Book:Seth Warner/Modern Algebra

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Seth Warner: Modern Algebra

Published $1965$, Dover Publications, Inc.

ISBN 0-486-66341-8.


Subject Matter


Contents

  • Preface


  • CHAPTER 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 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 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 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 (PAGES 1-457)
  • LIST OF SYMBOLS (PAGES 459-797)


  • INDEX (PAGES 1-457)
  • INDEX (PAGES 459-797)