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 variste 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. 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.
Author(s): Avner Ash, Robert Gross
Edition: New
Publisher: Princeton University Press
Year: 2008
Language: English
Pages: 307
Tags: Математика;Популярная математика;
Cover......Page 1
Contents......Page 10
Foreword......Page 16
Preface to the Paperback Edition......Page 22
Preface......Page 26
Acknowledgments......Page 32
Greek Alphabet......Page 34
PART ONE. ALGEBRAIC PRELIMINARIES......Page 36
The Bare Notion of Representation......Page 38
An Example: Counting......Page 40
Digression: Definitions......Page 41
Counting (Continued)......Page 42
Counting Viewed as a Representation......Page 43
The Definition of a Representation......Page 44
Counting and Inequalities as Representations......Page 45
Summary......Page 46
CHAPTER 2. GROUPS......Page 48
The Group of Rotations of a Sphere......Page 49
The General Concept of "Group"......Page 52
In Praise of Mathematical Idealization......Page 53
Digression: Lie Groups......Page 54
The abc of Permutations......Page 56
Permutations in General......Page 60
Cycles......Page 61
Digression: Mathematics and Society......Page 64
Cyclical Time......Page 66
Congruences......Page 68
Arithmetic Modulo a Prime......Page 71
Modular Arithmetic and Group Theory......Page 74
Modular Arithmetic and Solutions of Equations......Page 76
Overture to Complex Numbers......Page 77
Complex Arithmetic......Page 79
Algebraic Closure......Page 82
CHAPTER 6. EQUATIONS AND VARIETIES......Page 84
The History of Equations......Page 85
Z-Equations......Page 87
Varieties......Page 89
Systems of Equations......Page 91
Equivalent Descriptions of the Same Variety......Page 93
Finding Roots of Polynomials......Page 96
Are There General Methods for Finding Solutions to Systems of Polynomial Equations?......Page 97
Deeper Understanding Is Desirable......Page 100
The Simplest Polynomial Equations......Page 102
When is –1 a Square mod p?......Page 104
The Legendre Symbol......Page 106
Digression: Notation Guides Thinking......Page 107
Multiplicativity of the Legendre Symbol......Page 108
When Is 2 a Square mod p?......Page 109
When Is 3 a Square mod p?......Page 110
When Is 5 a Square mod p? ( Will This Go On Forever?)......Page 111
The Law of Quadratic Reciprocity......Page 113
Examples of Quadratic Reciprocity......Page 115
PART TWO. GALOIS THEORY AND REPRESENTATIONS......Page 120
CHAPTER 8. GALOIS THEORY......Page 122
Polynomials and Their Roots......Page 123
The Field of Algebraic Numbers Q[sup(alg)]......Page 124
The Absolute Galois Group of Q Defined......Page 127
A Conversation with s: A Playlet in Three Short Scenes......Page 128
How Elements of G Behave......Page 131
Summary......Page 136
Elliptic Curves Are "Group Varieties"......Page 138
An Example......Page 139
The Group Law on an Elliptic Curve......Page 142
A Much-Needed Example......Page 143
Digression: What Is So Great about Elliptic Curves?......Page 144
The Congruent Number Problem......Page 145
Torsion and the Galois Group......Page 146
Matrices and Matrix Representations......Page 149
Matrices and Their Entries......Page 150
Matrix Multiplication......Page 152
Linear Algebra......Page 155
Digression: Graeco-Latin Squares......Page 157
Square Matrices......Page 159
Matrix Inverses......Page 161
The General Linear Group of Invertible Matrices......Page 164
The Group GL(2, Z)......Page 165
Solving Matrix Equations......Page 167
Morphisms of Groups......Page 170
A[sub(4)], Symmetries of a Tetrahedron......Page 174
Representations of A[sub(4)]......Page 177
Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves......Page 181
The Field Generated by a Z-Polynomial......Page 184
Examples......Page 186
Digression: The Inverse Galois Problem......Page 189
Two More Things......Page 190
The Big Picture and the Little Pictures......Page 192
Basic Facts about the Restriction Morphism......Page 194
Examples......Page 196
CHAPTER 15. THE GREEKS HAD A NAME FOR IT......Page 197
Traces......Page 198
Conjugacy Classes......Page 200
Examples of Characters......Page 201
How the Character of a Representation Determines the Representation......Page 206
Digression: A Fact about Rotations of the Sphere......Page 210
Something for Nothing......Page 212
Good Prime, Bad Prime......Page 214
Algebraic Integers, Discriminants, and Norms......Page 215
A Working Definition of Frob[sub(p)]......Page 219
An Example of Computing Frobenius Elements......Page 220
Frob[sub(p)] and Factoring Polynomials modulo p......Page 221
Appendix: The Official Definition of the Bad Primes for a Galois Representation......Page 223
Appendix: The Official Definition of "Unramified" and Frob[sub(p)]......Page 224
PART THREE. RECIPROCITY LAWS......Page 226
The List of Traces of Frobenius......Page 228
Black Boxes......Page 230
Weak and Strong Reciprocity Laws......Page 231
Digression: Conjecture......Page 232
Kinds of Black Boxes......Page 234
Roots of Unity......Page 235
How Frob[sub(q)] Acts on Roots of Unity......Page 237
One-Dimensional Galois Representations......Page 239
Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve......Page 240
How Frob[sub(q)] Acts on p-Torsion Points......Page 242
An Example......Page 244
Another Example......Page 246
Yet Another Example......Page 247
The Proof......Page 249
CHAPTER 19. QUADRATIC RECIPROCITY REVISITED......Page 251
Simultaneous Eigenelements......Page 252
The Z-Variety x[sup(2)] – W......Page 253
A Weak Reciprocity Law......Page 255
A Strong Reciprocity Law......Page 256
A Derivation of Quadratic Reciprocity......Page 257
Vector Spaces and Linear Actions of Groups......Page 260
Linearization......Page 263
Ètale Cohomology......Page 264
Conjectures about Ètale Cohomology......Page 266
What Is Mathematics?......Page 268
Reciprocity......Page 270
Modular Forms......Page 271
Review of Reciprocity Laws......Page 274
A Physical Analogy......Page 275
CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS......Page 277
The Three Pieces of the Proof......Page 278
Frey Curves......Page 279
The Modularity Conjecture......Page 280
Lowering the Level......Page 282
Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves......Page 284
Bring on the Reciprocity Laws......Page 285
What Wiles and Taylor–Wiles Did......Page 287
Generalized Fermat Equations......Page 289
What Henri Darmon and Loïc Merel Did......Page 290
Prospects for Solving the Generalized Fermat Equations......Page 291
Topics Covered......Page 292
Back to Solving Equations......Page 293
Digression: Why Do Math?......Page 295
The Congruent Number Problem......Page 296
Peering Past the Frontier......Page 298
Bibliography......Page 300
E......Page 304
L......Page 305
R......Page 306
Z......Page 307
