Description

Book Synopsis

Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialised work by emphasising an understanding of the nature of algebraic structures.



Table of Contents
  • 0. Sets and Relations.
  • I. GROUPS AND SUBGROUPS.
  • 1. Introduction and Examples.
  • 2. Binary Operations.
  • 3. Isomorphic Binary Structures.
  • 4. Groups.
  • 5. Subgroups.
  • 6. Cyclic Groups.
  • 7. Generators and Cayley Digraphs.
  • I. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS.
  • 8. Groups of Permutations.
  • 9. Orbits, Cycles, and the Alternating Groups.
  • 10. Cosets and the Theorem of Lagrange.
  • 11. Direct Products and Finitely Generated Abelian Groups.
  • 12. Plane Isometries.
  • III. HOMOMORPHISMS AND FACTOR GROUPS.
  • 13. Homomorphisms.
  • 14. Factor Groups.
  • 15. Factor-Group Computations and Simple Groups.
  • 16. Group Action on a Set.
  • 17. Applications of G-Sets to Counting.
  • IV. RINGS AND FIELDS.
  • 18. Rings and Fields.
  • 19. Integral Domains.
  • 20. Fermat's and Euler's Theorems.
  • 21. The Field of Quotients of an Integral Domain.
  • 22. Rings of Polynomials.
  • 23. Factorization of Polynomials over a Field.
  • 24. Noncommutative Examples.
  • 25. Ordered Rings and Fields.
  • V. IDEALS AND FACTOR RINGS.
  • 26. Homomorphisms and Factor Rings.
  • 27. Prime and Maximal Ideas.
  • 28. Gröbner Bases for Ideals.
  • VI. EXTENSION FIELDS.
  • 29. Introduction to Extension Fields.
  • 30. Vector Spaces.
  • 31. Algebraic Extensions.
  • 32. Geometric Constructions.
  • 33. Finite Fields.
  • VII. ADVANCED GROUP THEORY.
  • 34. Isomorphism Theorems.
  • 35. Series of Groups.
  • 36. Sylow Theorems.
  • 37. Applications of the Sylow Theory.
  • 38. Free Abelian Groups.
  • 39. Free Groups.
  • 40. Group Presentations.
  • VIII.. AUTOMORPHISMS AND GALOIS THEORY.
  • 41. Automorphisms of Fields.
  • 42. The Isomorphism Extension Theorem.
  • 43. Splitting Fields.
  • 44. Separable Extensions.
  • 45. Totally Inseparable Extensions.
  • 46. Galois Theory.
  • 47. Illustrations of Galois Theory.
  • 48. Cyclotomic Extensions.
  • 49. Insolvability of the Quintic.
  • Appendix: Matrix Algebra.
  • Notations.
  • Index.

First Course in Abstract Algebra A

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    A Paperback by John B. Fraleigh

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      Publisher: Pearson Education
      Publication Date: Publication Date: 7/30/2013 12:00:00 AM
      ISBN13: 9781292024967, 978-1292024967
      ISBN10: 1292024968
      Also in:
      Mathematics Algebra

      Description

      Book Synopsis

      Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialised work by emphasising an understanding of the nature of algebraic structures.



      Table of Contents
      • 0. Sets and Relations.
      • I. GROUPS AND SUBGROUPS.
      • 1. Introduction and Examples.
      • 2. Binary Operations.
      • 3. Isomorphic Binary Structures.
      • 4. Groups.
      • 5. Subgroups.
      • 6. Cyclic Groups.
      • 7. Generators and Cayley Digraphs.
      • I. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS.
      • 8. Groups of Permutations.
      • 9. Orbits, Cycles, and the Alternating Groups.
      • 10. Cosets and the Theorem of Lagrange.
      • 11. Direct Products and Finitely Generated Abelian Groups.
      • 12. Plane Isometries.
      • III. HOMOMORPHISMS AND FACTOR GROUPS.
      • 13. Homomorphisms.
      • 14. Factor Groups.
      • 15. Factor-Group Computations and Simple Groups.
      • 16. Group Action on a Set.
      • 17. Applications of G-Sets to Counting.
      • IV. RINGS AND FIELDS.
      • 18. Rings and Fields.
      • 19. Integral Domains.
      • 20. Fermat's and Euler's Theorems.
      • 21. The Field of Quotients of an Integral Domain.
      • 22. Rings of Polynomials.
      • 23. Factorization of Polynomials over a Field.
      • 24. Noncommutative Examples.
      • 25. Ordered Rings and Fields.
      • V. IDEALS AND FACTOR RINGS.
      • 26. Homomorphisms and Factor Rings.
      • 27. Prime and Maximal Ideas.
      • 28. Gröbner Bases for Ideals.
      • VI. EXTENSION FIELDS.
      • 29. Introduction to Extension Fields.
      • 30. Vector Spaces.
      • 31. Algebraic Extensions.
      • 32. Geometric Constructions.
      • 33. Finite Fields.
      • VII. ADVANCED GROUP THEORY.
      • 34. Isomorphism Theorems.
      • 35. Series of Groups.
      • 36. Sylow Theorems.
      • 37. Applications of the Sylow Theory.
      • 38. Free Abelian Groups.
      • 39. Free Groups.
      • 40. Group Presentations.
      • VIII.. AUTOMORPHISMS AND GALOIS THEORY.
      • 41. Automorphisms of Fields.
      • 42. The Isomorphism Extension Theorem.
      • 43. Splitting Fields.
      • 44. Separable Extensions.
      • 45. Totally Inseparable Extensions.
      • 46. Galois Theory.
      • 47. Illustrations of Galois Theory.
      • 48. Cyclotomic Extensions.
      • 49. Insolvability of the Quintic.
      • Appendix: Matrix Algebra.
      • Notations.
      • Index.

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