Foreword

by Barry Mazur xv

Preface to the Paperback Edition xxi

Preface xxv

Acknowledgments xxxi

Greek Alphabet xxxiii

PART ONE. ALGEBRAIC PRELIMINARIES

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HAPTER 1. REPRESENTATIONS 3

The Bare Notion of Representation 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

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HAPTER 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

x CONTENTS

CHAPTER 3. PERMUTATIONS 21

The abc of Permutations 21

Permutations in General 25

Cycles 26

Digression: Mathematics and Society 29

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HAPTER 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

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HAPTER 5. COMPLEX NUMBERS 42

Overture to Complex Numbers 42

Complex Arithmetic 44

Complex Numbers and Solving Equations 47

Digression: Theorem 47

Algebraic Closure 47

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HAPTER 6. EQUATIONS AND VARIETIES 49

The Logic of Equality 50

The History of Equations 50

Z-Equations 52

Varieties 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

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HAPTER 7. QUADRATIC RECIPROCITY 67

The Simplest Polynomial Equations 67

When is −1 a Square mod p?69

The Legendre Symbol 71

Digression: Notation Guides Thinking 72

Multiplicativity of the Legendre Symbol 73

CONTENTS xi

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

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HAPTER 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

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HAPTER 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

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HAPTER 10. MATRICES 114

Matrices and Matrix Representations 114

Matrices and Their Entries 115

Matrix Multiplication 117

Linear Algebra 120

Digression: Graeco-Latin Squares 122

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HAPTER 11. GROUPS OF MATRICES 124

Square Matrices 124

Matrix Inverses 126

xii CONTENTS

The General Linear Group of Invertible Matrices 129

The Group GL(2, Z) 130

Solving Matrix Equations 132

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HAPTER 12. GROUP REPRESENTATIONS 135

Morphisms of Groups 135

A

4

, Symmetries of a Tetrahedron 139

Representations of A

4

142

Mod p Linear Representations of the Absolute Galois

Group from Elliptic Curves 146

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HAPTER 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

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HAPTER 14. THE RESTRICTION MORPHISM 157

The Big Picture and the Little Pictures 157

Basic Facts about the Restriction Morphism 159

Examples 161

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HAPTER 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

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HAPTER 16. FROBENIUS 177

Something for Nothing 177

Good Prime, Bad Prime 179

Algebraic Integers, Discriminants, and Norms 180

A Working Deﬁnition of Frob

p

184

An Example of Computing Frobenius Elements 185

Frob

p

and Factoring Polynomials modulo p 186

CONTENTS xiii

Appendix: The Ofﬁcial Deﬁnition of the Bad Primes for

a Galois Representation 188

Appendix: The Ofﬁcial Deﬁnition of “Unramiﬁed” and

Frob

p

189

PART THREE. RECIPROCITY LAWS

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HAPTER 17. RECIPROCITY LAW S 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

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HAPTER 18. ONE- AND TWO-DIMENSIONAL

REPRESENTATIONS 200

Roots of Unity 200

How Frob

q

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 Frob

q

Acts on p-Torsion Points 207

The 2-Torsion 209

An Example 209

Another Example 211

Yet Another Example 212

The Proof 214

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HAPTER 19. QUADRATIC RECIPROCITY REVISITED 216

Simultaneous Eigenelements 217

The Z-Variety x

2

− W 218

A Weak Reciprocity Law 220

A Strong Reciprocity Law 221

A Derivation of Quadratic Reciprocity 222

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HAPTER 20. A MACHINE FOR MAKING GALOIS

REPRESENTATIONS 225

Vector Spaces and Linear Actions of Groups 225

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