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ISBN13: 978-0618661411

ISBN10: 0618661417

Cover type:

Edition: 6TH 05

Copyright: 2005

Publisher: Houghton Mifflin Harcourt

Published: 2005

International: No

ISBN10: 0618661417

Cover type:

Edition: 6TH 05

Copyright: 2005

Publisher: Houghton Mifflin Harcourt

Published: 2005

International: No

Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.

**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 Direct Product

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

ISBN10: 0618661417

Cover type:

Edition: 6TH 05

Copyright: 2005

Publisher: Houghton Mifflin Harcourt

Published: 2005

International: No

Joseph Gallian is a well-known active researcher and award-winning teacher. His Contemporary Abstract Algebra, 6/e, includes challenging topics in abstract algebra as well as numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings that give the subject a current feel and makes the content interesting and relevant for students.

Table of Contents

**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 Direct Product

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