First Course in Abstract Algebra, A [Rental Edition]

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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