**Student Mathematical Library**

Volume: 82;
2017;
277 pp;
Softcover

MSC: Primary 00; 12; 13; 15; 20;

Print ISBN: 978-1-4704-3583-7

Product Code: STML/82

List Price: $52.00

AMS Member Price: $41.60

**Electronic ISBN: 978-1-4704-4044-2
Product Code: STML/82.E**

List Price: $52.00

AMS Member Price: $41.60

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#### Supplemental Materials

# Problems in Abstract Algebra

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*A. R. Wadsworth*

This is a book of problems in abstract algebra for strong undergraduates or beginning graduate students. It can be used as a supplement to a course or for self-study. The book provides more variety and more challenging problems than are found in most algebra textbooks. It is intended for students wanting to enrich their learning of mathematics by tackling problems that take some thought and effort to solve. The book contains problems on groups (including the Sylow Theorems, solvable groups, presentation of groups by generators and relations, and structure and duality for finite abelian groups); rings (including basic ideal theory and factorization in integral domains and Gauss's Theorem); linear algebra (emphasizing linear transformations, including canonical forms); and fields (including Galois theory). Hints to many problems are also included.

#### Readership

Undergraduate and graduate students interested in teaching and learning undergraduate algebra.

#### Reviews & Endorsements

A collection of very interesting problems covering almost all the topics of abstract algebra for advanced undergraduates or beginning graduate students is presented through five very main chapters of this book...It is surely a problem book that if it is correctly used can upgrade the level of the reader in abstract algebra.

-- Panayiotis Vlamos, Zentralblatt MATH

It is well known that students learn best by doing, and, in that vein, Wadsworth's new text will help many algebra students. As the title alludes, this book is a collection of exercises about the first-year abstract algebra sequence, but it focuses more on exploratory topics than on the basics. Readers willing to work out each exercise will gain a deep understanding of algebra...The most appreciative audience will be graduate students preparing for algebra exams, for which this is a perfect study guide.

-- A. Misseldine, CHOICE

I'll certainly be using some of the problems the next time I teach algebra. I'm even tempted to make it the only textbook for the course.

-- Fernando Q. Gouvêa, MAA Reviews

#### Table of Contents

# Table of Contents

## Problems in Abstract Algebra

- Cover Cover11
- Title page i2
- Contents iii4
- Preface vii8
- Introduction 110
- Chapter 1. Integers and Integers mod ๐ 716
- Chapter 2. Groups 1322
- 2.1. Groups, subgroups, and cosets 1322
- 2.2. Group homomorphisms and factor groups 2534
- 2.3. Group actions 3241
- 2.4. Symmetric and alternating groups 3645
- 2.5. ๐-groups 4150
- 2.6. Sylow subgroups 4352
- 2.7. Semidirect products of groups 4453
- 2.8. Free groups and groups by generators and relations 5362
- 2.9. Nilpotent, solvable, and simple groups 5867
- 2.10. Finite abelian groups 6675

- Chapter 3. Rings 7382
- 3.1. Rings, subrings, and ideals 7382
- 3.2. Factor rings and ring homomorphisms 8998
- 3.3. Polynomial rings and evaluation maps 97106
- 3.4. Integral domains, quotient fields 100109
- 3.5. Maximal ideals and prime ideals 103112
- 3.6. Divisibility and principal ideal domains 107116
- 3.7. Unique factorization domains 115124

- Chapter 4. Linear Algebra and Canonical Forms of Linear Transformations 125134
- 4.1. Vector spaces and linear dependence 125134
- 4.2. Linear transformations and matrices 132141
- 4.3. Dual space 139148
- 4.4. Determinants 142151
- 4.5. Eigenvalues and eigenvectors, triangulation and diagonalization 150159
- 4.6. Minimal polynomials of a linear transformation and primary decomposition 155164
- 4.7. ๐-cyclic subspaces and ๐-annihilators 161170
- 4.8. Projection maps 164173
- 4.9. Cyclic decomposition and rational and Jordan canonical forms 167176
- 4.10. The exponential of a matrix 177186
- 4.11. Symmetric and orthogonal matrices over \R 180189
- 4.12. Group theory problems using linear algebra 187196

- Chapter 5. Fields and Galois Theory 191200
- 5.1. Algebraic elements and algebraic field extensions 192201
- 5.2. Constructibility by compass and straightedge 199208
- 5.3. Transcendental extensions 202211
- 5.4. Criteria for irreducibility of polynomials 205214
- 5.5. Splitting fields, normal field extensions, and Galois groups 208217
- 5.6. Separability and repeated roots 216225
- 5.7. Finite fields 223232
- 5.8. Galois field extensions 226235
- 5.9. Cyclotomic polynomials and cyclotomic extensions 234243
- 5.10. Radical extensions, norms, and traces 244253
- 5.11. Solvability by radicals 253262

- Suggestions for Further Reading 257266
- Bibliography 259268
- Index of Notation 261270
- Subject and Terminology Index 267276
- Back Cover Back Cover1290