# Fearless Symmetry: Exposing the Hidden Patterns of Numbers

Princeton University Press, 2006 - 272 strani

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 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.

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### Vsebina

 Algebraic Preliminaries 1 REPRESENTATIONS 3 Counting 5 Definitions 6 Counting Continued 7 Counting Viewed as a Representation 8 The Definition of a Representation 9 Counting and Inequalities as Representations 10
 Representations of A4 142 Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves 146 THE GALOIS GROUP OF A POLYNOMIAL 149 Examples 151 The Inverse Galois Problem 154 Two More Things 155 THE RESTRICTION MORPHISM 157 Basic Facts about the Restriction Morphism 159

 Summary 11 GROUPS 13 The Group of Rotations of a Sphere 14 The General Concept of Group 17 In Praise of Mathematical Idealization 18 Lie Groups 19 PERMUTATIONS 21 Permutations in General 25 Cycles 26 Mathematics and Society 29 MODULAR ARITHMETIC 31 Congruences 33 Arithmetic Modulo a Prime 36 Modular Arithmetic and Group Theory 39 Modular Arithmetic and Solutions of Equations 41 COMPLEX NUMBERS 42 Complex Arithmetic 44 Complex Numbers and Solving Equations 47 EQUATIONS AND VARIETIES 49 The Logic of Equality 50 ZEquations 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 QUADRATIC RECIPROCITY 67 When Is 1 a Square mod p? 69 The Legendre Symbol 71 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 Galois Theory and Representations 85 GALOIS THEORY 87 Polynomials and Their Roots 88 The Field of Algebraic Numbers Qalg 89 The Absolute Galois Group of Q Defined 92 A Playlet in Three Short Scenes 93 Symmetry 96 Why Is G a Group? 101 ELLIPTIC CURVES 103 An Example 104 The Group Law on an Elliptic Curve 107 A MuchNeeded Example 108 What Is So Great about Elliptic Curves? 109 The Congruent Number Problem 110 Torsion and the Galois Group 111 MATRICES 114 Matrices and Their Entries 115 Matrix Multiplication 117 Linear Algebra 120 GraecoLatin Squares 122 GROUPS OF MATRICES 124 Matrix Inverses 126 The General Linear Group of Invertible Matrices 129 The Group GL2Z 130 Solving Matrix Equations 132 GROUP REPRESENTATIONS 135 A4 Symmetries of a Tetrahedron 139
 Examples 161 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 FROBENIUS 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 The Official Definition of the Bad Primes for a Galois Representation 188 The Official Definition of Unramified and Frobp 189 Reciprocity Laws 191 RECIPROCITY LAWS 193 Black Boxes 195 Weak and Strong Reciprocity Laws 196 Conjecture 197 Kinds of Black Boxes 199 ONE AND TWODIMENSIONAL REPRESENTATIONS 200 How Frobg Acts on Roots of Unity 202 OneDimensional Galois Representations 204 TwoDimensional Galois Representations Arising from the pTorsion Points of an Elliptic Curve 205 How Frobq Acts on pTorsion Points 207 The 2Torsion 209 Another Example 211 Yet Another Example 212 The Proof 214 QUADRATIC RECIPROCITY REVISITED 216 Simultaneous Eigenelements 217 The ZVariety x2 W 218 A Weak Reciprocity Law 220 A Strong Reciprocity Law 221 A Derivation of Quadratic Reciprocity 222 A MACHINE FOR MAKING GALOIS REPRESENTATIONS 225 Linearization 228 Étale Cohomology 229 Conjectures about Étale Cohomology 231 A LAST LOOK AT RECIPROCITY 233 Reciprocity 235 Modular Forms 236 Review of Reciprocity Laws 239 A Physical Analogy 240 FERMATS 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 TaylorWiles Did 252 Generalized Fermat Equations 254 What Henri Darmon and Loïc Merel Did 255 Prospects for Solving the Generalized Fermat Equations 256 RETROSPECT 257 Back to Solving Equations 258 Why Do Math? 260 The Congruent Number Problem 261 Peering Past the Frontier 263 Bibliography 265 Index 269 Avtorske pravice

### Priljubljeni odlomki

Stran vii - Tyger! Tyger! burning bright In the forests of the night, What immortal hand or eye Could frame thy fearful symmetry? In what distant deeps or skies Burnt the fire of thine eyes? On what wings dare he aspire? What the hand dare seize the fire? And what shoulder, & what art, Could twist the sinews of thy heart?
Stran xvi - If we had some exact language (like the one called Adamitic by some) or at least a kind of truly philosophic writing, in which the ideas were reduced to a kind of alphabet of human thought...
Stran vii - In seed time learn, in harvest teach, in winter enjoy.

### O avtorju (2006)

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.