9780136731627

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For courses in Abstract Algebra.

A comprehensive approach to abstract algebra

A First Course in Abstract Algebra, 8th Edition retains its hallmark goal of covering all the topics needed for an in-depth introduction to abstract algebra — and is designed to be relevant to future graduate students, future high school teachers, and students who intend to work in industry. New co-author Neal Brand has revised this classic text carefully and thoughtfully, drawing on years of experience teaching the course with this text to produce a meaningful and worthwhile update. This in-depth introduction gives students a firm foundation for more specialized work in algebra by including extensive explanations of the what, the how, and the why behind each method the authors choose. This revision also includes applied topics such as RSA encryption and coding theory, as well as examples of applying Gröbner bases.

0136731627 / 9780136731627 A FIRST COURSE IN ABSTRACT ALGEBRA [RENTAL EDITION], 8/e

Instructor's Preface

Dependence Chart

Student's Preface

0. Sets and Relations

I. GROUPS AND SUBGROUPS

1. Binary Operations

2. Groups

3. Abelian Groups

4. Nonabelian Examples

5. Subgroups

6. Cyclic Groups

7. Generating Sets and Cayley Digraphs

II. STRUCTURE OF GROUPS

8. Groups and Permutations

9. Finitely Generated Abelian Groups

10. Cosets and the Theorem of Lagrange

11. Plane Isometries

III. HOMOMORPHISMS AND FACTOR GROUPS

12. Factor Groups

13. Factor-Group Computations and Simple Groups

14. Groups Actions on a Set

15. Applications of G -Sets to Counting

IV. ADVANCED GROUP THEORY

16. Isomorphism Theorems

17. Sylow Theorems

18. Series of Groups

19. Free Abelian Groups

20. Free Groups

21. Group Presentations

V. RINGS AND FIELDS

22. Rings and Fields

23. Integral Domains

24. Fermat's and Euler's Theorems

25. Encryption

VI. CONSTRUCTING RINGS AND FIELDS

26. The Field of Quotients of an Integral Domain

27. Rings and Polynomials

28. Factorization of Polynomials over Fields

29. Algebraic Coding Theory

30. Homomorphisms and Factor Rings

31. Prime and Maximal Ideals

32. Noncommutative Examples

VII. COMMUTATIVE ALGEBRA

33. Vector Spaces

34. Unique Factorization Domains

35. Euclidean Domains

36. Number Theory

37. Algebraic Geometry

38. Gröbner Basis for Ideals

VIII. EXTENSION FIELDS

39. Introduction to Extension Fields

40. Algebraic Extensions

41. Geometric Constructions

42. Finite Fields

IX. Galois Theory

43. Introduction to Galois Theory

44. Splitting Fields

45. Separable Extensions

46. Galois Theory

47. Illustrations of Galois Theory

48. Cyclotomic Extensions

49. Insolvability of the Quintic

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First Course in Abstract Algebra, A [Rental Edition]

First Course in Abstract Algebra, A [Rental Edition]

0136731627

Summary

Table of Contents

Supplemental Materials