**Pure and Applied Undergraduate Texts**

Volume: 9;
2003;
227 pp;
Hardcover

MSC: Primary 12;

Print ISBN: 978-0-8218-4795-4

Product Code: AMSTEXT/9

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-1122-0
Product Code: AMSTEXT/9.E**

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

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# Abstract Algebra

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*Ronald Solomon*

This undergraduate text takes a novel approach to the standard introductory material on groups, rings, and fields. At the heart of the text is a semi-historical journey through the early decades of the subject as it emerged in the revolutionary work of Euler, Lagrange, Gauss, and Galois. Avoiding excessive abstraction whenever possible, the text focuses on the central problem of studying the solutions of polynomial equations. Highlights include a proof of the Fundamental Theorem of Algebra, essentially due to Euler, and a proof of the constructability of the regular 17-gon, in the manner of Gauss. Another novel feature is the introduction of groups through a meditation on the meaning of congruence in the work of Euclid. Everywhere in the text, the goal is to make clear the links connecting abstract algebra to Euclidean geometry, high school algebra, and trigonometry, in the hope that students pursuing a career as secondary mathematics educators will carry away a deeper and richer understanding of the high school mathematics curriculum. Another goal is to encourage students, insofar as possible in a textbook format, to build the course for themselves, with exercises integrally embedded in the text of each chapter.

#### Readership

Undergraduate students interested in abstract algebra.

#### Table of Contents

# Table of Contents

## Abstract Algebra

- Cover Cover11 free
- Title page v6 free
- Preface ix10
- Contents xi12 free
- Introduction 114
- Chapter 0. Background 720 free
- 1. What Is Congruence? 1427
- 2. Some Two-Dimensional Geometry 2235
- 3. Symmetry 3245
- 4. The Root of It All 4659
- 5. The Renaissance of Algebra 4962
- 6. Complex Numbers 5972
- 7. Symmetric Polynomials and The Fundamental Theorem of Algebra 6881
- 8. Permutations and Lagrange’s Theorem 7891
- 9. Orbits and Cauchy’s Formula 8699
- 9A. Hamilton’s Quaternions (Optional) 95108
- 10. Back to Euclid 102115
- 11. Euclid’s Lemma for Polynomials 113126
- 12. Fermat and the Rebirth of Number Theory 122135
- 13. Lagrange’s Theorem Revisited 136149
- 14. Rings and Squares 142155
- 14A. More Rings and More Squares 149162
- 15. Fermat’s Last Theorem (for Polynomials) 157170
- 15A. Still more Fermat’s Last Theorem (Optional) 165178
- 16. Constmctible Polygons and the Method of Mr. Gauss 170183
- 17. Cyclotomic Fields and Linear Algebra 177190
- 18. A Lagrange Theorem for Fields and Nonconstructibility 191204
- 19. Galois Fields and the Fundamental Theorem of Algebra Revisited 196209
- 20. Galois’ Theory of Equations 207220
- 21. The Galois Correspondence 212225
- 22. Constructible Numbers and Solvable Equations 217230
- Index 223236
- Back Cover Back Cover1241