Abstract Algebra An Introduction

ISBN-10: 1111569622
ISBN-13: 9781111569624
Edition: 3rd 2013
List price: $147.95 Buy it from $25.00
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Description: ABSTRACT ALGEBRA: AN INTRODUCTION is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging  More...

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Book details

List price: $147.95
Edition: 3rd
Copyright year: 2013
Publisher: Brooks/Cole
Publication date: 7/27/2012
Binding: Hardcover
Pages: 616
Size: 7.50" wide x 9.50" long x 1.25" tall
Weight: 2.618

ABSTRACT ALGEBRA: AN INTRODUCTION is intended for a first undergraduate course in modern abstract algebra. The flexible design of the text makes it suitable for courses of various lengths and different levels of mathematical sophistication, ranging from a traditional abstract algebra course to one with a more applied flavor. The emphasis is on clarity of exposition. The thematic development and organizational overview is what sets this book apart. The chapters are organized around three themes: arithmetic, congruence, and abstract structures. Each Them is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another.

Preface
To the Instructor
To the Student
Thematic Table of Contents for the Gore Course
The Core Course
Arithmetic in Z Revisited
The Division Algorithm
Divisibility
Primes and Unique Factorization
Congruence in Z and Modular Arithmetic
Congruence and Congruence Classes
Modular Arithmetic
The Structure of Z<sub>p</sub> (p Prime) and Z<sub>n</sub>
Rings
Definition and Examples of Rings
Basic Properties of Rings
Isomorphisms and Homomorphisms
Arithmetic in F[x]
Polynomial Arithmetic and the Division Algorithm
Divisibility in F[x]
Irreducibles and Unique Factorization
Polynomial Functions, Roots, and Reducibility
Irreducibility in Q[x]
Irreducibility in R[x] and C[x]
Congruence in F[x] and Congruence-Class Arithmetic
Congruence in F[x] and Congruence Classes
Congruence-Class Arithmetic
The Structure of F[x]/(p(x)) When p(x) Is Irreducible
Ideals and Quotient Rings
Ideals and Congruence
Quotient Rings and Homomorphisms
The Structure of R/1 When / Is Prime or Maximal
Groups
Definition and Examples of Groups
A Definition and Examples of Groups
Basic Properties of Groups
Subgroups
Isomorphisms and Homomorphisms
The Symmetric and Alternating Groups
Normal Subgroups and Quotient Groups
Congruence and Lagrange's Theorem
Normal Subgroups
Quotient Groups
Quotient Groups and Homomorphisms
The Simplicity of A<sub>n</sub>
Advanced Topics
Topics in Group Theory
Direct Products
Finite Abelian Groups
The Sylow Theorems
Conjugacy and the Proof of the Sylow Theorems
The Structure of Finite Groups
Arithmetic in Integral Domains
Euclidean Domains
Principal Ideal Domains and Unique Factorization Domains
Factorization of Quadratic Integers
The Field of Quotients of an Integral Domain
Unique Factorization in Polynomial Domains
Field Extensions
Vector Spaces
Simple Extensions
Algebraic Extensions
Splitting Fields
Separability
Finite Fields
Galois Theory
The Galois Group
The Fundamental Theorem of Galois Theory
Solvability by Radicals
Excursions and Applications
Public-Key Cryptography
Prerequisite: Section 2.3
The Chinese Remainder Theorem
Proof of the Chinese Remainder Theorem
Prerequisites: Section 2.1, Appendix C
Applications of the Chinese Remainder Theorem
Prerequisite: Section 3.1
The Chinese Remainder Theorem for Rings
Prerequisite: Section 6.2
Geometric Constructions
Prerequisites: Sections 4.1, 4.4, and 4.5
Algebraic Coding Theory
Linear Codes
Prerequisites: Section 7.4, Appendix F
Decoding Techniques
Prerequisite: Section 8.4
BCH Codes
Prerequisite: Section 11.6
Appendices
Logic and Proof
Sets and Functions
Well Ordering and Induction
Equivalence Relations
The Binomial Theorem
Matrix Algebra
Polynomials
Bibliography
Answers and Suggestions for Selected Odd-Numbered Exercises
Index

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