Contemporary Abstract Algebra

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Notations
Preface
0. Preliminaries
	Properties of Integers
	Modular Arithmetic
	Complex Numbers
	Mathematical Induction
	Equivalence Relations
	Functions (Mappings)
	Exercises
1. Introduction to Groups
	Symmetries of a Square
	The Dihedral Groups
	Bibliography of Niels Abel
2. Groups
	Definition and Examples of Groups
	Elementary Properties of Groups
	Historical Note
	Exercises
3. Finite Groups; Subgroups
	Terminology and Notation
	Subgroup Tests
	Examples of Subgroups
	Exercises
4. Cyclic Groups
	Properties of Cyclic Groups
	Classification of Subgroups of Cyclic Groups
	Bibliography of James Joseph Sylvester
5. Permutation Groups
	Definitions and Notation
	Cycle Notation
	Properties of Permutations
	A Check-Digit Scheme Based on D5
	Bibliography of Augustin Cauchy
	Bibliography of Alan Turing
6. Isomorphisms
	Motivation
	Definition and Examples
	Properties of Isomorphisms
	Automorphisms
	Cayley’s Theorem
	Exercises
	Bibliography of Arthur Cayley
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
	Bibliography of Joseph Lagrange
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
	Bibliography of Leonard Adleman
9. Normal Subgroups and Factor Groups
	Normal Subgroups
	Factor Groups
	Applications of Factor Groups
	Internal Direct Products
	Exercises
	Bibliography of Évariste Galois
10. Group Homomorphisms
	Definition and Examples
	Properties of Homomorphisms
	The First Isomorphism Theorem
	Exercises
	Bibliography of Camille Jordan
11. Fundamental Theorem of Finite Abelian Groups
	The Fundamental Theorem
	The Isomorphism Classes of Abelian Groups
	Proof of the Fundamental Theorem
	Exercises
12. Introduction to Rings
	Motivation and Definition
	Examples of Rings
	Properties of Rings
	Subrings
	Exercises
	Bibliography of I. N. Herstein
13. Integral Domains
	Definition and Examples
	Fields
	Characteristic of a Ring
	Exercises
14. Ideals and Factor Rings
	Ideals
	Factor Rings
	Prime Ideals and Maximal Ideals
	Exercises
	Bibliography of Richard Dedekind
	Bibliography of Emmy Noether
15. Ring Homomorphisms
	Definition and Examples
	Properties of Ring Homomorphisms
	The Field of Quotients
	Exercises
16. Polynomial Rings
	Notation and Terminology
	The Division Algorithm and Consequences
	Exercises
17. Factorization of Polynomials
	Reducibility Tests
	Irreducibility Tests
	Unique Factorization in Z[x]
	Weird Dice: An Application of Unique Factorization
	Exercises
	Bibliography of Serge Lang
18. Divisibility in Integral Domains
	Irreducibles, Primes
	Historical Discussion of Fermat’s Last Theorem
	Unique Factorization Domains
	Euclidean Domains
	Exercises
	Bibliography of Sophie Germain
	Bibliography of Andrew Wiles
	Bibliography of Pierre de Fermat
19. Extension Fields
	The Fundamental Theorem of Field Theory
	Splitting Fields
	Zeros of an Irreducible Polynomial
	Exercises
	Bibliography of Leopold Kronecker
20. Algebraic Extensions
	Characterization of Extensions
	Finite Extensions
	Properties of Algebraic Extensions
	Exercises
	Bibliography of Ernst Steinitz
21. Finite Fields
	Classification of Finite Fields
	Structure of Finite Fields
	Subfields of a Finite Field
	Exercises
	Bibliography of L. E. Dickson
	Bibliography of E. H. Moore
22. Geometric Constructions
	Historical Discussion of Geometric Constructions
	Constructible Numbers
	Angle-Trisectors and Circle-Squarers
	Exercises
23. Sylow Theorems
	Conjugacy Classes
	The Class Equation
	The Sylow Theorems
	Applications of Sylow Theorems
	Exercises
	Bibliography of Ludwig Sylow
24. Finite Simple Groups
	Historical Background
	Nonsimplicity Tests
	The Simplicity of A5
	The Fields Medal
	The Cole Prize
	Exercises
	Bibliography of Michael Aschbacher
	Bibliography of Daniel Gorenstein
	Bibliography of John Thompson
25. Generators and Relations
	Motivation
	Definitions and Notation
	Free Group
	Generators and Relations
	Classification of Groups of Order Up to 15
	Characterization of Dihedral Groups
	Exercises
	Bibliography of Marshall Hall, Jr.
26. Symmetry Groups
	Isometries
	Classification of Finite Plane Symmetry Groups
	Classification of Finite Groups of Rotations in R3
	Exercises
27. Symmetry and Counting
	Motivation
	Burnside’s Theorem
	Applications
	Group Action
	Exercises
	Bibliography of William Burnside
28. Cayley Digraphs of Groups
	Motivation
	The Cayley Digraph of a Group
	Hamiltonian Circuits and Paths
	Some Applications
	Exercises
	Bibliography of William Rowan Hamilton
	Bibliography of Paul Erdos
29. Introduction to Algebraic Coding Theory
	Motivation
	Linear Codes
	Parity-Check Matrix Decoding
	Coset Decoding
	Historical Note
	Exercises
	Bibliography of Richard W. Hamming
	Bibliography of Jessie MacWilliams
	Bibliography of Vera Pless
30. An Introduction to Galois Theory
	Fundamental Theorem of Galois Theory
	Solvability of Polynomials by Radicals
	Insolvability of a Quintic
	Exercises
31. Cyclotomic Extensions
	Motivation
	Cyclotomic Polynomials
	The Constructible Regular n-gons
	Exercises
	Bibliography of Carl Friedrich Gauss
	Bibliography of Manjul Bhargava
Selected Answers
Index