Fearless symmetry: exposing the hidden patterns of numbers

Foreword

Preface to the Paperback Edition

Preface

Acknowledgments

Greek Alphabet

Part 1. 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 2. Galois Theory and Representations

Chapter 8. Galois Theory

Polynomials and Their Roots

The Field of Algebraic Numbers Q[superscript 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[subscript 4], Symmetries of a Tetrahedron

Representations of A[subscript 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[subscript p]

An Example of Computing Frobenius Elements

Frob[subscript 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[subscript p]

Part 3. 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[subscript 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[subscript 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[superscript 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

Etale Cohomology

Conjectures about Etale 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 Loic 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