Book:Seth Warner/Modern Algebra

Subject Matter

 * Set Theory
 * Abstract Algebra

Contents

 * Preface


 * CHAPTER $\text {I}$. ALGEBRAIC STRUCTURES
 * 1. The Language of Set Theory
 * 2. Compositions
 * 3. Unions and Intersections of Sets
 * 4. Neutral Elements and Inverses
 * 5. Compositions and Inverses of Functions
 * 6. Isomorphisms of Algebraic Structure
 * 7. Semigroups and Groups


 * CHAPTER $\text {II}$. 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 $\text {III}$. THE NATURAL NUMBERS
 * 14. Orderings
 * 15. Ordered Semigroups
 * 16. The Natural Numbers
 * 17. Finite Sets
 * 18. Induced $N$-ary Operations
 * 19. Combinatorial Analysis


 * CHAPTER $\text {IV}$. 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 $\text {V}$. 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 $\text {VI}$. 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 $\text {VII}$. THE REAL AND COMPLEX NUMBER FIELDS
 * 41. Dedekind and Archimedean Ordered Fields
 * 42. The Construction of a Dedekind Ordered Field
 * 43. Isomorphisms of Archimedean Ordered Groups
 * 44. The Field of Complex Numbers
 * 45. The Algebra of Quaternions


 * CHAPTER $\text {VIII}$. 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 $\text {IX}$. LINEAR OPERATORS
 * 54. Diagonalizable Operators
 * 55. Primary and Torsion-free Modules
 * 56. Finitely Generated Modules
 * 57. Decompositions of Linear Operators
 * 58. Determinants


 * CHAPTER $\text {X}$. INNER PRODUCT SPACES
 * 59. Inner Products
 * 60. Orthonormal Bases
 * 61. Adjoints
 * 62. The Spectral Theorem
 * 63. Linear Operators on Inner Product Spaces


 * CHAPTER $\text {XI}$. THE AXIOM OF CHOICE
 * 64. The Axiom of Choice
 * 65. Zorn's Lemma
 * 66. Algebraic Closures


 * LIST OF SYMBOLS (PAGES 1-457)


 * LIST OF SYMBOLS (PAGES 459-797)


 * INDEX (PAGES 1-457)


 * INDEX (PAGES 459-797)



Critical View
Thorough, but many of the exercises are busy-work, and hence are not documented on.

Source work progress
* : $\S 30$: Transpose of Row Matrix is Column Matrix


 * Another pass as follows:


 * : Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.3$


 * The above example has several subexamples, some of which have been glossed over. Needs to be rationalised.


 * There is a missing result in the below:


 * : Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 1^\circ$