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 Evariste Galois. They have been used extensively in the oldest and largest branch of mathematics – number theory – for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive 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. *Fearless Symmetry* 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, *Fearless Symmetry* will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

"The authors are to be admired for taking a very difficult topic and making it [...] more accessible than it was before."

– Timothy Gowers, *Nature*

"The authors [...] outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields."

– *Science News*

"The book [...] does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject."

– William M. McGovern, *SIAM Review*

"Unique [...] [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics."

– Lindsay N. Childs, *Mathematical Reviews*

"To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program."

– Lindsay N. Childs, MathSciNet

"All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different – by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat's Last Theorem."

– Peter Galison, Harvard University

PART ONE: ALGEBRAIC PRELIMINARIES

CHAPTER 1. REPRESENTATIONS 3

The Bare NotionofRepresentation 3

An Example: Counting 5

Digression: Definitions 6

Counting (Continued)7

Counting Viewed as a Representation 8

The Definition of a Representation 9

Counting and Inequalities as Representations 10

Summary 11

CHAPTER 2. GROUPS 13

The Group of Rotations of a Sphere 14

The General Concept of "Group" 17

In Praise of Mathematical Idealization 18

Digression: Lie Groups 19

CHAPTER 3. PERMUTATIONS 21

The abc of Permutations 21

Permutations in General 25

Cycles 26

Digression: Mathematics and Society 29

CHAPTER 4. MODULAR ARITHMETIC 31

Cyclical Time 31

Congruences 33

Arithmetic Modulo a Prime 36

Modular Arithmetic and Group Theory 39

Modular Arithmetic and Solutions of Equations 41

CHAPTER 5. COMPLEX NUMBERS 42

Overture to Complex Numbers 42

Complex Arithmetic 44

Complex Numbers and Solving Equations 47

Digression: Theorem 47

Algebraic Closure 47

CHAPTER 6. EQUATIONS AND VARIETIES 49

The Logic of Equality 50

The History of Equations 50

Z-Equations 52

Vari eti es 54

Systems of Equations 56

Equivalent Descriptions of the Same Variety 58

Finding Roots of Polynomials 61

Are There General Methods for Finding Solutions to

Systems of Polynomial Equations? 62

Deeper Understanding Is Desirable 65

CHAPTER 7. QUADRATIC RECIPROCITY 67

The Simplest Polynomial Equations 67

When is -1 aSquaremodp? 69

The Legendre Symbol 71

Digression: Notation Guides Thinking 72

Multiplicativity of the Legendre Symbol 73

When Is 2 a Square mod p?74

When Is 3 a Square mod p?75

When Is 5 a Square mod p? (Will This Go On Forever?) 76

The Law of Quadratic Reciprocity 78

Examples of Quadratic Reciprocity 80

PART TWO. GALOIS THEORY AND REPRESENTATIONS

CHAPTER 8. GALOIS THEORY 87

Polynomials and Their Roots 88

The Field of Algebraic Numbers Q alg 89

The Absolute Galois Group of Q Defined 92

A Conversation with s: A Playlet in Three Short Scenes 93

Digression: Symmetry 96

How Elements of G Behave 96

Why Is G a Group? 101

Summary 101

CHAPTER 9. ELLIPTIC CURVES 103

Elliptic Curves Are "Group Varieties" 103

An Example 104

The Group Law on an Elliptic Curve 107

A Much-Needed Example 108

Digression: What Is So Great about Elliptic Curves? 109

The Congruent Number Problem 110

Torsion and the Galois Group 111

CHAPTER 10. MATRICES 114

Matrices and Matrix Representations 114

Matrices and Their Entries 115

Matrix Multiplication 117

Linear Algebra 120

Digression: Graeco-Latin Squares 122

CHAPTER 11. GROUPS OF MATRICES 124

Square Matrices 124

Matrix Inverses 126

The General Linear Group of Invertible Matrices 129

The Group GL(2, Z) 130

Solving Matrix Equations 132

CHAPTER 12. GROUP REPRESENTATIONS 135

Morphisms of Groups 135

A4, Symmetries of a Tetrahedron 139

Representations of A4 142

Mod p Linear Representations of the Absolute Galois

Group from Elliptic Curves 146

CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL 149

The Field Generated by a Z-Polynomial 149

Examples 151

Digression: The Inverse Galois Problem 154

Two More Things 155

CHAPTER 14. THE RESTRICTION MORPHISM 157

The BigPicture andthe Little Pictures 157

Basic Facts about the Restriction Morphism 159

Examples 161

CHAPTER 15. THE GREEKS HAD A NAME FOR IT 162

Traces 163

Conjugacy Classes 165

Examples of Characters 166

How the Character of a Representation Determines the

Representation 171

Prelude to the Next Chapter 175

Digression: A Fact about Rotations of the Sphere 175

CHAPTER 16. FROBENIUS 177

Something for Nothing 177

Good Prime, Bad Prime 179

Algebraic Integers, Discriminants, and Norms 180

A Working Definition of Frobp 184

An Example of Computing Frobenius Elements 185

Frobp and Factoring Polynomials modulo p 186

Appendix: The Official Definition of the Bad Primes fora Galois Representation 188

Appendix: The Official Definition of "Unramified" and Frobp 189

PART THREE. RECIPROCITY LAWS

CHAPTER 17. RECIPROCITY LAWS 193

The List of Traces of Frobenius 193

Black Boxes 195

Weak and Strong Reciprocity Laws 196

Digression: Conjecture 197

Kinds of Black Boxes 199

CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS 200

Roots of Unity 200

How Frobq Acts on Roots of Unity 202

One-Dimensional Galois Representations 204

Two-Dimensional Galois Representations Arising from

the p-Torsion Points of an Elliptic Curve 205

How Frobq Acts on p-Torsion Points 207

The 2-Torsion 209

An Example 209

Another Example 211

Yet Another Example 212

The Proof 214

CHAPTER 19. QUADRATIC RECIPROCITY REVISITED 216

Simultaneous Eigenelements 217

The Z-Variety x2-W 218

A Weak Reciprocity Law 220

A Strong Reciprocity Law 221

A Derivation of Quadratic Reciprocity 222

CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS 225

Vector Spaces and Linear Actions of Groups 225

Linearization 228

Etale Cohomology 229

Conjectures about Étale Cohomology 231

CHAPTER 21. A LAST LOOK AT RECIPROCITY 233

What Is Mathematics? 233

Reciprocity 235

Modular Forms 236

Review of Reciprocity Laws 239

A Physical Analogy 240

CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS 242

The Three Pieces of the Proof 243

Frey Curves 244

The Modularity Conjecture 245

Lowering the Level 247

Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves 249

Bring on the Reciprocity Laws 250

What Wiles and Taylor-Wiles Did 252

Generalized Fermat Equations 254

What Henri Darmon and Loyc Merel Did 255

Prospects for Solving the Generalized Fermat Equations 256

CHAPTER 23. RETROSPECT 257

Topics Covered 257

Back to Solving Equations 258

Digression: Why Do Math? 260

The Congruent Number Problem 261

Peering Past the Frontier 263

Bibliography 265

Index 269

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**Avner Ash** is Professor of Mathematics at Boston College. He is the author (with D. Mumford, M. Rapoport, and Y. Tai) of *Smooth Compactification of Locally Symmetric Varieties*. **Robert Gross** is Associate Professor of Mathematics at Boston College.

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