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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 | |