Modern Algebra: (Dover Books on Mathematics)

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.

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