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Abstract Algebra (Dover Books on Mathematics)

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Synopses & Reviews

Publisher Comments:

This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.

In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diophantine problems, and congruences. Chapters 6 through 9 examine groups, rings, domains, fields, polynomial rings, and quadratic domains.

Chapters 10 through 13 cover modular systems, modules and vector spaces, linear transformations and matrices, and the elementary theory of matrices. The author, Professor of Mathematics at the University of Pittsburgh, includes many examples and, at the end of each chapter, a large number of problems of varying levels of difficulty.

Synopsis:

Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.

Synopsis:

Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.

Table of Contents

PREFACE

1. A COMMON LANGUAGE

  1.1. Sets

  1.2. "Ordered pairs, products, and relations"

  1.3. Functions and mappings

  1.4. Binary operations

  1.5. Abstract systems

  1.6. Suggested reading

2. THE BASIC NUMBER SYSTEMS

  2.1 The natural number system

  2.2 Order and cancellation

  2.3. Well-ordering

  2.4. Counting and finite sets

  2.5. The integers defined

  2.6. Ordering the integers

  2.7. Isomorphic systems and extensions

  2.8. Another extension

  2.9. Order and density

  2.10. * The real number system

  2.11. Power of the abstract approach

  2.12. Remarks

  2.13. Suggested reading

3. DECOMPOSITIONS OF INTEGERS

  3.1. Divisor theorem

  3.2. Congruence and factors

  3.3. Primes

  3.4. Greatest common factor

  3.5. Unique factorization again

  3.6. Euler's totient

  3.7. Suggested reading

4. * DIOPHANTINE PROBLEMS

  4.1. Linear Diophantine equations

  4.2. More linear Diophantine equations

  4.3. Linear congruences

  4.4. Pythagorean triples

  4.5. Method of descent

  4.6. Sum of two squares

  4.7. Suggested reading

5. ANOTHER LOOK AT CONGRUENCES

  5.1. The system of congruence classes modulo m

  5.2. Homomorphisms

  5.3. Subsystems and quotient systems

  5.4. * System of ideals

  5.5. * Remarks

  5.6. Suggested reading

6. GROUPS

  6.1. Definitions and examples

  6.2. Elementary properties

  6.3. Subgroups and cyclic groups

  6.4. Cosets

  6.5. Abelian groups

  6.6. * Finite Abelian groups

  6.7. * Normal subgroups

  6.8. * Sylow's theorem

  6.9. * Additional remarks

  6.10. Suggested reading

7. "RINGS, DOMAINS, AND FIELDS"

  7.1. Definitions and examples

  7.2. Elementary properties

  7.3. Exponentiation and scalar product

  7.4. Subsystems and characteristic

  7.5. Isomorphisms and extensions

  7.6. Homomorphisms and ideals

  7.7. Ring of functions

  7.8. Suggested reading

8. POLYNOMIAL RINGS

  8.1. Polynomial rings

  8.2. Polynomial domains

  8.3. Reducibility in the domain of a field

  8.4. Reducibility over the rational field

  8.5. Ideals and extensions

  8.6. Root fields and splitting fields

  8.7. * Automorphisms and Galois groups

  8.8. * An application to geometry

  8.9. * Transcendental extensions and partial fractions

  8.10. Suggested reading

9. * QUADRATIC DOMAINS

  9.1. Quadratic fields and integers

  9.2. Factorization in quadratic domains

  9.3. Gaussian integers

  9.4. Ideals and integral bases

  9.5. The semigroup of ideals

  9.6. Factorization of ideals

  9.7. Unique factorization and primes

  9.8. Quadratic residues

  9.9. Principal ideal domains

  9.10. Remarks

  9.11. Suggested reading

10. * MODULAR SYSTEMS

  10.1. The polynomial ring of J/(m)

  10.2. Zeros modulo a prime

  10.3. Zeros modulo a prime power

  10.4. Zeros modulo a composite

  10.5. Galois fields

  10.6. Automorphisms of a Galois field

  10.7 Suggested reading

11. MODULES AND VECTOR SPACES

  11.1. Definitions and examples

  11.2. Subspaces

  11.3. Linear independence and bases

  11.4. Dimension and isomorphism

  11.5. Row echelon form

  11.6. Uniqueness

  11.7. Systems of linear equations

  11.8. Column rank

  11.9. Suggested reading

12. LINEAR TRANSFORMATIONS AND MATRICES

  12.1. Homomorphisms and linear transformations

  12.2. Bases and matrices

  12.3. Addition

  12.4. Multiplication

  12.5. Rings of linear transformations of matrices

  12.6. Nonsingular matrices

  12.7. Change of basis

  12.8. * Ideals and algebras

  12.9 Suggested reading

13. ELEMENTARY THEORY OF MATRICES

  13.1. Special types of matrices

  13.2. A factorization

  13.3. On the right side

  13.4. Over a polynomial domain

  13.5. Determinants

  13.6. Determinant of a product

  13.7. Characteristic polynomial

  13.8. Triangularization and diagonalization

  13.9. Nilpotent matrices and transformations

  13.10. Jordan form

  13.11. Remarks

  13.12. Suggested reading

GENERAL REFERENCES

INDEX

Product Details

ISBN:
9780486688886
Author:
Deskins, W. E.
Publisher:
Dover Publications
Author:
Mathematics
Location:
New York :
Subject:
Algebra
Subject:
Algebra - General
Subject:
Algebra - Linear
Subject:
Algebras, linear
Subject:
Algebra - Abstract
Subject:
General Mathematics
Subject:
Mathematics - Algebra
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
68
Publication Date:
19960131
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
656
Dimensions:
8.5 x 5.38 in 1.58 lb

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Related Subjects


Reference » Science Reference » Technology
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Abstract Algebra (Dover Books on Mathematics) New Trade Paper
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Product details 656 pages Dover Publications - English 9780486688886 Reviews:
"Synopsis" by ,
Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.
"Synopsis" by ,
Excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. Features many examples and problems.

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