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Abstract Algebra (Dover Books on Mathematics)by W. E. Deskins
Synopses & ReviewsPublisher 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 ContentsPREFACE
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. Wellordering 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 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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