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