Abstract Algebra An Introduction to Groups, Rings and Fields

Name: Abstract Algebra An Introduction to Groups, Rings and Fields
Price: 56.11 USD
Availability: InStock
ISBN: 9789814340281

ISBN-10: 9814340286

ISBN-13: 9789814340281

Edition: 2011

Authors: Clive Reis

List price: $58.00

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Description:

This book is appropriate for second to fourth year undergraduates. In addition to the material traditionally taught at this level, the book contains several applications: PolyaBurnside Enumeration, Mutually Orthogonal Latin Squares, Error-Correcting Codes and a classification of the finite groups of isometries of the plane and the finite rotation groups in Euclidean 3-space. It is hoped that these applications will help the reader achieve a better grasp of the rather abstract ideas presented and convince him/her that pure mathematics, in addition to having an austere beauty of its own, can be applied to solving practical problems.An entire chapter is devoted to studying the algebraic system…

Book details

List price: $58.00
Copyright year: 2011
Publisher: World Scientific Publishing Co Pte Ltd
Publication date: 4/15/2011
Binding: Paperback
Pages: 520
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 1.584
Language: English



Preface


List of Tables


List of Figures



Logic and Proofs



Introduction



Statements, Connectives and Truth Tables



Relations Between Statements



Quantifiers



Methods of proof



Exercises



Set Theory



Definitions



Relations Between Sets



Operations Defined on Sets - Or New sets from Old



Exercises



Cartesian Products, Relations, Maps and Binary Operations



Introduction



Cartesian Products



Maps



Binary Operations



Exercises



The Integers



Introduction



Elementary Properties



Divisibility



The Fundamental Theorem of Arithmetic



The Algebraic System (Z<sub>n</sub>,+, ) and Congruences



Congruences in Z and Equations in Z<sub>n</sub>



Exercises



Groups



Introduction



Definitions and Elementary Properties



Alternative Axioms for Groups



Subgroups



Cyclic Groups



Exercises



Further Properties of Groups



Introduction



Cosets



Isomorphisms and Homomorphisms



Normal Subgroups and Factor Groups



Direct Products of Groups



Exercises



The Symmetric Groups



Introduction



The Cayley Representation Theorem



Permutations as Products of Disjoint Cycles



Odd and Even Permutations



Conjugacy Classes of a Group



Exercises



Rings, Integral Domains and Fields



Rings



Homomorphisms, Isomorphisms and Ideals



Isomorphism Theorems



Direct Sums of Rings



Integral Domains and Fields



Embedding an Integral Domain in a Field



The Characteristic of an Integral Domain



Exercjses



Polynomial Rings



Introduction



Definitions and Elementary Properties



The Division Algorithm and Applications



Irreducibility and Factorization of Polynomials



Polynomials Over More Familiar Fields



Factor Rings of the Form F[x]/(g(x)), F a Field



Exercises



Field Extensions



Introduction



Definitions and Elementary Results



Algebraic and Transcendental Elements



Algebraic Extensions



Finite Fields



Exercises



Latin Squares and Magic Squares



Latin Squares



Magic Squares



Exercises



Group Actions, the Class Equation and the Sylow Theorems



Group Actions



The Class Equation of a Finite Group



The Sylow Theorems



Applications of the Sylow Theorems



Exercises



Isometries



Isometries of R<sup>n</sup>



Finite Subgroups of E(2)



The Platonic Solids



Rotations in R<sup>3</sup>



Exercises



Polya-Burnside Enumeration



Introduction



A Theorem of Polya



Exercises



Group Codes



Introduction



Definitions and Notation



Group Codes



Construction of Group Codes



At the Receiving End



Nearest Neighbor Decoding for Group Codes



Hamming Codes



Exercises



Polynomial Codes



Definitions and Elementary Results



BCH Codes



Exercises



Rational, Real and Complex Numbers



Introduction



The Real and Rational Number Systems



Decimal Representation of Rational Numbers



Complex Numbers



Polar Form of a Complex Number



Exercises



Linear Algebra



Vector Spaces



Linear Transformations



Inner Product Spaces



Orthogonal Linear Transformations and Orthogonal Matrices



Determinants



Eigenvalues and Eigenvectors



Exercises


Index