O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Fearless Symmetry

Book Description

Mathematicians 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.

Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.

The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

Table of Contents

  1. Cover (1/2)
  2. Cover (2/2)
  3. Contents (1/2)
  4. Contents (2/2)
  5. Foreword (1/2)
  6. Foreword (2/2)
  7. Preface to the Paperback Edition
  8. Preface (1/2)
  9. Preface (2/2)
  10. Acknowledgments
  11. Greek Alphabet
  12. PART ONE. ALGEBRAIC PRELIMINARIES
    1. CHAPTER 1. REPRESENTATIONS
      1. The Bare Notion of Representation
      2. An Example: Counting
      3. Digression: Definitions
      4. Counting (Continued)
      5. Counting Viewed as a Representation
      6. The Definition of a Representation
      7. Counting and Inequalities as Representations
      8. Summary
    2. CHAPTER 2. GROUPS
      1. The Group of Rotations of a Sphere
      2. The General Concept of "Group"
      3. In Praise of Mathematical Idealization
      4. Digression: Lie Groups
    3. CHAPTER 3. PERMUTATIONS
      1. The abc of Permutations
      2. Permutations in General
      3. Cycles
      4. Digression: Mathematics and Society
    4. CHAPTER 4. MODULAR ARITHMETIC
      1. Cyclical Time
      2. Congruences
      3. Arithmetic Modulo a Prime
      4. Modular Arithmetic and Group Theory
      5. Modular Arithmetic and Solutions of Equations
    5. CHAPTER 5. COMPLEX NUMBERS
      1. Overture to Complex Numbers
      2. Complex Arithmetic
      3. Complex Numbers and Solving Equations
      4. Digression: Theorem
      5. Algebraic Closure
    6. CHAPTER 6. EQUATIONS AND VARIETIES
      1. The Logic of Equality
      2. The History of Equations
      3. Z-Equations
      4. Varieties
      5. Systems of Equations
      6. Equivalent Descriptions of the Same Variety
      7. Finding Roots of Polynomials
      8. Are There General Methods for Finding Solutions to Systems of Polynomial Equations?
      9. Deeper Understanding Is Desirable
    7. CHAPTER 7. QUADRATIC RECIPROCITY
      1. The Simplest Polynomial Equations
      2. When is –1 a Square mod p?
      3. The Legendre Symbol
      4. Digression: Notation Guides Thinking
      5. Multiplicativity of the Legendre Symbol
      6. When Is 2 a Square mod p?
      7. When Is 3 a Square mod p?
      8. When Is 5 a Square mod p? ( Will This Go On Forever?)
      9. The Law of Quadratic Reciprocity
      10. Examples of Quadratic Reciprocity
  13. PART TWO. GALOIS THEORY AND REPRESENTATIONS
    1. CHAPTER 8. GALOIS THEORY
      1. Polynomials and Their Roots
      2. The Field of Algebraic Numbers Q[sup(alg)]
      3. The Absolute Galois Group of Q Defined
      4. A Conversation with s: A Playlet in Three Short Scenes
      5. Digression: Symmetry
      6. How Elements of G Behave
      7. Why Is G a Group?
      8. Summary
    2. CHAPTER 9. ELLIPTIC CURVES
      1. Elliptic Curves Are "Group Varieties"
      2. An Example
      3. The Group Law on an Elliptic Curve
      4. A Much-Needed Example
      5. Digression: What Is So Great about Elliptic Curves?
      6. The Congruent Number Problem
      7. Torsion and the Galois Group
    3. CHAPTER 10. MATRICES
      1. Matrices and Matrix Representations
      2. Matrices and Their Entries
      3. Matrix Multiplication
      4. Linear Algebra
      5. Digression: Graeco-Latin Squares
    4. CHAPTER 11. GROUPS OF MATRICES
      1. Square Matrices
      2. Matrix Inverses
      3. The General Linear Group of Invertible Matrices
      4. The Group GL(2, Z)
      5. Solving Matrix Equations
