Cover -- Contents -- Foreword -- Preface to the Paperback Edition -- Preface -- Acknowledgments -- Greek Alphabet -- PART ONE. ALGEBRAIC PRELIMINARIES -- CHAPTER 1. REPRESENTATIONS -- The Bare Notion of Representation -- An Example: Counting -- Digression: Definitions -- Counting (Continued) -- Counting Viewed as a Representation -- The Definition of a Representation -- Counting and Inequalities as Representations -- Summary -- CHAPTER 2. GROUPS -- The Group of Rotations of a Sphere -- The General Concept of "Group" -- In Praise of Mathematical Idealization -- Digression: Lie Groups -- CHAPTER 3. PERMUTATIONS -- The abc of Permutations -- Permutations in General -- Cycles -- Digression: Mathematics and Society -- CHAPTER 4. MODULAR ARITHMETIC -- Cyclical Time -- Congruences -- Arithmetic Modulo a Prime -- Modular Arithmetic and Group Theory -- Modular Arithmetic and Solutions of Equations -- CHAPTER 5. COMPLEX NUMBERS -- Overture to Complex Numbers -- Complex Arithmetic -- Complex Numbers and Solving Equations -- Digression: Theorem -- Algebraic Closure -- CHAPTER 6. EQUATIONS AND VARIETIES -- The Logic of Equality -- The History of Equations -- Z-Equations -- Varieties -- Systems of Equations -- Equivalent Descriptions of the Same Variety -- Finding Roots of Polynomials -- Are There General Methods for Finding Solutions to Systems of Polynomial Equations? -- Deeper Understanding Is Desirable -- CHAPTER 7. QUADRATIC RECIPROCITY -- The Simplest Polynomial Equations -- When is -1 a Square mod p? -- The Legendre Symbol -- Digression: Notation Guides Thinking -- Multiplicativity of the Legendre Symbol -- When Is 2 a Square mod p? -- When Is 3 a Square mod p? -- When Is 5 a Square mod p? ( Will This Go On Forever?) -- The Law of Quadratic Reciprocity -- Examples of Quadratic Reciprocity -- PART TWO. GALOIS THEORY AND REPRESENTATIONS.

CHAPTER 8. GALOIS THEORY -- Polynomials and Their Roots -- The Field of Algebraic Numbers Q[sup(alg)] -- The Absolute Galois Group of Q Defined -- A Conversation with s: A Playlet in Three Short Scenes -- Digression: Symmetry -- How Elements of G Behave -- Why Is G a Group? -- Summary -- CHAPTER 9. ELLIPTIC CURVES -- Elliptic Curves Are "Group Varieties" -- An Example -- The Group Law on an Elliptic Curve -- A Much-Needed Example -- Digression: What Is So Great about Elliptic Curves? -- The Congruent Number Problem -- Torsion and the Galois Group -- CHAPTER 10. MATRICES -- Matrices and Matrix Representations -- Matrices and Their Entries -- Matrix Multiplication -- Linear Algebra -- Digression: Graeco-Latin Squares -- CHAPTER 11. GROUPS OF MATRICES -- Square Matrices -- Matrix Inverses -- The General Linear Group of Invertible Matrices -- The Group GL(2, Z) -- Solving Matrix Equations -- CHAPTER 12. GROUP REPRESENTATIONS -- Morphisms of Groups -- A[sub(4)], Symmetries of a Tetrahedron -- Representations of A[sub(4)] -- Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves -- CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL -- The Field Generated by a Z-Polynomial -- Examples -- Digression: The Inverse Galois Problem -- Two More Things -- CHAPTER 14. THE RESTRICTION MORPHISM -- The Big Picture and the Little Pictures -- Basic Facts about the Restriction Morphism -- Examples -- CHAPTER 15. THE GREEKS HAD A NAME FOR IT -- Traces -- Conjugacy Classes -- Examples of Characters -- How the Character of a Representation Determines the Representation -- Prelude to the Next Chapter -- Digression: A Fact about Rotations of the Sphere -- CHAPTER 16. FROBENIUS -- Something for Nothing -- Good Prime, Bad Prime -- Algebraic Integers, Discriminants, and Norms -- A Working Definition of Frob[sub(p)] -- An Example of Computing Frobenius Elements.

Frob[sub(p)] and Factoring Polynomials modulo p -- Appendix: The Official Definition of the Bad Primes for a Galois Representation -- Appendix: The Official Definition of "Unramified" and Frob[sub(p)] -- PART THREE. RECIPROCITY LAWS -- CHAPTER 17. RECIPROCITY LAWS -- The List of Traces of Frobenius -- Black Boxes -- Weak and Strong Reciprocity Laws -- Digression: Conjecture -- Kinds of Black Boxes -- CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS -- Roots of Unity -- How Frob[sub(q)] Acts on Roots of Unity -- One-Dimensional Galois Representations -- Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve -- How Frob[sub(q)] Acts on p-Torsion Points -- The 2-Torsion -- An Example -- Another Example -- Yet Another Example -- The Proof -- CHAPTER 19. QUADRATIC RECIPROCITY REVISITED -- Simultaneous Eigenelements -- The Z-Variety x[sup(2)] - W -- A Weak Reciprocity Law -- A Strong Reciprocity Law -- A Derivation of Quadratic Reciprocity -- CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS -- Vector Spaces and Linear Actions of Groups -- Linearization -- Ètale Cohomology -- Conjectures about Ètale Cohomology -- CHAPTER 21. A LAST LOOK AT RECIPROCITY -- What Is Mathematics? -- Reciprocity -- Modular Forms -- Review of Reciprocity Laws -- A Physical Analogy -- CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS -- The Three Pieces of the Proof -- Frey Curves -- The Modularity Conjecture -- Lowering the Level -- Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves -- Bring on the Reciprocity Laws -- What Wiles and Taylor-Wiles Did -- Generalized Fermat Equations -- What Henri Darmon and Loïc Merel Did -- Prospects for Solving the Generalized Fermat Equations -- CHAPTER 23. RETROSPECT -- Topics Covered -- Back to Solving Equations -- Digression: Why Do Math?.

The Congruent Number Problem -- Peering Past the Frontier -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.