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Fearless Symmetry Exposing the Hidden Patterns of Numbers 1st Edition by Avner Ash, Robert Gross ISBN 1400837774 9781400837779

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Fearless Symmetry Exposing the Hidden Patterns of Numbers New Edition Avner Ash Digital Instant Download

Author(s): Avner Ash; Robert Gross
ISBN(s): 9781400837779, 1400837774
Edition: New edition with a New preface by the authors
File Details: PDF, 1.22 MB
Year: 2008
Language: english
SKU: EB-51951620 Category: Tags: ,

Fearless Symmetry Exposing the Hidden Patterns of Numbers 1st Edition by Avner Ash, Robert Gross – Ebook PDF Instant Download/Delivery: 1400837774, 9781400837779
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ISBN 10: 1400837774 
ISBN 13: 9781400837779
Author: Avner Ash, Robert Gross

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.

Fearless Symmetry Exposing the Hidden Patterns of Numbers 1st Table of contents:

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

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