A First Course in Abstract Algebra

by John Fraleigh

Overview

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

KEY TOPICS: Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Gröbner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra

MARKET: For all readers interested in abstract algebra.

Product Details

ISBN-13:	9780201763904
Publisher:	Pearson Education
Publication date:	11/06/2002
Series:	Featured Titles for Abstract Algebra Series
Edition description:	REV
Pages:	544
Product dimensions:	7.85(w) x 9.50(h) x 1.00(d)

Table of Contents

(*) Not required for the remainder of the text. (**) This section is required only for Chapters 17 and 36.).

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.

II. 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. *GROUPS IN TOPOLOGY.

41. Simplicial Complexes and Homology Groups.

42. Computations of Homology Groups.

43. More Homology Computations and Applications.

44. Homological Algebra.

IX. Factorization.

45. Unique Factorization Domains.

46. Euclidean Domains.

47. Gaussian Integers and Multiplicative Norms.

X. AUTOMORPHISMS AND GALOIS THEORY.

48. Automorphisms of Fields.

49. The Isomorphism Extension Theorem.

50. Splitting Fields.

51. Separable Extensions.

52. *Totally Inseparable Extensions.

53. Galois Theory.

54. Illustrations of Galois Theory.

55. Cyclotomic Extensions.

56. Insolvability of the Quintic.

Appendix: Matrix Algebra.

Notations.

Answers to odd-numbered exercises not asking for definitions or proofs.

Index.

Editorial Reviews

An introductory textbook for the first course in abstract algebra. This edition (4th ed., 1989) includes applications to coding, finite- state machines (automata), graph theory, and isometry groups of the plane with the attendant Escher art works. A number of new examples and exercises are drawn from linear algebra, which many students have studied. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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