Introduction to Abstract Algebra

ISBN-10: 1118135350

ISBN-13: 9781118135358

Edition: 4th 2012

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Description:

This Fourth Edition of Introduction to Abstract Algebra  is a self-contained introduction to the basic structures of abstract algebra: groups, rings, and fields.   This book is intended for a one or two semester abstract algebra course.  The writing style is appealing to students, and great effort has been made to motivate and be very clear about how the topics and applications relate to one another.  Over 500 solved examples are included to aid reader comprehension as well as to demonstrate how results in the theory are obtained.  Many applications (particularly to coding theory, cryptography, and to combinatorics) are provided to illustrate how the abstract structures relate to real-world problems.  In addition, historical notes and biographies of mathematicians put the subject into perspective.  Abstract thinking is difficult when first encountered and this is addressed in this book by presenting concrete examples (induction, number theory, integers modulo n, permutations) before the abstract structures are defined.  With this approach, readers can complete computations immediately using concepts that will be seen again later in the abstract setting.  Special topics such as symmetric polynomials, nilpotent groups, and finite dimensional algebras are also discussed.
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Book details

List price: $221.95
Edition: 4th
Copyright year: 2012
Publisher: John Wiley & Sons, Incorporated
Publication date: 3/6/2012
Binding: Hardcover
Pages: 560
Size: 7.00" wide x 10.00" long x 1.50" tall
Weight: 2.728
Language: English

Preface
Acknowledgments
Notation Used in The Text
A Sketch of the History of Algebra to 1929
Preliminaries
Proofs
Sets
Mappings
Equivalences
Integers and Permutations
Induction
Divisors and Prime Factorization
Integers Modulo n
Permutations
An Application to Cryptography
Groups
Binary Operations
Groups
Subgroups
Cyclic Groups and the Order of an Element
Homomorphisms and Isomorphisms
Cosets and Lagrange's Theorem
Groups of Motions and Symmetries
Normal Subgroups
Factor Groups
The Isomorphism Theorem
An Application to Binary Linear Codes
Rings
Examples and Basic Properties
Integral Domains and Fields
Ideals and Factor Rings
Homomorphisms
Ordered Integral Domains
Polynomials
Polynomials
Factorization of Polynomials Over a Field
Factor Rings of Polynomials Over a Field
Partial Fractions
Symmetric Polynomials
Formal Construction of Polynomials
Factorization in Integral Domains
Irreducibles and Unique Factorization
Principal Ideal Domains
Fields
Vector Spaces
Algebraic Extensions
Splitting Fields
Finite Fields
Geometric Constructions
The Fundamental Theorem of Algebra
An Application to Cyclic and BCH Codes
Modules over Principal Ideal Domains
Modules
Modules Over a PID
p-Groups and the Sylow Theorems
Products and Factors
Cauchy's Theorem
Group Actions
The Sylow Theorems
Semidirect Products
An Application to Combinatorics
Series of Subgroups
The Jordan-H�lder Theorem
Solvable Groups
Nilpotent Groups
Galois Theory
Galois Groups and Separability
The Main Theorem of Galois Theory
Insolvability of Polynomials
Cyclotomic Polynomials and Wedderburn's Theorem
Finiteness Conditions for Rings and Modules
Wedderburn's Theorem
The Wedderburn-Artin Theorem
Appendices
Complex Numbers
Matrix Algebra
Zorn's Lemma
Proof of the Recursion Theorem
Bibliography
Selected Answers
Index
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