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| Foreword | |
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| Preface to the Paperback Edition | |
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| Preface | |
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| Acknowledgments | |
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| Greek Alphabet | |
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| Algebraic Preliminaries | |
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| Representations | |
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| The Bare Notion of Representation | |
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| An Example: Counting | |
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| Digression: Definitions | |
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| Counting (Continued) | |
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| Counting Viewed as a Representation | |
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| The Definition of a Representation | |
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| Counting and Inequalities as Representations | |
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| Summary | |
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| Groups | |
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| The Group of Rotations of a Sphere | |
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| The General Concept of "Group" | |
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| In Praise of Mathematical Idealization | |
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| Digression: Lie Groups | |
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| Permutations | |
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| The abc of Permutations | |
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| Permutations in General | |
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| Cycles | |
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| Digression: Mathematics and Society | |
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| Modular Arithmetic | |
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| Cyclical Time | |
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| Congruences | |
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| Arithmetic Modulo a Prime | |
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| Modular Arithmetic and Group Theory | |
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| Modular Arithmetic and Solutions of Equations | |
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| Complex Numbers | |
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| Overture to Complex Numbers | |
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| Complex Arithmetic | |
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| Complex Numbers and Solving Equations | |
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| Digression: Theorem | |
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| Algebraic Closure | |
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| Equations and Varieties | |
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| The Logic of Equality | |
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| The History of Equations | |
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| Z-Equations | |
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| Varieties | |
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| Systems of Equations | |
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| Equivalent Descriptions of the Same Variety | |
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| Finding Roots of Polynomials | |
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| Are There General Methods for Finding Solutions to Systems of Polynomial Equations? | |
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| Deeper Understanding Is Desirable | |
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| Quadratic Reciprocity | |
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| The Simplest Polynomial Equations | |
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| When is -1 a Square mod p? | |
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| The Legendre Symbol | |
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| Digression: Notation Guides Thinking | |
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| Multiplicativity of the Legendre Symbol | |
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| When Is 2 a Square mod p? | |
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| When Is 3 a Square mod p? | |
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| When Is 5 a Square mod p? (Will This Go On Forever?) | |
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| The Law of Quadratic Reciprocity | |
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| Examples of Quadratic Reciprocity | |
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| Galois Theory and Representations | |
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| Galois Theory | |
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| Polynomials and Their Roots | |
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| The Field of Algebraic Numbers Q[superscript alg] | |
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| The Absolute Galois Group of Q Defined | |
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| A Conversation with s: A Playlet in Three Short Scenes | |
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| Digression: Symmetry | |
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| How Elements of G Behave | |
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| Why Is G a Group? | |
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| Summary | |
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| Elliptic Curves | |
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| Elliptic Curves Are "Group Varieties" | |
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| An Example | |
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| The Group Law on an Elliptic Curve | |
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| A Much-Needed Example | |
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| Digression: What Is So Great about Elliptic Curves? | |
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| The Congruent Number Problem | |
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| Torsion and the Galois Group | |
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| Matrices | |
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| Matrices and Matrix Representations | |
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| Matrices and Their Entries | |
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| Matrix Multiplication | |
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| Linear Algebra | |
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| Digression: Graeco-Latin Squares | |
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| Groups of Matrices | |
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| Square Matrices | |
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| Matrix Inverses | |
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| The General Linear Group of Invertible Matrices | |
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| The Group GL(2, Z) | |
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| Solving Matrix Equations | |
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| Group Representations | |
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| Morphisms of Groups | |
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| A[subscript 4], Symmetries of a Tetrahedron | |
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| Representations of A[subscript 4] | |
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| Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves | |
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| The Galois Group of a Polynomial | |
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| The Field Generated by a Z-Polynomial | |
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| Examples | |
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| Digression: The Inverse Galois Problem | |
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| Two More Things | |
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| The Restriction Morphism | |
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| The Big Picture and the Little Pictures | |
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| Basic Facts about the Restriction Morphism | |
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| Examples | |
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| The Greeks Had a Name for It | |
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| Traces | |
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| Conjugacy Classes | |
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| Examples of Characters | |
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| How the Character of a Representation Determines the Representation | |
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| Prelude to the Next Chapter | |
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| Digression: A Fact about Rotations of the Sphere | |
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| Frobenius | |
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| Something for Nothing | |
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| Good Prime, Bad Prime | |
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| Algebraic Integers, Discriminants, and Norms | |
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| A Working Definition of Frob[subscript p] | |
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| An Example of Computing Frobenius Elements | |
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| Frob[subscript p] and Factoring Polynomials modulo p | |
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| The Official Definition of the Bad Primes for a Galois Representation | |
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| The Official Definition of "Unramified" and Frob[subscript p] | |
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| Reciprocity Laws | |
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| Reciprocity Laws | |
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| The List of Traces of Frobenius | |
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| Black Boxes | |
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| Weak and Strong Reciprocity Laws | |
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| Digression: Conjecture | |
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| Kinds of Black Boxes | |
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| One- and Two-Dimensional Representations | |
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| Roots of Unity | |
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| How Frob[subscript q] Acts on Roots of Unity | |
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| One-Dimensional Galois Representations | |
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| Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve | |
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| How Frob[subscript q] Acts on p-Torsion Points | |
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| The 2-Torsion | |
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| An Example | |
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| Another Example | |
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| Yet Another Example | |
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| The Proof | |
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| Quadratic Reciprocity Revisited | |
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| Simultaneous Eigenelements | |
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| The Z-Variety x[superscript 2] - W | |
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| A Weak Reciprocity Law | |
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| A Strong Reciprocity Law | |
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| A Derivation of Quadratic Reciprocity | |
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| A Machine for Making Galois Representations | |
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| Vector Spaces and Linear Actions of Groups | |
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| Linearization | |
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| Etale Cohomology | |
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| Conjectures about Etale Cohomology | |
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| A Last Look at Reciprocity | |
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| What Is Mathematics? | |
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| Reciprocity | |
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| Modular Forms | |
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| Review of Reciprocity Laws | |
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| A Physical Analogy | |
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| Fermat's Last Theorem and Generalized Fermat Equations | |
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| The Three Pieces of the Proof | |
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| Frey Curves | |
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| The Modularity Conjecture | |
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| Lowering the Level | |
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| Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves | |
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| Bring on the Reciprocity Laws | |
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| What Wiles and Taylor-Wiles Did | |
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| Generalized Fermat Equations | |
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| What Henri Darmon and Loic Merel Did | |
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| Prospects for Solving the Generalized Fermat Equations | |
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| Retrospect | |
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| Topics Covered | |
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| Back to Solving Equations | |
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| Digression: Why Do Math? | |
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| The Congruent Number Problem | |
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| Peering Past the Frontier | |
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| Bibliography | |
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| Index | |