# Fearless Symmetry

, by Ash, Avner**Note:**Supplemental materials are not guaranteed with Rental or Used book purchases.

- ISBN: 9780691138718 | 0691138710
- Cover: Paperback
- Copyright: 8/4/2008

Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss symmetric patterns of numbers and the ingenious techniques mathematicians use to uncover them. The book starts with basic properties of integers and permutations and ends with current research in number theory. Along the way, it takes delightful historical and philosophical digressions on French mathematician Evariste Galois and well-known problems such as Fermat's Last Theorem, the Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Required reading for all math buffs, Fearless Symmetry will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life. Book jacket.

Avner Ash is professor of mathematics at Boston College and the coauthor of "Smooth Compactification of Locally Symmetric Varieties". Robert Gross is associate professor of mathematics at Boston College.

Foreword | p. xv |

Preface to the Paperback Edition | p. xxi |

Preface | p. xxv |

Acknowledgments | p. xxxi |

Greek Alphabet | p. xxxiii |

Algebraic Preliminaries | |

Representations | p. 3 |

The Bare Notion of Representation | p. 3 |

An Example: Counting | p. 5 |

Digression: Definitions | p. 6 |

Counting (Continued) | p. 7 |

Counting Viewed as a Representation | p. 8 |

The Definition of a Representation | p. 9 |

Counting and Inequalities as Representations | p. 10 |

Summary | p. 11 |

Groups | p. 13 |

The Group of Rotations of a Sphere | p. 14 |

The General Concept of "Group" | p. 17 |

In Praise of Mathematical Idealization | p. 18 |

Digression: Lie Groups | p. 19 |

Permutations | p. 21 |

The abc of Permutations | p. 21 |

Permutations in General | p. 25 |

Cycles | p. 26 |

Digression: Mathematics and Society | p. 29 |

Modular Arithmetic | p. 31 |

Cyclical Time | p. 31 |

Congruences | p. 33 |

Arithmetic Modulo a Prime | p. 36 |

Modular Arithmetic and Group Theory | p. 39 |

Modular Arithmetic and Solutions of Equations | p. 41 |

Complex Numbers | p. 42 |

Overture to Complex Numbers | p. 42 |

Complex Arithmetic | p. 44 |

Complex Numbers and Solving Equations | p. 47 |

Digression: Theorem | p. 47 |

Algebraic Closure | p. 47 |

Equations and Varieties | p. 49 |

The Logic of Equality | p. 50 |

The History of Equations | p. 50 |

Z-Equations | p. 52 |

Varieties | p. 54 |

Systems of Equations | p. 56 |

Equivalent Descriptions of the Same Variety | p. 58 |

Finding Roots of Polynomials | p. 61 |

Are There General Methods for Finding Solutions to Systems of Polynomial Equations? | p. 62 |

