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
Chapter
The Group of Rotations of a Sphere
The General Concept of "Group"
In Praise of Mathematical Idealization
Digression: Mathematics and Society
CHAPTER 4. MODULAR ARITHMETIC
Arithmetic Modulo a Prime
Modular Arithmetic and Group Theory
Modular Arithmetic and Solutions of Equations
CHAPTER 5. COMPLEX NUMBERS
Overture to Complex Numbers
Complex Numbers and Solving Equations
CHAPTER 6. EQUATIONS AND VARIETIES
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?
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
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
CHAPTER 9. ELLIPTIC CURVES
Elliptic Curves Are "Group Varieties"
The Group Law on an Elliptic Curve
Digression: What Is So Great about Elliptic Curves?
The Congruent Number Problem
Torsion and the Galois Group
Matrices and Matrix Representations
Matrices and Their Entries
Digression: Graeco-Latin Squares
CHAPTER 11. GROUPS OF MATRICES
The General Linear Group of Invertible Matrices
CHAPTER 12. GROUP REPRESENTATIONS
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
Digression: The Inverse Galois Problem
CHAPTER 14. THE RESTRICTION MORPHISM
The Big Picture and the Little Pictures
Basic Facts about the Restriction Morphism
CHAPTER 15. THE GREEKS HAD A NAME FOR IT
How the Character of a Representation Determines the Representation
Prelude to the Next Chapter
Digression: A Fact about Rotations of the Sphere
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
Weak and Strong Reciprocity Laws
CHAPTER 18. ONE- AND TWO-DIMENSIONAL REPRESENTATIONS
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
CHAPTER 19. QUADRATIC RECIPROCITY REVISITED
Simultaneous Eigenelements
The Z-Variety x[sup(2)] – W
A Derivation of Quadratic Reciprocity
CHAPTER 20. A MACHINE FOR MAKING GALOIS REPRESENTATIONS
Vector Spaces and Linear Actions of Groups
Conjectures about Ètale Cohomology
CHAPTER 21. A LAST LOOK AT RECIPROCITY
Review of Reciprocity Laws
CHAPTER 22. FERMAT'S LAST THEOREM AND GENERALIZED FERMAT EQUATIONS
The Three Pieces of the Proof
The Modularity Conjecture
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
Back to Solving Equations
The Congruent Number Problem
Peering Past the Frontier
Fearless Symmetry
：Exposing the Hidden Patterns of Numbers
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