Preface ix

Part 1. Numbers and Equations

Lesson 1. What Algebra Is

1. Numbers in disguise

1.1. Classical and modern algebra

2. Arithmetic and algebra

3. The environment of algebra: Number systems

4. Important concepts and principles in this lesson

5. Problems and questions

6. Further reading

Lesson 2. Equations and Their Solutions

1. Polynomial equations, coefficients, and roots

1.1. Geometric interpretations

2. The classification of equations

2.1. Diophantine equations

3. Numerical and formulaic approaches to equations

3.1. The numerical approach

3.2. The formulaic approach

4. Important concepts and principles in this lesson

5. Problems and questions

6. Further reading

Lesson 3. Where Algebra Comes From

1. An Egyptian problem

2. A Mesopotamian problem

3. A Chinese problem

4. An Arabic problem

5. A Japanese problem

6. Problems and questions

7. Further reading

Lesson 4. Why Algebra Is Important

1. Example: An ideal pendulum

2. Problems and questions

3. Further reading

v

vi CONTENTS

Lesson 5. Numerical Solution of Equations

1. A simple but crude method

2. Ancient Chinese methods of calculating

2.1. A linear problem in three unknowns

3. Systems of linear equations

4. Polynomial equations

4.1. Noninteger solutions

5. The cubic equation

6. Problems and questions

7. Further reading

Part 2. The Formulaic Approach to Equations

Lesson 6. Combinatoric Solutions I: Quadratic Equations

1. Why not set up tables of solutions?

2. The quadratic formula

3. Problems and questions

4. Further reading

Lesson 7. Combinatoric Solutions II: Cubic Equations

1. Reduction from four parameters to one

2. Graphical solutions of cubic equations

3. Efforts to find a cubic formula

3.1. Cube roots of complex numbers

4. Alternative forms of the cubic formula

5. The \irreducible case

5.1. Imaginary numbers

6. Problems and questions

7. Further reading

Part 3. Resolvents

Lesson 8. From Combinatorics to Resolvents

1. Solution of the irreducible case using complex numbers

2. The quartic equation

3. Viµete's solution of the irreducible case of the cubic

3.1. Comparison of the Viµete and Cardano solutions

4. The Tschirnhaus solution of the cubic equation

5. Lagrange's reflections on the cubic equation

5.1. The cubic formula in terms of the roots

5.2. A test case: The quartic

6. Problems and questions

7. Further reading

Lesson 9. The Search for Resolvents

1. Coefficients and roots

2. A unified approach to equations of all degrees

CONTENTS vii

2.1. A resolvent for the cubic equation

3. A resolvent for the general quartic equation

4. The state of polynomial algebra in 1770

4.1. Seeking a resolvent for the quintic

5. Permutations enter algebra

6. Permutations of the variables in a function

6.1. Two-valued functions

7. Problems and questions

8. Further reading

Part 4. Abstract Algebra

Lesson 10. Existence and Constructibility of Roots

1. Proof that the complex numbers are algebraically closed

2. Solution by radicals: General considerations

2.1. The quadratic formula

2.2. The cubic formula

2.3. Algebraic functions and algebraic formulas

3. Abel's proof

3.1. Taking the formula apart

3.2. The last step in the proof

3.3. The verdict on Abel's proof

4. Problems and questions

5. Further reading

Lesson 11. The Breakthrough: Galois Theory

1. An example of a solving an equation by radicals

2. Field automorphisms and permutations of roots

2.1. Subgroups and cosets

2.2. Normal subgroups and quotient groups

2.3. Further analysis of the cubic equation

2.4. Why the cubic formula must have the form it does

2.5. Why the roots of unity are important

2.6. The birth of Galois theory

3. A sketch of Galois theory

4. Solution by radicals

4.1. Abel's theorem

5. Some simple examples for practice

6. The story of polynomial algebra: a recap

7. Problems and questions

8. Further reading

Epilogue: Modern Algebra

1. Groups

2. Rings

2.1. Associative rings

viii CONTENTS

2.2. Lie rings

2.3. Special classes of rings

3. Division rings and fields

4. Vector spaces and related structures

4.1. Modules

4.2. Algebras

5. Conclusion

Appendix: Some Facts about Polynomials

Answers to the Problems and Questions

Subject Index

Name Index