Table of Contents

Chapter I. Algebraic Structures
1. The Language of Set Theory
2. Compositions
3. Unions and Intersections of Sets
4. Neutral Elements and Inverses
5. Composites and Inverses of Functions
6. Isomorphisms of Algebraic Structures
7. Semigroups and Groups
Chapter II. New Structures from Old
8. Compositions Induced on Subsets
9. Compositions Induced on the Set of All Subsets
10. Equivalence Relations
11. Quotient Structures
12. Homomorphisms
13. Compositions Induced on Cartesian Products and Function Spaces
Chapter III. The Natural Numbers
14. Orderings
15. Ordered Semigroups
16. The Natural Numbers
17. Finite Sets
18. Induced N-ary Operations
19. Combinatorial Analysis
Chapter IV. Rings and Fields
20. The Integers
21. Rings and Integral Domains
22. New Rings from Old
23. The Field of Rational Numbers
24. The Division Algorithm
25. Cyclic Groups and Lagrange's Theorem
Chapter V. Vector Spaces
26. Vector Spaces and Modules
27. Subspaces and Bases
28. Linear Transformations
29. Matrices
30. Linear Equations
31. Direct Sums and Quotient Spaces
32. Rings of Linear Operators
Chapter VI. Polynomials
33. Algebras
34. The Algebra of Polynomials
35. Principal Ideal Domains
36. Substitution
37. Irreducibility Criteria
38. Adjoining Roots
39. Finite Fields and Division Rings
40. Polynomials in Several Indeterminates
Chapter VII. The Real and Complex Number Fields
41. Dedekind and Archimedian Ordered Fields
42. The Construction of a Dedekind Ordered Field
43. Isomorphisms of Archimedian Ordered Groups
44. The field of Complex Numbers
45. The Algebra of Quaternions
Chapter VIII. Algebraic Extensions of Fields
46. Algebraic Extensions
47. Constructions by Ruler and Compass
48. Galois Theory
49. Separable and Normal Extensions
50. Roots of Unity
51. Solving Quadratics, Cubics, and Quartics
52. Permutation Groups
53. Solving Polynomials by Radicals
Chapter IX. Linear Operators
54. Diagonalizable Operators
55. Primary and Torsion-free Modules
56. Finitely Generated Modules
57. Decompositions of Linear Operators
58. Determinants
Chapter X. Inner Product Spaces
59. Inner Products
60. Orthonormal Bases
61. Adjoints
62. The Spectral Theorem
63. Linear Operators on Inner Product Spaces
Chapter XI. The Axiom of Choice
64. The Axiom of Choice
65. Zorn's Lemma
66. Algebraic Closures
List of Symbols, Index