**Graduate Studies in Mathematics**

Volume: 114;
2010;
1008 pp;

MSC: Primary 11; 12; 13; 15; 16; 18; 19; 20;

**Electronic ISBN: 978-1-4704-1176-3
Product Code: GSM/114.E**

List Price: $99.00

AMS Member Price: $79.20

MAA member Price: $89.10

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#### Supplemental Materials

# Advanced Modern Algebra: Second Edition

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*Joseph J. Rotman*

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.

#### Reviews & Endorsements

[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

…a highly welcome enhancement to the existing textbook literature in the field of algebra.

-- Zentralblatt fur Mathematik

#### Table of Contents

# Table of Contents

## Advanced Modern Algebra: Second Edition

- Cover Cover11 free
- Title page i2 free
- Contents v6 free
- Preface to the second edition ix10 free
- Special notation xiii14 free
- Groups I 118 free
- Commutative rings I 8198
- Fields 173190
- Groups II 223240
- Commutative rings II 295312
- Rings 391408
- Representation theory 525542
- Advanced linear algebra 635652
- Homology 751768
- Commutative rings III 873890
- Bibliography 9851,002
- Index 9911,008 free
- Back Cover Back Cover11,026