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Instructor’s Solutions Manual
to accompany

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 lengthy complete solutions are given in the answers at the back of the text. Then reference is simply given to the textanswers 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 J.B.F

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CONTENTS
0. Sets and Relations 1

I. Groupsand Subgroups
1. 2. 3. 4. 5. 6. 7. Introduction and Examples 4 Binary Operations 7 Isomorphic Binary Structures 9 Groups 13 Subgroups 17 Cyclic Groups 21 Generators and Cayley Digraphs 24

II. Permutations, Cosets, and Direct Products
8. 9. 10. 11. 12. Groups of Permutations 26 Orbits, Cycles, and the Alternating Groups 30 Cosets and the Theorem of Lagrange 34 Direct Products and FinitelyGenerated Abelian Groups Plane Isometries 42

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III. Homomorphisms and Factor Groups
13. 14. 15. 16. 17. Homomorphisms 44 Factor Groups 49 Factor-Group Computations and Simple Groups Group Action on a Set 58 Applications of G-Sets to Counting 61

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IV. Rings and Fields
18. 19. 20. 21. 22. 23. 24. 25. Rings and Fields 63 Integral Domains 68 Fermat’s and Euler’s Theorems 72 The Field ofQuotients of an Integral Domain 74 Rings of Polynomials 76 Factorization of Polynomials over a Field 79 Noncommutative Examples 85 Ordered Rings and Fields 87

V. Ideals and Factor Rings
26. Homomorphisms and Factor Rings 27. Prime and Maximal Ideals 94 28. Gr¨bner Bases for Ideals 99 o 89

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VI. Extension Fields
29. 30. 31. 32. 33. Introduction to Extension Fields Vector Spaces 107Algebraic Extensions 111 Geometric Constructions 115 Finite Fields 116 103

VII. Advanced Group Theory
34. 35. 36. 37. 38. 39. 40. Isomorphism Theorems 117 Series of Groups 119 Sylow Theorems 122 Applications of the Sylow Theory Free Abelian Groups 128 Free Groups 130 Group Presentations 133

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VIII. Groups in Topology
41. 42. 43. 44. Simplicial Complexes and Homology Groups 136Computations of Homology Groups 138 More Homology Computations and Applications 140 Homological Algebra 144

IX. Factorization
45. Unique Factorization Domains 148 46. Euclidean Domains 151 47. Gaussian Integers and Multiplicative Norms

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X. Automorphisms and Galois Theory
48. 49. 50. 51. 52. 53. 54. 55. 56. Automorphisms of Fields 159 The Isomorphism Extension Theorem Splitting Fields 165Separable Extensions 167 Totally Inseparable Extensions 171 Galois Theory 173 Illustrations of Galois Theory 176 Cyclotomic Extensions 183 Insolvability of the Quintic 185 187 164

APPENDIX Matrix Algebra

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0. Sets and Relations

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0. Sets and Relations
√ √ 1. { 3, − 3} 2. The set is empty. 3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30, 60,−60} 4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} 5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every element exceeds only a finite number of other elements but is exceeded by an infinite number of other elements. Such people might claim the answer should be ∅.) 6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64. 9. Q

8. Itis not a well-defined set.

10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1, 1/2, or 1/3. 11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)} 12. a. It is a function. It is not one-to-one since there are two pairs with second member 4. It is not onto B because there is no pair with second member 2. b. (Same answer as...