**Graduate Studies in Mathematics**

Volume: 56;
2003;
511 pp;
Softcover

MSC: Primary 13; 15; 16; 20;

**Print ISBN: 978-0-8218-3413-8
Product Code: GSM/56.S**

List Price: $70.00

AMS Member Price: $56.00

MAA Member Price: $63.00

**Electronic ISBN: 978-1-4704-1149-7
Product Code: GSM/56.E**

List Price: $93.00

AMS Member Price: $74.40

MAA Member Price: $83.70

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#### Supplemental Materials

# A Course in Algebra

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*E. B. Vinberg*

This is a comprehensive textbook on modern algebra written by an
internationally renowned specialist. It covers material traditionally found in
advanced undergraduate and basic graduate courses and presents it in a lucid
style. The author includes almost no technically difficult proofs, and
reflecting his point of view on mathematics, he tries wherever possible to
replace calculations and difficult deductions with conceptual proofs and to
associate geometric images to algebraic objects. The effort spent on the part
of students in absorbing these ideas will pay off when they turn to solving
problems outside of this textbook.

Another important feature is the presentation of most topics on several
levels, allowing students to move smoothly from initial acquaintance with the
subject to thorough study and a deeper understanding. Basic topics are
included, such as algebraic structures, linear algebra, polynomials, and
groups, as well as more advanced topics, such as affine and projective spaces,
tensor algebra, Galois theory, Lie groups, and associative algebras and their
representations. Some applications of linear algebra and group theory to
physics are discussed.

The book is written with extreme care and contains over 200 exercises and 70
figures. It is ideal as a textbook and also suitable for independent study for advanced
undergraduates and graduate students.

#### Readership

Advanced undergraduates, graduate students and research mathematicians interested in algebra.

#### Reviews & Endorsements

This is a masterly textbook on basic algebra. It is, at the same time, demanding and down-to-earth, challenging and user-friendly, abstract and concrete, concise and comprehensible, and above all extremely educating, inspiring and enlightening.

-- Zentralblatt MATH

Great book! The author's teaching experience shows in every chapter.

-- E. Zelmanov, University of California, San Diego

Vinberg has written an algebra book that is excellent, both as a classroom text or for self-study. It starts with the most basic concepts and builds in orderly fashion to moderately advanced topics … Well motivated examples help the student … to master the material thoroughly, and exercises test one's growing skill in addition to covering useful auxiliary facts … years of teaching abstract algebra have enabled Vinberg to say the right thing at the right time.

-- Irving Kaplansky, MSRI

#### Table of Contents

# Table of Contents

## A Course in Algebra

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface ix10 free
- Chapter 1. Algebraic Structures 112 free
- Chapter 2. Elements of Linear Algebra 3546
- Chapter 3. Elements of Polynomial Algebra 8192
- §3.1. Polynomial Algebra: Construction and Basic Properties 8192
- §3.2. Roots of Polynomials: General Properties 8798
- §3.3. Fundamental Theorem of Algebra of Complex Numbers 93104
- §3.4. Roots of Polynomials with Real Coefficients 98109
- §3.5. Factorization in Euclidean Domains 103114
- §3.6. Polynomials with Rational Coefficients 109120
- §3.7. Polynomials in Several Variables 112123
- §3.8. Symmetric Polynomials 116127
- §3.9. Cubic Equations 123134
- §3.10. Field of Rational Fractions 129140

- Chapter 4. Elements of Group Theory 137148
- Chapter 5. Vector Spaces 171182
- Chapter 6. Linear Operators 201212
- Chapter 7. Affine and Projective Spaces 239250
- Chapter 8. Tensor Algebra 295306
- Chapter 9. Commutative Algebra 325336
- Chapter 10. Groups 385396
- Chapter 11. Linear Representations and Associative Algebras 419430
- Chapter 12. Lie Groups 471482
- Answers to Selected Exercises 495506
- Bibliography 501512
- Index 503514
- Back Cover Back Cover1526