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Fearless Symmetry Exposing the Hidden Patterns of Numbers

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ISBN-10: 0691138710

ISBN-13: 9780691138718

Edition: 2008 (Revised)

Authors: Avner Ash, Robert Gross

List price: $27.95
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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 Symmetryis 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 Eacute;variste 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…    
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Book details

List price: $27.95
Copyright year: 2008
Publisher: Princeton University Press
Publication date: 8/24/2008
Binding: Paperback
Pages: 312
Size: 6.00" wide x 9.50" long x 0.75" tall
Weight: 1.188
Language: English

Preface to the Paperback Edition
Greek Alphabet
Algebraic Preliminaries
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
The Group of Rotations of a Sphere
The General Concept of "Group"
In Praise of Mathematical Idealization
Digression: Lie Groups
The abc of Permutations
Permutations in General
Digression: Mathematics and Society
Modular Arithmetic
Cyclical Time
Arithmetic Modulo a Prime
Modular Arithmetic and Group Theory
Modular Arithmetic and Solutions of Equations
Complex Numbers
Overture to Complex Numbers
Complex Arithmetic
Complex Numbers and Solving Equations
Digression: Theorem
Algebraic Closure
Equations and Varieties
The Logic of Equality
The History of Equations
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
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
Galois Theory and Representations
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?
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
Matrices and Matrix Representations
Matrices and Their Entries
Matrix Multiplication
Linear Algebra
Digression: Graeco-Latin Squares
Groups of Matrices
Square Matrices
Matrix Inverses
The General Linear Group of Invertible Matrices
The Group GL(2, Z)
Solving Matrix Equations
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
The Galois Group of a Polynomial
The Field Generated by a Z-Polynomial
Digression: The Inverse Galois Problem
Two More Things
The Restriction Morphism
The Big Picture and the Little Pictures
Basic Facts about the Restriction Morphism
The Greeks Had a Name for It
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
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
The Official Definition of the Bad Primes for a Galois Representation
The Official Definition of "Unramified" and Frob[subscript p]
Reciprocity Laws
Reciprocity Laws
The List of Traces of Frobenius
Black Boxes
Weak and Strong Reciprocity Laws
Digression: Conjecture
Kinds of Black Boxes
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
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
A Machine for Making Galois Representations
Vector Spaces and Linear Actions of Groups
Etale Cohomology
Conjectures about Etale Cohomology
A Last Look at Reciprocity
What Is Mathematics?
Modular Forms
Review of Reciprocity Laws
A Physical Analogy
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
Topics Covered
Back to Solving Equations
Digression: Why Do Math?
The Congruent Number Problem
Peering Past the Frontier