Modern Algebra An Introduction

ISBN-10: 0470384433
ISBN-13: 9780470384435
Edition: 6th 2009
Authors: John R. Durbin
List price: $218.95 Buy it from $52.61
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Description: This book presents an introduction to modern (abstract) algebra covering the basic ideas of groups, rings, and fields. The first part of the book treats ideas that are important but neither abstract nor complicated, and provides practice in handling  More...

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Book details

List price: $218.95
Edition: 6th
Copyright year: 2009
Publisher: John Wiley & Sons, Incorporated
Publication date: 12/31/2008
Binding: Hardcover
Pages: 352
Size: 7.25" wide x 10.00" long x 0.75" tall
Weight: 1.430
Language: English

This book presents an introduction to modern (abstract) algebra covering the basic ideas of groups, rings, and fields. The first part of the book treats ideas that are important but neither abstract nor complicated, and provides practice in handling mathematical statements - their meaning, quantification, negation, and proof. This edition features a new section to give more substance to the introduction to Galois theory, updated lists of references, and discussions of topics such as Fermats Last Theorem and the finite simple groups.

Introduction
Mappings and Operations
Mappings
Composition. Invertible Mappings
Operations
Composition as an Operation
Introduction to Groups
Definition and Examples
Permutations
Subgroups
Groups and Symmetry
Equivalence. Congruence. Divisibility
Equivalence Relations
Congruence. The Division Algorithm
Integers Modulo n
Greatest Common Divisors. The Euclidean Algorithm
Factorization. Euler's Phi-Function
Groups
Elementary Properties
Generators. Direct Products
Cosets
Lagrange's Theorem. Cyclic Groups
Isomorphism
More on Isomorphism
Cayley's Theorem
RSA Algorithm
Group Homomorphisms
Homomorphisms of Groups. Kernels
Quotient Groups
The Fundamental Homomorphism Theorem
Introduction to Rings
Definition and Examples
Integral Domains. Subrings
Fields
Isomorphism. Characteristic
The Familiar Number Systems
Ordered Integral Domains
The Integers
Field of Quotients. The Field of Rational Numbers
Ordered Fields. The Field of Real Numbers
The Field of Complex Numbers
Complex Roots of Unity
Polynomials
Definition and Elementary Properties
Appendix to Section 34
The Division Algorithm
Factorization of Polynomials
Unique Factorization Domains
Quotient Rings
Homomorphisms of Rings. Ideals
Quotient Rings
Quotient Rings of F[X]
Factorization and Ideals
Galois Theory: Overview
Simple Extensions. Degree
Roots of Polynomials
Fundamental Theorem: Introduction
Galois Theory
Algebraic Extensions
Splitting Fields. Galois Groups
Separability and Normality
Fundamental Theorem of Galois Theory
Solvability by Radicals
Finite Fields
Geometric Constructions
Three Famous Problems
Constructible Numbers
Impossible Constructions
Solvable and Alternating Groups
Isomorphism Theorems and Solvable Groups
Alternating Groups
Applications of Permutation Groups
Groups Acting on Sets
Burnside's Counting Theorem
Sylow's Theorem
Symmetry
Finite Symmetry Groups
Infinite Two-Dimensional Symmetry Groups
On Crystallographic Groups
The Euclidean Group
Lattices and Boolean Algebras
Partially Ordered Sets
Lattices
Boolean Algebras
Finite Boolean Algebras
Sets
Proofs
Mathematical Induction
Linear Algebra
Solutions to Selected Problems
Photo Credit List
Index of Notation
Index

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