    5. CHAPTER 12. GROUP REPRESENTATIONS
      1. Morphisms of Groups
      2. A[sub(4)], Symmetries of a Tetrahedron
      3. Representations of A[sub(4)]
      4. Mod p Linear Representations of the Absolute Galois Group from Elliptic Curves
    6. CHAPTER 13. THE GALOIS GROUP OF A POLYNOMIAL
      1. The Field Generated by a Z-Polynomial
      2. Examples
      3. Digression: The Inverse Galois Problem
      4. Two More Things
    7. CHAPTER 14. THE RESTRICTION MORPHISM
      1. The Big Picture and the Little Pictures
      2. Basic Facts about the Restriction Morphism
      3. Examples
    8. CHAPTER 15. THE GREEKS HAD A NAME FOR IT
      1. Traces
      2. Conjugacy Classes
      3. Examples of Characters
      4. How the Character of a Representation Determines the Representation
      5. Prelude to the Next Chapter
      6. Digression: A Fact about Rotations of the Sphere
    9. CHAPTER 16. FROBENIUS
      1. Something for Nothing
      2. Good Prime, Bad Prime
      3. Algebraic Integers, Discriminants, and Norms
      4. A Working Definition of Frob[sub(p)]
      5. An Example of Computing Frobenius Elements
      6. Frob[sub(p)] and Factoring Polynomials modulo p
      7. Appendix: The Official Definition of the Bad Primes for a Galois Representation
      8. Appendix: The Official Definition of "Unramified" and Frob[sub(p)]
  14. PART THREE. RECIPROCITY LAWS
    1. CHAPTER 17. RECIPROCITY LAWS
      1. The List of Traces of Frobenius
      2. Black Boxes
      3. Weak and Strong Reciprocity Laws
      4. Digression: Conjecture
      5. Kinds of Black Boxes
    2. CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS
      1. Roots of Unity
      2. How Frob[sub(q)] Acts on Roots of Unity
      3. One-Dimensional Galois Representations
      4. Two-Dimensional Galois Representations Arising from the p-Torsion Points of an Elliptic Curve
      5. How Frob[sub(q)] Acts on p-Torsion Points
      6. The 2-Torsion
      7. An Example
      8. Another Example
      9. Yet Another Example
      10. The Proof
    3. CHAPTER 19. QUADRATIC RECIPROCITY REVISITED
      1. Simultaneous Eigenelements
      2. The Z-Variety x[sup(2)] – W
      3. A Weak Reciprocity Law
      4. A Strong Reciprocity Law
      5. A Derivation of Quadratic Reciprocity
    4. CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS
      1. Vector Spaces and Linear Actions of Groups
      2. Linearization
      3. Ètale Cohomology
      4. Conjectures about Ètale Cohomology
    5. CHAPTER 21. A LAST LOOK AT RECIPROCITY
      1. What Is Mathematics?
      2. Reciprocity
      3. Modular Forms
      4. Review of Reciprocity Laws
      5. A Physical Analogy
    6. CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS
      1. The Three Pieces of the Proof
      2. Frey Curves
      3. The Modularity Conjecture
      4. Lowering the Level
      5. Proof of FLT Given the Truth of the Modularity Conjecture for Certain Elliptic Curves
      6. Bring on the Reciprocity Laws
      7. What Wiles and Taylor–Wiles Did
      8. Generalized Fermat Equations
      9. What Henri Darmon and Loïc Merel Did
      10. Prospects for Solving the Generalized Fermat Equations
    7. CHAPTER 23. RETROSPECT
      1. Topics Covered
      2. Back to Solving Equations
      3. Digression: Why Do Math?
      4. The Congruent Number Problem
      5. Peering Past the Frontier
  15. Bibliography
  16. Index
    1. A
    2. B
    3. C
    4. D
    5. E
    6. F
    7. G
    8. I
    9. K
    10. L
    11. M
    12. N
    13. O
    14. P
    15. Q
    16. R
    17. S
    18. T
    19. U
    20. V
    21. W
    22. Z