Book Description
Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.
Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematicsnumber theoryfor such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack wellknown problems such as Fermat's Last Theorem, Pythagorean Triples, and the everelusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.
The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.
Table of Contents
 Cover (1/2)
 Cover (2/2)
 Contents (1/2)
 Contents (2/2)
 Foreword (1/2)
 Foreword (2/2)
 Preface to the Paperback Edition
 Preface (1/2)
 Preface (2/2)
 Acknowledgments
 Greek Alphabet

PART ONE. ALGEBRAIC PRELIMINARIES
 CHAPTER 1. REPRESENTATIONS
 CHAPTER 2. GROUPS
 CHAPTER 3. PERMUTATIONS
 CHAPTER 4. MODULAR ARITHMETIC
 CHAPTER 5. COMPLEX NUMBERS
 CHAPTER 6. EQUATIONS AND VARIETIES

CHAPTER 7. QUADRATIC RECIPROCITY
 The Simplest Polynomial Equations
 When is –1 a Square mod p?
 The Legendre Symbol
 Digression: Notation Guides Thinking
 Multiplicativity of the Legendre Symbol
 When Is 2 a Square mod p?
 When Is 3 a Square mod p?
 When Is 5 a Square mod p? ( Will This Go On Forever?)
 The Law of Quadratic Reciprocity
 Examples of Quadratic Reciprocity

PART TWO. GALOIS THEORY AND REPRESENTATIONS
 CHAPTER 8. GALOIS THEORY
 CHAPTER 9. ELLIPTIC CURVES
 CHAPTER 10. MATRICES
 CHAPTER 11. GROUPS OF MATRICES
 CHAPTER 12. GROUP REPRESENTATIONS
 CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL
 CHAPTER 14. THE RESTRICTION MORPHISM
 CHAPTER 15. THE GREEKS HAD A NAME FOR IT

CHAPTER 16. FROBENIUS
 Something for Nothing
 Good Prime, Bad Prime
 Algebraic Integers, Discriminants, and Norms
 A Working Definition of Frob[sub(p)]
 An Example of Computing Frobenius Elements
 Frob[sub(p)] and Factoring Polynomials modulo p
 Appendix: The Official Definition of the Bad Primes for a Galois Representation
 Appendix: The Official Definition of "Unramified" and Frob[sub(p)]

PART THREE. RECIPROCITY LAWS
 CHAPTER 17. RECIPROCITY LAWS

CHAPTER 18. ONE AND TWODIMENSIONAL REPRESENTATIONS
 Roots of Unity
 How Frob[sub(q)] Acts on Roots of Unity
 OneDimensional Galois Representations
 TwoDimensional Galois Representations Arising from the pTorsion Points of an Elliptic Curve
 How Frob[sub(q)] Acts on pTorsion Points
 The 2Torsion
 An Example
 Another Example
 Yet Another Example
 The Proof
 CHAPTER 19. QUADRATIC RECIPROCITY REVISITED
 CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS
 CHAPTER 21. A LAST LOOK AT RECIPROCITY

CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS
 The Three Pieces of the Proof
 Frey Curves
 The Modularity Conjecture
 Lowering the Level
 Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves
 Bring on the Reciprocity Laws
 What Wiles and Taylor–Wiles Did
 Generalized Fermat Equations
 What Henri Darmon and Loïc Merel Did
 Prospects for Solving the Generalized Fermat Equations
 CHAPTER 23. RETROSPECT
 Bibliography
 Index
Product Information
 Title: Fearless Symmetry
 Author(s):
 Release date: August 2008
 Publisher(s): Princeton University Press
 ISBN: 9781400837779