Table of Contents

Note: Each chapter concludes with Exercises.

• I. Integers and Equivalence Relations
• Preliminaries
  Properties of Integers
  Modular Arithmetic
  Mathematical Induction
  Equivalence Relations
  Functions (Mappings)
  Computer Exercises

  II. Groups

  1. Introduction to Groups

  Symmetries of a Square

  The Dihedral Groups

  Biography of Neils Abel

  2. Groups

  Definition and Examples of Groups

  Elementary Properties of Groups

  Historical Note

  Computer Exercises

  3. Finite Groups; Subgroups

  Terminology and Notation

  Subgroup Tests

  Examples of Subgroups

  Computer Exercises

  4. Cyclic Groups

  Properties of Cyclic Groups

  Classification of Subgroups of Cyclic Groups

  Computer Exercises

  Biography of J. J. Sylvester

  Supplementary Exercises for Chapters 1–4

  5. Permutation Groups

  Definition and Notation

  Cycle Notation

  Properties of Permutations

  A Check-Digit Scheme Based on D5

  Computer Exercises

  Biography of Augustin Cauchy

  6. Isomorphisms

  Motivation

  Definition and Examples

  Cayley's Theorem

  Properties of Isomorphisms

  Automorphisms

  Biography 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

  Computer Exercises

  Biography of Joseph Lagrange

  8. External Direct Products

  Definition and Examples

  Properties of External Direct Products

  The Group of Units Modulo n as an External DirectProduct

  Applications

  Computer Exercises

  Biography of Leonard Adleman

  Supplementary Exercises for Chapters 5–8

  9. Normal Subgroups and Factor Groups

  Normal Subgroups

  Factor Groups

  Applications of Factor Groups

  Internal Direct Products

  Biography of Évariste Galois

  10. Group Homomorphisms

  Definition and Examples

  Properties of Homomorphisms

  The First Isomorphism Theorem

  Computer Exercises

  Biography of Camille Jordan

  11. Fundamental Theorem of Finite Abelian Groups

  The Fundamental Theorem

  The Isomorphism Classes of Abelian Groups

  Proof of the Fundamental Theorem

  Computer Exercises

  Supplementary Exercises for Chapters 9–11

  III. Rings

  12. Introduction to Rings

  Motivation and Definition

  Examples of Rings

  Properties of Rings

  Subrings

  Computer Exercises

  Biography of I. N. Herstein

  13. Integral Domains

  Definition and Examples

  Fields

  Characteristic of a Ring

  Computer Exercises

  Biography of Nathan Jacobson

  14. Ideals and Factor Rings

  Ideals

  Factor Rings

  Prime Ideals and Maximal Ideals

  Biography of Richard Dedekind

  Biography of Emmy Noether

  Supplementary Exercises for Chapters 12–14

  15. Ring Homomorphisms

  Definition and Examples

  Properties of Ring Homomorphisms

  The Field of Quotients

  16. Polynomial Rings

  Notation and Terminology

  The Division Algorithm and Consequences

  Biography of Saunders Mac Lane

  17. Factorization of Polynomials

  Reducibility Tests

  Irreducibility Tests

  Unique Factorization in Z [x]

  Weird Dice: An Application of Unique Factorization

  Computer Exercises

  18. Divisibility in Integral Domains

  Irreducibles, Primes

  Historical Discussion of Fermat's Last Theorem

  Unique Factorization Domains

  Euclidean Domains

  Biography of Sophie Germain

  Biography of Andrew Wiles

  Supplementary Exercises for Chapters 15–18

  IV. Fields

  19. Vector Spaces

  Definition and Examples

  Subspaces

  Linear Independence

  Biography of Emil Artin

  Biography of Olga Taussky-Todd

  20. Extension Fields

  The Fundamental Theorem of Field Theory

  Splitting Fields

  Zeros of an Irreducible Polynomial

  Biography of Leopold Kronecker

  21. Algebraic Extensions

  Characterization of Extensions

  Finite Extensions

  Properties of Algebraic Extensions

  Biography of Irving Kaplansky

  22. Finite Fields

  Classification of Finite Fields

  Structure of Finite Fields

  Subfields of a Finite Field

  Computer Exercises

  Biography of L. E. Dickson

  23. Geometric Constructions

  Historical Discussion of Geometric Constructions

  Constructible Numbers

  Angle-Trisectors and Circle-Squarers

  Supplementary Exercises for Chapters 19–23

  V. Special Topics

  24. Sylow Theorems

  Conjugacy Classes

  The Class Equation

  The Probability That Two Elements Commute

  The Sylow Theorems

  Applications of Sylow Theorems

  Biography of Ludvig Sylow

  25. Finite Simple Groups

  Historical Background

  Nonsimplicity Tests

  The Simplicity of A5

  The Fields Medal

  The Cole Prize

  Computer Exercises

  Biography of Michael Aschbacher

  Biography of Daniel Gorenstein

  Biography of John Thompson

  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

  Biography of Marshall Hall, Jr.

  27. Symmetry Groups

  Isometries

  Classification of Finite Plane Symmetry Groups

  Classification of Finite Group of Rotations in R3

  28. Frieze Groups and Crystallographic Groups

  The Frieze Groups

  The Crystallographic Groups

  Identification of Plane Periodic Patterns

  Biography of M. C. Escher

  Biography of George Pólya

  Biography of John H. Conway

  29. Symmetry and Counting

  Motivation

  Burnside's Theorem

  Applications

  Group Action

  Biography of William Burnside

  30. Cayley Digraphs of Groups

  Motivation

  The Cayley Digraph of a Group

  Hamiltonian Circuits and Paths

  Some Applications

  Biography of William Rowan Hamilton

  Biography of Paul Erdös

  31. Introduction to Algebraic Coding Theory

  Motivation

  Linear Codes

  Parity-Check Matrix Decoding

  Coset Decoding

  Historical Note: Reed-Solomon Codes

  Biography of Richard W. Hamming

  Biography of Jessie MacWilliams

  Biography of Vera Pless

  32. An Introduction to Galois Theory

  Fundamental Theorem of Galois Theory

  Solvability of Polynomials by Radicals

  Insolvability of a Quintic

  Biography of Philip Hall

  33. Cyclotomic Extensions

  Motivation

  Cyclotomic Polynomials

  The Constructible Regular n-gons

  Computer Exercise

  Biography of Carl Friedrich Gauss

  Biography of Manjul Bhargava

  Supplementary Exercises for Chapters 24–33