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Fearless Symmetry by Robert Gross, Avner Ash

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Foreword
by Barry Mazur xv
Preface to the Paperback Edition xxi
Preface xxv
Acknowledgments xxxi
Greek Alphabet xxxiii
PART ONE. ALGEBRAIC PRELIMINARIES
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
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
C
HAPTER 14. THE RESTRICTION MORPHISM 157
The Big Picture and the Little Pictures 157
Basic Facts about the Restriction Morphism 159
Examples 161
C
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
C
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
C
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
C
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
C
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
C
HAPTER 20. A MACHINE FOR MAKING GALOIS
REPRESENTATIONS 225
Vector Spaces and Linear Actions of Groups 225

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