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Contemporary Abstract Algebra

ISBN-10: 1133599702
ISBN-13: 9781133599708
Edition: 8th 2013
Authors: Joseph Gallian
List price: $207.95 Buy it from $19.15
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Description: CONTEMPORARY ABSTRACT ALGEBRA, EIGHTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists,  More...

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Book details

List price: $207.95
Edition: 8th
Copyright year: 2013
Publisher: Brooks/Cole
Publication date: 7/9/2012
Binding: Hardcover
Pages: 656
Size: 6.50" wide x 9.50" long x 1.25" tall
Weight: 1.936
Language: English

CONTEMPORARY ABSTRACT ALGEBRA, EIGHTH EDITION provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students.

Preface
Integers and Equivalence Relations
Preliminaries
Properties of Integers
Modular Arithmetic
Complex Numbers
Mathematical Induction
Equivalence Relations
Functions (Mappings)
Exercises
Groups
Introduction to Groups
Symmetries of a Square
The Dihedral Groups
Exercises
Biography of Niels Abel
Groups
Definition and Examples of Groups
Elementary Properties of Groups
Historical Note
Exercises
Finite Groups; Subgroups
Terminology and Notation
Subgroup Tests
Examples of Subgroups
Exercises
Cyclic Groups
Properties of Cyclic Groups
Classification of Subgroups of Cyclic Groups
Exercises
Biography of James Joseph Sylvester
Supplementary Exercises for Chapters 1-4
Permutation Groups
Definition and Notation
Cycle Notation
Properties of Permutations
A Check-Digit Scheme Based on D<sub>5</sub>
Exercises
Biography of Augustin Cauchy
Isomorphisms
Motivation
Definition and Examples
Cayley's Theorem
Properties of Isomorphisms
Automorphisms
Exercises
Biography of Arthur Cayley
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
An Application of Cosets to the Rubik's Cube
Exercises
Biography of Joseph Lagrange
External Direct Products
Definition and Examples
Properties of External Direct Products
The Group of Units Modulo n as an External Direct Product
Applications
Exercises
Biography of Leonard Adleman
Supplementary Exercises for Chapters 5-8
Normal Subgroups and Factor Groups
Normal Subgroups
Factor Groups
Applications of Factor Groups
Internal Direct Products
Exercises
Biography of �variste Galois
Group Homomorphisms
Definition and Examples
Properties of Homomorphisms
The First Isomorphism Theorem
Exercises
Biography of Camille Jordan
Fundamental Theorem of Finite Abelian Groups
The Fundamental Theorem
The Isomorphism Classes of Abelian Groups
Proof of the Fundamental Theorem
Exercises
Supplementary Exercises for Chapters 9-11
Rings
Introduction to Rings
Motivation and Definition
Examples of Rings
Properties of Rings
Subrings
Exercises
Biography of I. N. Herstein
Integral Domains
Definition and Examples
Fields
Characteristic of a Ring
Exercises
Biography of Nathan Jacobson
Ideals and Factor Rings
Ideals
Factor Rings
Prime Ideals and Maximal Ideals
Exercises
Biography of Richard Dedekind
Biography of Emmy Noether
Supplementary Exercises for Chapters 12-14
Ring Homomorphisms
Definition and Examples
Properties of Ring Homomorphisms
The Field of Quotients
Exercises
Polynomial Rings
Notation and Terminology
The Division Algorithm and Consequences
Exercises
Biography of Saunders Mac Lane
Factorization of Polynomials
Reducibility Tests
Irreducibility Tests
Unique Factorization in Z[x]
Weird Dice: An Application of Unique Factorization
Exercises
Biography of Serge Lang
Divisibility in Integral Domains
Irreducibles, Primes
Historical Discussion of Fermat's Last Theorem
Unique Factorization Domains
Euclidean Domains
Exercises
Biography of Sophie Germain
Biography of Andrew Wiles
Supplementary Exercises for Chapters 15-18
Fields
Vector Spaces
Definition and Examples
Subspaces
Linear Independence
Exercises
Biography of Emil Artin
Biography of Olga Taussky-Todd
Extension Fields
The Fundamental Theorem of Field Theory
Splitting Fields
Zeros of an Irreducible Polynomial
Exercises
Biography of Leopold Kronecker
Algebraic Extensions
Characterization of Extensions
Finite Extensions
Properties of Algebraic Extensions
Exercises
Biography of Irving Kaplansky
Finite Fields
Classification of Finite Fields
Structure of Finite Fields
Subfields of a Finite Field
Exercises
Biography of L. E. Dickson
Geometric Constructions
Historical Discussion of Geometric Constructions
Constructible Numbers
Angle-Trisectors and Circle-Squarers
Exercises
Supplementary Exercises for Chapters 19-23
Special Topics
Sylow Theorems
Conjugacy Classes
The Class Equation
The Probability That Two Elements Commute
The Sylow Theorems
Applications of Sylow Theorems
Exercises
Biography of Ludwig Sylow
Finite Simple Groups
Historical Background
Nonsimplicity Tests
The Simplicity of A<sub>5</sub>
The Fields Medal
The Cole Prize
Exercises
Biography of Michael Aschbacher
Biography of Daniel Gorenstein
Biography of John Thompson
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
Exercises
Biography of Marshall Hall, Jr.
Symmetry Groups
Isometries
Classification of Finite Plane Symmetry Groups
Classification of Finite Groups of Rotations in R<sup>3</sup>
Exercises
Frieze Groups and Grystallographic Groups
The Frieze Groups
The Crystallographic Groups
Identification of Plane Periodic Patterns
Exercises
Biography of M. C. Escher
Biography of George P�lya
Biography of John H. Conway
Symmetry and Counting
Motivation
Burnside's Theorem
Applications
Group Action
Exercises
Biography of William Burnside
Cayley Digraphs of Groups
Motivation
The Cayley Digraph of a Group
Hamiltonian Circuits and Paths
Some Applications
Exercises
Biography of William Rowan Hamilton
Biography of Paul Erdos
Introduction to Algebraic Coding Theory
Motivation
Linear Codes
Parity-Check Matrix Decoding
Coset Decoding
Historical Note: The Ubiquitous Reed-Solomon Codes
Exercises
Biography of Richard W. Hamming
Biography of Jessie MacWilliams
Biography of Vera Pless
An Introduction to Galois Theory
Fundamental Theorem of Galois Theory
Solvability of Polynomials by Radicals
Insolvability of a Quintic
Exercises
Biography of Philip Hall
Cyclotomic Extensions
Motivation
Cyclotomic Polynomials
The Constructible Regular n-gons
Exercises
Biography of Carl Friedrich Gauss
Biography of Manjul Bhargava
Supplementary Exercises for Chapters 24-33
Selected Answers
Index of Mathematicians
Index of Terms

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