Modern Algebra

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Overview

This standard text, written for junior and senior undergraduates, is unusual in that its presentation is accessible enough for the beginner, yet its thoroughness and mathematical rigor provide the more advanced student with an exceptionally comprehensive treatment of every aspect of modern algebra. It especially lends itself to use by beginning graduate students unprepared in modern algebra.
The presentation opens with a study of algebraic structures in general; the first part ...

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Modern Algebra

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Overview

This standard text, written for junior and senior undergraduates, is unusual in that its presentation is accessible enough for the beginner, yet its thoroughness and mathematical rigor provide the more advanced student with an exceptionally comprehensive treatment of every aspect of modern algebra. It especially lends itself to use by beginning graduate students unprepared in modern algebra.
The presentation opens with a study of algebraic structures in general; the first part then carries the development from natural numbers through rings and fields, vector spaces, and polynomials. The second part (originally published as a separate volume) is made up of five chapters on the real and complex number fields, algebraic extensions of fields, linear operations, inner product spaces, and the axiom of choice.
For the benefit of the beginner who can best absorb the principles of algebra by solving problems, the author has provided over 1300 carefully selected exercises. "There is a vast amount of material in these books and a great deal is either new or presented in a new form." — Mathematical Reviews. Preface. List of Symbols. Exercises. Index. 28 black-and-white line illustrations.

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Product Details

  • ISBN-13: 9780486663418
  • Publisher: Dover Publications
  • Publication date: 6/1/1990
  • Series: Dover Books on Mathematics Series
  • Edition description: Unabridged
  • Pages: 832
  • Product dimensions: 5.40 (w) x 8.41 (h) x 1.54 (d)

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
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Customer Reviews

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Sort by: Showing all of 4 Customer Reviews
  • Anonymous

    Posted January 10, 2001

    A must for every mathematics dep. library

    One of the best introductory algebra textbooks around.It is aimed at undergraduate students but even postgrads can benefit from it.I won't give a list of contents because it can be found in the review at Mathscinet.But I should add that it also covers constructions using ruler and compass plus the unsolvability of the three famous geometrical problems of antiquity. There is a huge number of exercises which form an integral part of the book.Most of the exercises are really summaries of proofs of important theorems.For example the Sylow theorems for groups are mentioned in the exercises. The book does not simply teach algebra it also teaches rigorous mathematical thinking and it manages to be understandable as well as formal.A student who will work carefully through the detailed proofs and write detailed solutions for a large number of exercises will gain huge amounts of mathematical maturity as well as knowledge of the specific topic. Special mention should be made of the final chapter on the axiom of choice.If a student feels puzzled as to what the axiom of choice says or when it should be used then this chapter will make things a lot clearer for them.Also in just 4 pages it presents a detailed proof of the equivalence of the axiom of choice with Zorn's lemma , the well ordering principle and Hausdorff's maximal chain principle.These proofs use just elementary concepts from set theory without any mention of ordinals or transfinite induction.The chapter also presents the main applications of the axiom of choice in algebra like the proof that every vector space has a basis or that every field can be extended to an algebraically closed field. Thoroughly recommended !!

    1 out of 1 people found this review helpful.

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    Posted January 31, 2010

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    Posted October 24, 2009

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    Posted September 26, 2009

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