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Contemporary Abstract Algebra

Contemporary Abstract Algebra - 4th edition

Contemporary Abstract Algebra - 4th edition

ISBN13: 9780395861790

ISBN10: 0395861799

Contemporary Abstract Algebra by Joseph A. Gallian - ISBN 9780395861790
Edition: 4TH 98
Copyright: 1998
Publisher: Houghton Mifflin Harcourt
International: No
Contemporary Abstract Algebra by Joseph A. Gallian - ISBN 9780395861790

ISBN13: 9780395861790

ISBN10: 0395861799

Edition: 4TH 98

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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.

Table of Contents

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

Definition and Examples
Cayley's Theorem
Properties of Isomorphisms
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
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
Computer Exercises
Biography of I.N. Herstein

13. Integral Domains

Definition and Examples
Characteristic of a Ring
Computer Exercises
Biography of Nathan Jacobson

14. Ideals and Factor Rings

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
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

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

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

Burnside's Theorem
Group Action
Biography of William Burnside

30. Cayley Digraphs of Groups

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

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

Cyclotomic Polynomials
The Constructible Regular n-gons
Computer Exercise
Biography of Carl Friedrich Gauss
Supplementary Exercises Ch. 24-33