## Fearless Symmetry: Exposing the Hidden Patterns of NumbersMathematicians 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, 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, |

### What people are saying - Write a review

#### LibraryThing Review

User Review - ftong - LibraryThingAs a high school student, I found that this book struck an admirable balance between explaining in detail the simple concepts and explaining in essence the sweep of grand theorems. I will likely reread parts of this book for gems to contemplate. Read full review

#### LibraryThing Review

User Review - fpagan - LibraryThingEnjoyable, technical treatment of a fragment of modern number theory involving group representations, Galois theory, elliptic curves, reciprocity laws, and other esoterica that went into the proof of ... Read full review

### Contents

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 |

265 | |

269 | |

### Other editions - View all

Fearless Symmetry: Exposing the Hidden Patterns of Numbers Avner Ash,Robert Gross Limited preview - 2008 |