Contemporary Abstract Algebra

Contents
Preface
Part 1: Integers and Equivalence Relations
Ch 0: Preliminaries
Properties of Integers
Modular Arithmetic
Complex Numbers
Mathematical Induction
Equivalence Relations
Functions (Mappings)
Exercises
Computer Exercises
Suggested Readings
Part 2: Groups
Ch 1: Introduction to Groups
Symmetries of a Square
The Dihedral Groups
Exercises
Suggested Reading
Suggested Website
Ch 2: Groups
Definition and Examples of Groups
Elementary Properties of Groups
Historical Note
Exercises
Computer Exercises
References
Suggested Readings
Ch 3: Finite Groups; Subgroups
Terminology and Notation
Subgroup Tests
Examples of Subgroups
Exercises
Computer Exercises
Suggested Readings
Ch 4: Cyclic Groups
Properties of Cyclic Groups
Classification of Subgroups of Cyclic Groups
Exercises
Computer Exercises
Suggested Reading
Ch 5: Permutation Groups
Definition and Notation
Cycle Notation
Properties of Permutations
A Check-Digit Scheme Based on D5
Exercises
Computer Exercises
References
Suggested Readings
Ch 6: Isomorphisms
Motivation
Definition and Examples
Cayley's Theorem
Properties of Isomorphisms
Automorphisms
Exercises
Reference
Computer Exercises
Ch 7: Cosets and Lagrange's Theorem
Properties of Cosets
Lagrange's Theorem and Consequences
An Application of Cosets to Permutation Groups
The Rotation Group of a Cube and a Soccer Ball
An Application of Cosets to the Rubik's Cube
Exercises
Computer Exercises
Ch 8: External Direct Products
Definition and Examples
Properties of External Direct Products
The Group of Units Modulo n as an External Direct Product
Applications
Exercises
Computer Exercises
References
Suggested Readings
Ch 9: Normal Subgroups and Factor Groups
Normal Subgroups
Factor Groups
Applications of Factor Groups
Internal Direct Products
Exercises
Suggested Readings
Ch 10: Group Homomorphisms
Definition and Examples
Properties of Homomorphisms
The First Isomorphism Theorem
Exercises
Computer Exercise
Suggested Readings
Ch 11: Fundamental Theorem of Finite Abelian Groups
The Fundamental Theorem
The Isomorphism Classes of Abelian Groups
Proof of the Fundamental Theorem
Exercises
Computer Exercises
Reference
Suggested Readings
Suggested Website
Part 3: Rings
Ch 12: Introduction to Rings
Motivation and Definition
Examples of Rings
Properties of Rings
Subrings
Exercises
Computer Exercises
Suggested Reading
Ch 13: Integral Domains
Definition and Examples
Fields
Characteristic of a Ring
Exercises
Computer Exercises
Suggested Readings
Ch 14: Ideals and Factor Rings
Ideals
Factor Rings
Prime Ideals and Maximal Ideals
Exercises
Computer Exercises
Suggested Reading
Ch 15: Ring Homomorphisms
Definition and Examples
Properties of Ring Homomorphisms
The Field of Quotients
Exercises
Suggested Readings
Ch 16: Polynomial Rings
Notation and Terminology
The Division Algorithm and Consequences
Exercises
Suggested Reading
Ch 17: Factorization of Polynomials
Reducibility Tests
Irreducibility Tests
Unique Factorization in Z[x]
Weird Dice: An Application of Unique Factorization
Exercises
Computer Exercises
Reference
Suggested Readings
Ch 18: Divisibility in Integral Domains
Irreducibles, Primes
Historical Discussion of Fermat's Last Theorem
Unique Factorization Domains
Euclidean Domains
Exercises
Computer Exercise
References
Suggested Readings
Suggested Video
Suggested Websites
Part 4: Fields
Ch 19: Vector Spaces
Definition and Examples
Subspaces
Linear Independence
Exercises
Ch 20: Extension Fields
The Fundamental Theorem of Field Theory
Splitting Fields
Zeros of an Irreducible Polynomial
Exercises
Ch 21: Algebraic Extensions
Characterization of Extensions
Finite Extensions
Properties of Algebraic Extensions
Exercises
Suggested Readings
Ch 22: Finite Fields
Classification of Finite Fields
Structure of Finite Fields
Subfields of a Finite Field
Exercises
Computer Exercises
Suggested Reading
Ch 23: Geometric Constructions
Historical Discussion of Geometric Constructions
Constructible Numbers
Angle-Trisectors and Circle-Squarers
Exercises
References
Suggested Website
Part 5: Special Topics
Ch 24: Sylow Theorems
Conjugacy Classes
The Class Equation
The Sylow Theorems
Applications of Sylow Theorems
Exercises
Computer Exercises
Suggested Reading
Ch 25: Finite Simple Groups
Historical Background
Nonsimplicity Tests
The Simplicity of A5
The Fields Medal
The Cole Prize
Exercises
Computer Exercises
References
Suggested Readings
Ch 26: Generators and Relations
Motivation
Definitions and Notation
Free Group
Generators and Relations
Classification of Groups of Order Up to 15
Characterization of Dihedral Groups
Realizing the Dihedral Groups with Mirrors
Exercises
References
Suggested Readings
Ch 27: Symmetry Groups
Isometries
Classification of Finite Plane Symmetry Groups
Classification of Finite Groups of Rotations in R3
Exercises
References
Suggested Readings
Suggested Website
Ch 28: Frieze Groups and Crystallographic Groups
The Frieze Groups
The Crystallographic Groups
Identification of Plane Periodic Patterns
Exercises
References
Suggested Readings
Suggested Websites
Ch 29: Symmetry and Counting
Motivation
Burnside's Theorem
Applications
Group Action
Exercises
Suggested Readings
Ch 30: Cayley Digraphs of Groups
Motivation
The Cayley Digraph of a Group
Hamiltonian Circuits and Paths
Some Applications
Exercises
References
Suggested Readings
Suggested Website
Suggested DVD
Suggested Software
Ch 31: Introduction to Algebraic Coding Theory
Motivation
Linear Codes
Parity-Check Matrix Decoding
Coset Decoding
Exercises
Reference
Suggested Readings
Ch 32: An Introduction to Galois Theory
Fundamental Theorem of Galois Theory
Solvability of Polynomials by Radicals
Insolvability of a Quintic
Exercises
Reference
Suggested Readings
Suggested Website
Ch 33: Cyclotomic Extensions
Motivation
Cyclotomic Polynomials
The Constructible Regular n-gons
Exercises
Computer Exercises
Selected Answers
Index of Mathematicians
Index of Terms