Deeper Understanding Is Desirable | p. 65 |

Quadratic Reciprocity | p. 67 |

The Simplest Polynomial Equations | p. 67 |

When is -1 a Square mod p? | p. 69 |

The Legendre Symbol | p. 71 |

Digression: Notation Guides Thinking | p. 72 |

Multiplicativity of the Legendre Symbol | p. 73 |

When Is 2 a Square mod p? | p. 74 |

When Is 3 a Square mod p? | p. 75 |

When Is 5 a Square mod p? (Will This Go On Forever?) | p. 76 |

The Law of Quadratic Reciprocity | p. 78 |

Examples of Quadratic Reciprocity | p. 80 |

Galois Theory and Representations | |

Galois Theory | p. 87 |

Polynomials and Their Roots | p. 88 |

The Field of Algebraic Numbers Q[superscript alg] | p. 89 |

The Absolute Galois Group of Q Defined | p. 92 |

A Conversation with s: A Playlet in Three Short Scenes | p. 93 |

Digression: Symmetry | p. 96 |

How Elements of G Behave | p. 96 |

Why Is G a Group? | p. 101 |

Summary | p. 101 |

Elliptic Curves | p. 103 |

Elliptic Curves Are "Group Varieties" | p. 103 |

An Example | p. 104 |

The Group Law on an Elliptic Curve | p. 107 |

A Much-Needed Example | p. 108 |

Digression: What Is So Great about Elliptic Curves? | p. 109 |

The Congruent Number Problem | p. 110 |

Torsion and the Galois Group | p. 111 |

Matrices | p. 114 |

Matrices and Matrix Representations | p. 114 |

Matrices and Their Entries | p. 115 |

Matrix Multiplication | p. 117 |

Linear Algebra | p. 120 |

Digression: Graeco-Latin Squares | p. 122 |

Groups of Matrices | p. 124 |

Square Matrices | p. 124 |

Matrix Inverses | p. 126 |

The General Linear Group of Invertible Matrices | p. 129 |

The Group GL(2, Z) | p. 130 |

Solving Matrix Equations | p. 132 |

Group Representations | p. 135 |

Morphisms of Groups | p. 135 |

A[subscript 4], Symmetries of a Tetrahedron | p. 139 |

Representations of A[subscript 4] | p. 142 |

Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves | p. 146 |

The Galois Group of a Polynomial | p. 149 |

The Field Generated by a Z-Polynomial | p. 149 |

Examples | p. 151 |

Digression: The Inverse Galois Problem | p. 154 |

Two More Things | p. 155 |

The Restriction Morphism | p. 157 |

The Big Picture and the Little Pictures | p. 157 |

Basic Facts about the Restriction Morphism | p. 159 |

Examples | p. 161 |

The Greeks Had a Name for It | p. 162 |

Traces | p. 163 |

Conjugacy Classes | p. 165 |

Examples of Characters | p. 166 |

How the Character of a Representation Determines the Representation | p. 171 |

Prelude to the Next Chapter | p. 175 |

Digression: A Fact about Rotations of the Sphere | p. 175 |

Frobenius | p. 177 |

Something for Nothing | p. 177 |

Good Prime, Bad Prime | p. 179 |

Algebraic Integers, Discriminants, and Norms | p. 180 |

A Working Definition of Frob[subscript p] | p. 184 |

An Example of Computing Frobenius Elements | p. 185 |

Frob[subscript p] and Factoring Polynomials modulo p | p. 186 |

The Official Definition of the Bad Primes for a Galois Representation | p. 188 |

The Official Definition of "Unramified" and Frob[subscript p] | p. 189 |

Reciprocity Laws | |

Reciprocity Laws | p. 193 |

The List of Traces of Frobenius | p. 193 |

Black Boxes | p. 195 |

Weak and Strong Reciprocity Laws | p. 196 |

Digression: Conjecture | p. 197 |

Kinds of Black Boxes | p. 199 |

One- and Two-Dimensional Representations | p. 200 |

Roots of Unity | p. 200 |

How Frob[subscript q] Acts on Roots of Unity | p. 202 |

One-Dimensional Galois Representations | p. 204 |

Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve | p. 205 |

How Frob[subscript q] Acts on p-Torsion Points | p. 207 |

The 2-Torsion | p. 209 |

An Example | p. 209 |

Another Example | p. 211 |

Yet Another Example | p. 212 |

The Proof | p. 214 |

Quadratic Reciprocity Revisited | p. 216 |

Simultaneous Eigenelements | p. 217 |

The Z-Variety x[superscript 2] - W | p. 218 |

A Weak Reciprocity Law | p. 220 |

A Strong Reciprocity Law | p. 221 |

A Derivation of Quadratic Reciprocity | p. 222 |

A Machine for Making Galois Representations | p. 225 |

Vector Spaces and Linear Actions of Groups | p. 225 |

Linearization | p. 228 |

Etale Cohomology | p. 229 |

Conjectures about Etale Cohomology | p. 231 |

A Last Look at Reciprocity | p. 233 |

What Is Mathematics? | p. 233 |

Reciprocity | p. 235 |

Modular Forms | p. 236 |

Review of Reciprocity Laws | p. 239 |

A Physical Analogy | p. 240 |

Fermat's Last Theorem and Generalized Fermat Equations | p. 242 |

The Three Pieces of the Proof | p. 243 |

Frey Curves | p. 244 |

The Modularity Conjecture | p. 245 |

Lowering the Level | p. 247 |

Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves | p. 249 |

Bring on the Reciprocity Laws | p. 250 |

What Wiles and Taylor-Wiles Did | p. 252 |

Generalized Fermat Equations | p. 254 |

What Henri Darmon and Loic Merel Did | p. 255 |

Prospects for Solving the Generalized Fermat Equations | p. 256 |

Retrospect | p. 257 |

Topics Covered | p. 257 |

Back to Solving Equations | p. 258 |

Digression: Why Do Math? | p. 260 |

The Congruent Number Problem | p. 261 |

Peering Past the Frontier | p. 263 |

Bibliography | p. 265 |

Index | p. 269 |

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