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, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.

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