Book:W.E. Deskins/Abstract Algebra
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W.E. Deskins: Abstract Algebra
Published $\text {1964}$, Dover Publications, Inc.
- ISBN 0-486-68888-7
Subject Matter
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.
Source work progress
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.5$: Corollary $2.25.1$
- Redo from start
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Chapter $1$: A Common Language: $\S 1.1$ Sets
- Some gaps to fill just before here