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Contemporary abstract algebra / Joseph A. Gallian.

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Bibliographic Details
Main Author: Gallian, Joseph A.
Format: Book
Language:English
Published: Boston, MA : Houghton Mifflin, 2006.
Edition:Sixth edition.
Subjects:
Algebra, Abstract > Textbooks.
Algebra, Abstract.
Textbooks.
Table of Contents:
  • Part 1. Integers and Equivalence Relations
  • Chapter 0. Preliminaries
  • Properties of Integers
  • Modular Arithmetic
  • Mathematical Induction
  • Equivalence Relations
  • Functions (Mappings)
  • Part 2. Groups
  • Chapter 1. Introduction to Groups
  • Symmetries of a Square
  • The Dihedral Groups
  • Chapter 2. Groups
  • Definition and Examples of Groups
  • Elementary Properties of Groups
  • Historical Note
  • Chapter 3. Finite Groups; Subgroups
  • Terminology and Notation
  • Subgroup Tests
  • Examples of Subgroups
  • Chapter 4. Cyclic Groups
  • Properties of Cyclic Groups
  • Classification of Subgroups of Cyclic Groups
  • Chapter 5. Permutation Groups
  • Definition and Notation
  • Cycle Notation
  • Properties of Permutations
  • A Check-Digit Scheme Based on D[subscript 5]
  • Chapter 6. Isomorphisms
  • Motivation
  • Definition and Examples
  • Cayley's Theorem
  • Properties of Isomorphisms
  • Automorphisms
  • Chapter 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
  • Chapter 8. External Direct Products
  • Definition and Examples
  • Properties of External Direct Products
  • The Group of Units Modulo n as an External Direct Product
  • Applications
  • Chapter 9. Normal Subgroups and Factor Groups
  • Normal Subgroups
  • Factor Groups
  • Applications of Factor Groups
  • Internal Direct Products
  • Chapter 10. Group Homomorphisms
  • Definition and Examples
  • Properties of Homomorphisms
  • The First Isomorphism Theorem
  • Chapter 11. Fundamental Theorem of Finite Abelian Groups
  • The Fundamental Theorem
  • The Isomorphism Classes of Abelian Groups
  • Proof of the Fundamental Theorem
  • Part 3. Rings
  • Chapter 12. Introduction to Rings
  • Motivation and Definition
  • Examples of Rings
  • Properties of Rings
  • Subrings
  • Chapter 13. Integral Domains
  • Definition and Examples
  • Fields
  • Characteristic of a Ring
  • Chapter 14. Ideals and Factor Rings
  • Ideals
  • Factor Rings
  • Prime Ideals and Maximal Ideals
  • Chapter 15. Ring Homomorphisms
  • Definition and Examples
  • Properties of Ring Homomorphisms
  • The Field of Quotients
  • Chapter 16. Polynomial Rings
  • Notation and Terminology
  • The Division Algorithm and Consequences
  • Chapter 17. Factorization of Polynomials
  • Reducibility Tests
  • Irreducibility Tests
  • Unique Factorization in Z[x]
  • Weird Dice: An Application of Unique Factorization
  • Chapter 18. Divisibility in Integral Domains
  • Irreducibles, Primes
  • Historical Discussion of Fermat's Last Theorem
  • Unique Factorization Domains
  • Euclidean Domains
  • Part 4. Fields
  • Chapter 19. Vector Spaces
  • Definition and Examples
  • Subspaces
  • Linear Independence
  • Chapter 20. Extension Fields
  • The Fundamental Theorem of Field Theory
  • Splitting Fields
  • Zeros of an Irreducible Polynomial
  • Chapter 21. Algebraic Extensions
  • Characterization of Extensions
  • Finite Extensions
  • Properties of Algebraic Extensions
  • Chapter 22. Finite Fields
  • Classification of Finite Fields
  • Structure of Finite Fields
  • Subfields of a Finite Field
  • Chapter 23. Geometric Constructions
  • Historical Discussion of Geometric Constructions
  • Constructible Numbers
  • Angle-Trisectors and Circle-Squarers
  • Part 5. Special Topics
  • Chapter 24. Sylow Theorems
  • Conjugacy Classes
  • The Class Equation
  • The Probability That Two Elements Commute
  • The Sylow Theorems
  • Applications of Sylow Theorems
  • Chapter 25. Finite Simple Groups
  • Historical Background
  • Nonsimplicity Tests
  • The Simplicity of A[subscript 5]
  • The Fields Medal
  • The Cole Prize
  • Chapter 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
  • Chapter 27. Symmetry Groups
  • Isometries
  • Classification of Finite Plane Symmetry Groups
  • Classification of Finite Groups of Rotations in R[superscript 3]
  • Chapter 28. Frieze Groups and Crystallographic Groups
  • The Frieze Groups
  • The Crystallographic Groups
  • Identification of Plane Periodic Patterns
  • Chapter 29. Symmetry and Counting
  • Motivation
  • Burnside's Theorem
  • Applications
  • Group Action
  • Chapter 30. Cayley Digraphs of Groups
  • Motivation
  • The Cayley Digraph of a Group
  • Hamiltonian Circuits and Paths
  • Some Applications
  • Chapter 31. Introduction to Algebraic Coding Theory
  • Motivation
  • Linear Codes
  • Parity-Check Matrix Decoding
  • Coset Decoding
  • Historical Note: Reed-Solomon Codes
  • Chapter 32. An Introduction to Galois Theory
  • Fundamental Theorem of Galois Theory
  • Solvability of Polynomials by Radicals
  • Insolvability of a Quintic
  • Chapter 33. Cyclotomic Extensions
  • Motivation
  • Cyclotomic Polynomials
  • The Constructible Regular n-gons.

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