文档介绍：Instructor’s
Solutions Manual
to pany
A First Course in
Abstract Algebra
Seventh Edition
John B. Fraleigh
University of Rhode Island
Preface
This manual contains solutions to all exercises in the text, except those odd-numbered exercises for which
fairly plete solutions are given in the answers at the back of the text. Then reference is simply
given to the text answers to save typing.
I prepared these solutions myself. While I tried to be accurate, there are sure to be the inevitable
mistakes and typos. An author reading proof rends to see what he or she wants to see. However, the
instructor should find this manual adequate for the purpose for which it is intended.
Morgan, Vermont
July, 2002
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CONTENTS
0. Sets and Relations 1
I. Groups and Subgroups
1. Introduction and Examples 4
2. Binary Operations 7
3. Isomorphic Binary Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
7. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
8. Groups of Permutations 26
9. Orbits, Cycles, and the Alternating Groups 30
10. Cosets and the Theorem of Lagrange 34
11. Direct Products and Finitely Generated Abelian Groups 37
12. Plane Isometries 42
III. Homomorphisms and Factor Groups
13. Homomorphisms 44
14. Factor Groups 49
15. Factor-putations and Simple Groups 53
16. Group Action on a Set 58
17. Applications of G-Sets to Counting 61
IV. Rings and Fields
18. Rings and Fields 63
19. Integral Domains 68
20. Fermat’s and Euler’s Theorems 72
21. The Field of Quotients of an Integral Domain 74
22. Rings of Polynomials 76
23. Factorization of Polynomials over a Field 79
24. mutative Examples 85
25. Ordered Rings and Fields 87
V. Ideals and Factor Rings
26. Homomorphisms and Factor Rings 89
27. Prime and Maximal Ideals 94
28. Gr¨obner Bases for Ideals 99
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VI. Extension Fields
29. Introduction to Extension Fields 103
30. Vector Spaces 107
31. Algebraic Extensions 11