Introduction to Algebra

Title
Preface
Contents
Index of common symbols
Greek alphabet
I. Fundamental Concepts
	1.1. Sets
	1.2. Union and Intersection
	1.3. Mappings
	1.4. One-to-One Maps
	1.5. Products
	1.6. Operations
	1.7. On Statements and Theorems
	1.8. Relations
	1.9. Equivalence Relations and Partitions
	1.10. Some Information on Integers
	1.11. Generalized Associativity
II. Linear Equations and Matrices
	2.1. Closure of Subsets Under an Operation
	2.2. Integral Primes
	2.3. Fields of Complex Numbers
	2.4. Linear Equations Over a Field
	2.5. Vector Spaces of n-tuples
	2.6. Subspaces
	2.7. Linear Dependence
	2.8. Linear Systems and Matrices
	2.9. Matrices and Elementary Row Operations
	2.10. Application to Linear Systems
	2.11. Matrix Operations
	2.12. Elementary Properties of Matrices
	2.13. Linear Systems Reexamined
	2.14. Special Matrices
	2.15. The Transpose
	2.16. Column Operations and Equivalence
	2.17. Nonsingularity
	2.18. Exponents in Associative Systems
III. Groups
	3.1. The Group Concept
	3.2. Groups of Mappings
	3.3. Permutation Groups
	3.4. Even and Odd Permutations
	3.5. Elementary Properties
	3.6. Subgroups
	3.7. Generators
	3.8. Cyclic Groups
	3.9. Integers Modulo m
	3.10. Coset Decomposition
	3.11. Some Consequences
	3.12. Isomorphism
IV. Rings
	4.1. Definitions and Additive Properties
	4.2. Properties Involving Multiplication
	4.3. Integers modulo n
	4.4. Natural Multiples
	4.5. Zero-Divisors and Characteristic
	4.6. Subrings
	4.7. Quaternions
	4.8. Polynomials Over a Ring: The Concept
	4.9. Polynomials Over a Ring: Existence
	4.10. Indeterminates
	4.11. A Generalization of Integers Modulo n
	4.12. Isomorphism
	4.13. Homomorphisms
	4.14. The Kernel
	4.15. Direct Sums
V. Integral Domains
	5.1. Cancellation and Zero-Divisors
	5.2. Subdomains
	5.3. Isomorphism and Characteristic
	5.4. Binomial Formula
	5.5. Polynomials
	5.6. Positive Elements
	5.7. Ordering a Domain
	5.8. Greater Than
	5.9. Well-ordering
	5.10. Induction
	5.11. The Integers
VI. Fields
	6.1. The Universal Division Property
	6.2. Construction of Quotients
	6.3. Quotient Fields
	6.4. Extension of Order
	6.5. Fields in General
	6.6. Subfields
	6.7. Quotient Field of a Subdomain
	6.8. Prime Subfield
	6.9. Division Algorithm for Polynomials
VII. Divisibility
	7.1. The Basic Language of Divisibility
	7.2. Greatest Common Divisors
	7.3. The Domain of Integers
	7.4. Unique Factorization of Integers
	7.5. Computational Matters
	7.6. A New Look at GCDs
	7.7. GCDs for Polynomials
	7.8. Unique Factorization for Polynomials
	7.9. Generalizations
VIII. Classical Algebra
	8.1. The Real Problem
	8.2. Order in the Reals
	8.3. Complex Numbers
	8.4. Algebraic Closure
	8.5. Roots of Polynomials
	8.6. Multiplicity
	8.7. Isomorphism over R
	8.8. The Complex Plane
	8.9. Roots of Complex Numbers
	8.10. Irreducibility
	8.11. Theorems on Integers
IX. Vector Spaces
	9.1. The Concept
	9.2. Subspaces
	9.3. Linear Dependence
	9.4. Bases
	9.5. Minimal and Maximal Properties
	9.6. Dimension
	9.7. Coordinates
	9.8. Isomorphism
	9.9. Row Spaces
	9.10. Echelon Bases
	9.11. Rank
	9.12. Equivalence
	9.13. Bilinear Forms
	9.14. The Meaning of Canonical Sets
	9.15. Linear Systems
	9.16. Minimal Polynomial
X. Extension Fields
	10.1. Quadratic Cases
	10.2. Generation of Intermediate Fields
	10.3. Algebraic Elements
	10.4. Simple Algebraic Extensions: First View
	10.5. Extensions as Vector Spaces
	10.6. Successive Extensions
	10.7. Constructible Numbers
	10.8. Trisection of Angles
	10.9. Simple Algebraic Extensions: Second View
	10.10. Construction of Extension Fields
	10.11. Root Fields
	10.12. Finite Fields
	10.13. Algebraic Integers
	10.14. Factorization of Algebraic Integers
XI. Determinants
	11.1. The Dimension
	11.2. Some Important Types
	11.3. Two Basic Properties
	11.4. Functions of n Vectors
	11.5. Elementary Operations and Products
	11.6. Cofactors
	11.7. Vandermonde's Matrix
	11.8. The Adjoint
	11.9. Cramer's Rule
	11.10. Determinants and Rank
	11.11. The Cayley-Hamilton Theorem
XII. Linear Transformations
	12.1. Definition and Examples
	12.2. Finite-Dimensional Domains
	12.3. The Image Space
	12.4. The Kernel
	12.5. Non-Singularity
	12.6. Sums of Linear Transformations
	12.7. Multiplication by Scalars
	12.8. Products
	12.9. Representation by Matrices
	12.10. Matrix Calculation of Transformations
	12.11. Matrix Interpretations
	12.12. Change of Basis
	12.13. Rotations in V_2(R)
	12.14. An Application of Similarity
XIII. Forms and Matrices
	13.1. Forms
	13.2. Connection with Vectors
	13.3. Substitutions and Change of Basis
	13.4. Diagonalization
	13.5. Real Symmetric Matrices
	13.6. Semi-Definite Matrices and Forms
XIV. Length and Orthogonality
	14.1. Length in V_n(R)
	14.2. Angles in V_n(R)
	14.3. Orthogonal Matrices
	14.4. Projections and Orthogonal Complements
	14.5. Orthonormal Bases
	14.6. Projection on a Subspace
	14.7. Characteristic Vectors and Roots
	14.8. Triangular and Diagonal Matrices
	14.9. Diagonalization Criteria
	14.10. Distinct Characteristic Roots
	14.11. Algebraic and Geometric Multiplicities
	14.12. Invariants and Criteria
	14.13. Real Quadratic Forms
	14.14. Pairs of Real Quadratic Forms
Index