Abstract Algebra
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
1 online resource
9781306369077, 9780486158464, 130636907X, 0486158462
868966062
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Cover; 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. Remarks2.13. 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 systems5.4. *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 functions7.8. 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 ideals9.6. 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
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