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Edition: 4TH 98

Copyright: 1998

Publisher: Houghton Mifflin Harcourt

Published: 1998

International: No

Copyright: 1998

Publisher: Houghton Mifflin Harcourt

Published: 1998

International: No

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One of the best-selling texts for the course, *Abstract Algebra,* 4/e, emphasizes theory and includes challenging topics in abstract algebra. Numerous figures, tables, photographs, charts, biographies, and computer exercises highlight the currency of the subject, making it interesting and relevant for students. The author is an active researcher and award-winning teacher.

- A new chapter on cyclotomic polynomials replaces the chapter on Boolean algebra.
- About 150 new exercises, examples, and real-world applications appear throughout the text.
- Several new biographical sketches of modern-day mathematicians such as Vera Pless and Jessie MacWilliams highlight the relevance of mathematics to the contemporary world.

**I. Integers and Equivalence Relations**

**0. Preliminaries**

Properties of Integers

Modular Arithmetic

Mathematical Induction

Equivalence Relations

Functions (Mappings)

**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 Exercise

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

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 5-7

**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

Biography of Camille Jordan

**11. Fundamental Theorem of Finite Abelian Groups**

The Fundamental Theorem

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

**17. Factorization of Polynomials**

Reducibility Tests

Irreducibility Tests

Unique Factorization in *Z [x] *Weird Dice: An Application of Unique Factorization

Computer Exercises

Biography of Carl Friedrich Gauss

**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 Tausslay-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

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 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 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

Supplementary Exercises Ch. 24-33

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Summary

One of the best-selling texts for the course, *Abstract Algebra,* 4/e, emphasizes theory and includes challenging topics in abstract algebra. Numerous figures, tables, photographs, charts, biographies, and computer exercises highlight the currency of the subject, making it interesting and relevant for students. The author is an active researcher and award-winning teacher.

- A new chapter on cyclotomic polynomials replaces the chapter on Boolean algebra.
- About 150 new exercises, examples, and real-world applications appear throughout the text.
- Several new biographical sketches of modern-day mathematicians such as Vera Pless and Jessie MacWilliams highlight the relevance of mathematics to the contemporary world.

Table of Contents

**I. Integers and Equivalence Relations**

**0. Preliminaries**

Properties of Integers

Modular Arithmetic

Mathematical Induction

Equivalence Relations

Functions (Mappings)

**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 Exercise

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

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 5-7

**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

Biography of Camille Jordan

**11. Fundamental Theorem of Finite Abelian Groups**

The Fundamental Theorem

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

**17. Factorization of Polynomials**

Reducibility Tests

Irreducibility Tests

Unique Factorization in *Z [x] *Weird Dice: An Application of Unique Factorization

Computer Exercises

Biography of Carl Friedrich Gauss

**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 Tausslay-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

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 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 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

Supplementary Exercises Ch. 24-33

Publisher Info

Publisher: Houghton Mifflin Harcourt

Published: 1998

International: No

Published: 1998

International: No

One of the best-selling texts for the course, *Abstract Algebra,* 4/e, emphasizes theory and includes challenging topics in abstract algebra. Numerous figures, tables, photographs, charts, biographies, and computer exercises highlight the currency of the subject, making it interesting and relevant for students. The author is an active researcher and award-winning teacher.

- A new chapter on cyclotomic polynomials replaces the chapter on Boolean algebra.
- About 150 new exercises, examples, and real-world applications appear throughout the text.
- Several new biographical sketches of modern-day mathematicians such as Vera Pless and Jessie MacWilliams highlight the relevance of mathematics to the contemporary world.

**I. Integers and Equivalence Relations**

**0. Preliminaries**

Properties of Integers

Modular Arithmetic

Mathematical Induction

Equivalence Relations

Functions (Mappings)

**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 Exercise

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

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 5-7

**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

Biography of Camille Jordan

**11. Fundamental Theorem of Finite Abelian Groups**

The Fundamental Theorem

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

**17. Factorization of Polynomials**

Reducibility Tests

Irreducibility Tests

Unique Factorization in *Z [x] *Weird Dice: An Application of Unique Factorization

Computer Exercises

Biography of Carl Friedrich Gauss

**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 Tausslay-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

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 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 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

Supplementary Exercises Ch. 24-33