Hardcover ISBN:  9780821847954 
Product Code:  AMSTEXT/9 
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AMS Member Price:  $63.20 
eBook ISBN:  9781470411220 
Product Code:  AMSTEXT/9.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $60.00 
Hardcover ISBN:  9780821847954 
eBook: ISBN:  9781470411220 
Product Code:  AMSTEXT/9.B 
List Price:  $154.00 $116.50 
MAA Member Price:  $138.60 $104.85 
AMS Member Price:  $123.20 $93.20 
Hardcover ISBN:  9780821847954 
Product Code:  AMSTEXT/9 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $63.20 
eBook ISBN:  9781470411220 
Product Code:  AMSTEXT/9.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $60.00 
Hardcover ISBN:  9780821847954 
eBook ISBN:  9781470411220 
Product Code:  AMSTEXT/9.B 
List Price:  $154.00 $116.50 
MAA Member Price:  $138.60 $104.85 
AMS Member Price:  $123.20 $93.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 9; 2003; 227 ppMSC: Primary 12
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 semihistorical 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 17gon, 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.
ReadershipUndergraduate students interested in abstract algebra.

Table of Contents

Cover

Title page

Preface

Contents

Introduction

Chapter 0. Background

1. What Is Congruence?

2. Some TwoDimensional Geometry

3. Symmetry

4. The Root of It All

5. The Renaissance of Algebra

6. Complex Numbers

7. Symmetric Polynomials and The Fundamental Theorem of Algebra

8. Permutations and Lagrange’s Theorem

9. Orbits and Cauchy’s Formula

9A. Hamilton’s Quaternions (Optional)

10. Back to Euclid

11. Euclid’s Lemma for Polynomials

12. Fermat and the Rebirth of Number Theory

13. Lagrange’s Theorem Revisited

14. Rings and Squares

14A. More Rings and More Squares

15. Fermat’s Last Theorem (for Polynomials)

15A. Still more Fermat’s Last Theorem (Optional)

16. Constmctible Polygons and the Method of Mr. Gauss

17. Cyclotomic Fields and Linear Algebra

18. A Lagrange Theorem for Fields and Nonconstructibility

19. Galois Fields and the Fundamental Theorem of Algebra Revisited

20. Galois’ Theory of Equations

21. The Galois Correspondence

22. Constructible Numbers and Solvable Equations

Index

Back Cover


Additional Material

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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 semihistorical 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 17gon, 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.
Undergraduate students interested in abstract algebra.

Cover

Title page

Preface

Contents

Introduction

Chapter 0. Background

1. What Is Congruence?

2. Some TwoDimensional Geometry

3. Symmetry

4. The Root of It All

5. The Renaissance of Algebra

6. Complex Numbers

7. Symmetric Polynomials and The Fundamental Theorem of Algebra

8. Permutations and Lagrange’s Theorem

9. Orbits and Cauchy’s Formula

9A. Hamilton’s Quaternions (Optional)

10. Back to Euclid

11. Euclid’s Lemma for Polynomials

12. Fermat and the Rebirth of Number Theory

13. Lagrange’s Theorem Revisited

14. Rings and Squares

14A. More Rings and More Squares

15. Fermat’s Last Theorem (for Polynomials)

15A. Still more Fermat’s Last Theorem (Optional)

16. Constmctible Polygons and the Method of Mr. Gauss

17. Cyclotomic Fields and Linear Algebra

18. A Lagrange Theorem for Fields and Nonconstructibility

19. Galois Fields and the Fundamental Theorem of Algebra Revisited

20. Galois’ Theory of Equations

21. The Galois Correspondence

22. Constructible Numbers and Solvable Equations

Index

Back Cover