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.
Author(s): W. E. Deskins
Series: Dover Books on Mathematics Series
Edition: 1
Publisher: Dover
Year: 1995
Language: English
Pages: 917
City: New York
Tags: Abstract Algebra, Algebra
Title Page
Copyright Page
Dedication
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 and 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
