SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. SetsA andARelations 1
I. Groups and Subgroups
A A
2. IntroductionA andA Examples 4
3. BinaryA Operations 7
4. IsomorphicA BinaryA Structures 9
5. Groups 13
6. Subgroups 17
7. CyclicAA Groups 21
8. GeneratorsA andA CayleyA Digraphs 24
II. Permutations, Cosets, and Direct Products
A A A A
9. GroupsAofAPermutations 26
10. Orbits,ACycles,AandAtheAAlternatingAGroups
30
11. CosetsAandAtheATheoremA ofALagrange 34
12. DirectA ProductsA andA FinitelyA GeneratedA AbelianA Groups 37
13. PlaneA Isometries 42
III. Homomorphisms and Factor Groups A A A
14. Homomorphisms 44
15. FactorA Groups 49
16. Factor-GroupA ComputationsA andA SimpleA Groups 53
17. GroupAActionAonAaASet 58
18. ApplicationsAofAG-SetsAtoACounting 61
IV. Rings and Fields
A A
19. RingsAandAFields 63
20. IntegralA Domains 68
21. Fermat’sA andA Euler’sA Theorems 72
22. TheA FieldA ofA QuotientsA ofA anA IntegralA Domain 74
23. RingsA ofA Polynomials 76
24. FactorizationAofAPolynomialsAoverAaAField 79
25. NoncommutativeAExamples 85
26. OrderedA RingsA andA Fields 87
V. Ideals and Factor Rings
A A A
27. HomomorphismsAandAFactorARings 89
28. PrimeAandAMaximalAIdeals 94
,29. GröbnerABasesAforAIdeals 99
, VI. Extension Fields A
30. IntroductionAtoAExtensionAFields 103
31. VectorA Spaces 107
32. AlgebraicA Extensions 111
33. GeometricAConstructions 115
34. FiniteA Fields 116
VII. Advanced Group Theory A A
35. IsomorphismATheorems 117
36. SeriesAofAGroups 119
37. SylowA Theorems 122
38. ApplicationsA ofA theA SylowA Theory 124
39. FreeA AbelianA Groups 128
40. FreeAGroups 130
41. GroupA Presentations 133
VIII. Groups in TopologyA A
42. SimplicialA ComplexesA andA HomologyA Groups 136
43. ComputationsAofAHomologyAGroups 138
44. MoreAHomologyAComputationsAandAApplications 140
45. HomologicalAAlgebra 144
IX. Factorization
46. UniqueA FactorizationA Domains 148
47. EuclideanA Domains 151
48. GaussianA IntegersA andA MultiplicativeA Norms 154
X. Automorphisms and Galois Theory
A A A
49. AutomorphismsAofAFields 159
50. TheA IsomorphismA ExtensionA Theorem 164
51. SplittingA Fields 165
52. SeparableAExtensions 167
53. TotallyAInseparableAExtensions 171
54. GaloisA Theory 173
55. IllustrationsAofAGaloisATheory 176
56. CyclotomicAExtensions 183
57. InsolvabilityA ofA theA Quintic 185
APPENDIXAA MatrixAA Algebra 187
iv
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. SetsA andARelations 1
I. Groups and Subgroups
A A
2. IntroductionA andA Examples 4
3. BinaryA Operations 7
4. IsomorphicA BinaryA Structures 9
5. Groups 13
6. Subgroups 17
7. CyclicAA Groups 21
8. GeneratorsA andA CayleyA Digraphs 24
II. Permutations, Cosets, and Direct Products
A A A A
9. GroupsAofAPermutations 26
10. Orbits,ACycles,AandAtheAAlternatingAGroups
30
11. CosetsAandAtheATheoremA ofALagrange 34
12. DirectA ProductsA andA FinitelyA GeneratedA AbelianA Groups 37
13. PlaneA Isometries 42
III. Homomorphisms and Factor Groups A A A
14. Homomorphisms 44
15. FactorA Groups 49
16. Factor-GroupA ComputationsA andA SimpleA Groups 53
17. GroupAActionAonAaASet 58
18. ApplicationsAofAG-SetsAtoACounting 61
IV. Rings and Fields
A A
19. RingsAandAFields 63
20. IntegralA Domains 68
21. Fermat’sA andA Euler’sA Theorems 72
22. TheA FieldA ofA QuotientsA ofA anA IntegralA Domain 74
23. RingsA ofA Polynomials 76
24. FactorizationAofAPolynomialsAoverAaAField 79
25. NoncommutativeAExamples 85
26. OrderedA RingsA andA Fields 87
V. Ideals and Factor Rings
A A A
27. HomomorphismsAandAFactorARings 89
28. PrimeAandAMaximalAIdeals 94
,29. GröbnerABasesAforAIdeals 99
, VI. Extension Fields A
30. IntroductionAtoAExtensionAFields 103
31. VectorA Spaces 107
32. AlgebraicA Extensions 111
33. GeometricAConstructions 115
34. FiniteA Fields 116
VII. Advanced Group Theory A A
35. IsomorphismATheorems 117
36. SeriesAofAGroups 119
37. SylowA Theorems 122
38. ApplicationsA ofA theA SylowA Theory 124
39. FreeA AbelianA Groups 128
40. FreeAGroups 130
41. GroupA Presentations 133
VIII. Groups in TopologyA A
42. SimplicialA ComplexesA andA HomologyA Groups 136
43. ComputationsAofAHomologyAGroups 138
44. MoreAHomologyAComputationsAandAApplications 140
45. HomologicalAAlgebra 144
IX. Factorization
46. UniqueA FactorizationA Domains 148
47. EuclideanA Domains 151
48. GaussianA IntegersA andA MultiplicativeA Norms 154
X. Automorphisms and Galois Theory
A A A
49. AutomorphismsAofAFields 159
50. TheA IsomorphismA ExtensionA Theorem 164
51. SplittingA Fields 165
52. SeparableAExtensions 167
53. TotallyAInseparableAExtensions 171
54. GaloisA Theory 173
55. IllustrationsAofAGaloisATheory 176
56. CyclotomicAExtensions 183
57. InsolvabilityA ofA theA Quintic 185
APPENDIXAA MatrixAA Algebra 187
iv
