Book:W.E. Deskins/Abstract Algebra

Subject Matter

 * Abstract Algebra

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. Uniquefactorization 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

An asterisk (*) indicates sections or chapters containing material which is not essential to the understanding of the principal ideas in subsequent chapters.