- Fernando Q. Gouvêa, MAA Reviews
This new edition, now in two parts, has been significantly
reorganized and many sections have been rewritten. The first part,
designed for a first year of graduate algebra, consists of two
courses: Galois theory and Module theory. Topics covered in the first
course are classical formulas for solutions of cubic and quartic
equations, classical number theory, commutative algebra, groups, and
Galois theory. Topics in the second course are Zorn's lemma, canonical
forms, inner product spaces, categories and limits, tensor products,
projective, injective, and flat modules, multilinear algebra, affine
varieties, and Gröbner bases.
The second part presents many topics mentioned in the first part in
greater depth and in more detail. The five chapters of the book are
devoted to group theory, representation theory, homological algebra,
categories, and commutative algebra, respectively. The book can be
used as a text for a second abstract algebra graduate course, as a
source of additional material to a first abstract algebra graduate
course, or for self-study.
Graduate students and researchers interested in learning and teaching algebra.
This new edition, now in two parts, has been significantly reorganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory. Topics covered in the first course are classical formulas for solutions of cubic and quartic equations, classical number theory, commutative algebra, groups, and Galois theory. Topics in the second course are Zorn's lemma, canonical forms, inner product spaces, categories and limits, tensor products, projective, injective, and flat modules, multilinear algebra, affine varieties, and Gröbner bases.
Graduate students and researchers interested in learning and teaching algebra.
The organization of the text is clear and the pace is leisurely...recently I taught the second part of a first-year graduate course in algebra, and in the process dipped into this book at various points. I enjoyed the presentation, and the commentary on constructions and results aimed to place them in context, and I found the numerous examples and exercises helpful. I will be using this book when teaching such courses again, if not always as the main text, certainly as a useful supplement.
-- Srikanth B. Iyengar, Mathematical Reviews
Rotman is a wonderful expositor, and the two courses in this book strike me as well thought out and well presented.
-- Fernando Q. Gouvêa, MAA
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.