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Abstract Algebra

ISBN-10: 0471433349
ISBN-13: 9780471433347
Edition: 3rd 2004 (Revised)
List price: $132.95 Buy it from $22.55 Rent it from $28.41
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Description: Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic  More...

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Book details

List price: $132.95
Edition: 3rd
Copyright year: 2004
Publisher: John Wiley & Sons, Incorporated
Publication date: 7/14/2003
Binding: Hardcover
Pages: 944
Size: 7.78" wide x 9.51" long x 1.45" tall
Weight: 3.564
Language: English

Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings. * The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible.

Preface
Preliminaries
Basics
Properties of the Integers
Z / n Z: The Integers Modulo n
Group Theory
Introduction to Groups
Basic Axioms and Examples
Dihedral Groups
Symmetric Groups
Matrix Groups
The Quaternion Group
Homomorphisms and Isomorphisms
Group Actions
Subgroups
Definition and Examples
Centralizers and Normalizers, Stabilizers and Kernels
Cyclic Groups and Cyclic Subgroups
Subgroups Generated by Subsets of a Group
The Lattice of Subgroups of a Group
Quotient Groups and Homomorphisms
Definitions and Examples
More on Cosets and Lagrange's Theorem
The Isomorphism Theorems
Composition Series and the Holder Program
Transpositions and the Alternating Group
Group Actions
Group Actions and Permutation Representations
Groups Acting on Themselves by Left Multiplication--Cayley's Theorem
Groups Acting on Themselves by Conjugation--The Class Equation
Automorphisms
The Sylow Theorems
The Simplicity of A[subscript n]
Direct and Semidirect Products and Abelian Groups
Direct Products
The Fundamental Theorem of Finitely Generated Abelian Groups
Table of Groups of Small Order
Recognizing Direct Products
Semidirect Products
Further Topics in Group Theory
p-groups, Nilpotent Groups, and Solvable Groups
Applications in Groups of Medium Order
A Word on Free Groups
Ring Theory
Introduction to Rings
Basic Definitions and Examples
Examples: Polynomial Rings, Matrix Rings, and Group Rings
Ring Homomorphisms an Quotient Rings
Properties of Ideals
Rings of Fractions
The Chinese Remainder Theorem
Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains
Euclidean Domains
Principal Ideal Domains (P.I.D.s)
Unique Factorization Domains (U.F.D.s)
Polynomial Rings
Definitions and Basic Properties
Polynomial Rings over Fields I
Polynomial Rings that are Unique Factorization Domains
Irreducibility Criteria
Polynomial Rings over Fields II
Polynomials in Several Variables over a Field and Grobner Bases
Modules and Vector Spaces
Introduction to Module Theory
Basic Definitions and Examples
Quotient Modules and Module Homomorphisms
Generation of Modules, Direct Sums, and Free Modules
Tensor Products of Modules
Exact Sequences--Projective, Injective, and Flat Modules
Vector Spaces
Definitions and Basic Theory
The Matrix of a Linear Transformation
Dual Vector Spaces
Determinants
Tensor Algebras, Symmetric and Exterior Algebras
Modules over Principal Ideal Domains
The Basic Theory
The Rational Canonical Form
The Jordan Canonical Form
Field Theory and Galois Theory
Field Theory
Basic Theory of Field Extensions
Algebraic Extensions
Classical Straightedge and Compass Constructions
Splitting Fields and Algebraic Closures
Separable and Inseparable Extensions
Cyclotomic Polynomials and Extensions
Galois Theory
Basic Definitions
The Fundamental Theorem of Galois Theory
Finite Fields
Composite Extensions and Simple Extensions
Cyclotomic Extensions and Abelian Extensions over Q
Galois Groups of Polynomials
Solvable and Radical Extensions: Insolvability of the Quintic
Computation of Galois Groups over Q
Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
An Introduction to Commutative Rings, Algebraic Geometry, and Homological Algebra
Commutative Rings and Algebraic Geometry
Noetherian Rings and Affine Algebraic Sets
Radicals and Affine Varieties
Integral Extensions and Hilbert's Nullstellensatz
Localization
The Prime Spectrum of a Ring
Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
Artinian Rings
Discrete Valuation Rings
Dedekind Domains
Introduction to Homological Algebra and Group Cohomology
Introduction to Homological Algebra--Ext and Tor
The Cohomology of Groups
Crossed Homomorphisms and H[superscript 1](G, A)
Group Extensions, Factor Sets and H[superscript 2](G, A)
Introduction to the Representation Theory of Finite Groups
Representation Theory and Character Theory
Linear Actions and Modules over Group Rings
Wedderburn's Theorem and Some Consequences
Character Theory and the Orthogonality Relations
Examples and Applications of Character Theory
Characters of Groups of Small Order
Theorems of Burnside and Hall
Introduction to the Theory of Induced Characters
Cartesian Products and Zorn's Lemma
Category Theory
Index

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