# Book:W.E. Deskins/Abstract Algebra

## W.E. Deskins: Abstract Algebra

Published $\text {1964}$, Dover Publications, Inc.

ISBN 0-486-68888-7.

### 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
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
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
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
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
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
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
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
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
9.9. Principal ideal domains
9.10. Remarks
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
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
12. LINEAR TRANSFORMATIONS AND MATRICES
12.1. Homomorphisms and linear transformations
12.2. Bases and matrices
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
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