SOLUTION tMANUAL
First tCourse tin tAbstract tAlgebra
tA t t
t8th tEdition tby tJohn tB.
tFraleigh t t
tAll tChapters tFull tComplete
,
, CONTENTS
1. Sets tand tRelations 1
I. Groups t and t Subgroups
2. Introduction tand tExamples 4
3. Binary t Operations 7
4. Isomorphic t Binary t Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic t tGroups 21
8. Generators tand tCayley tDigraphs 24
II. Permutations, tCosets, tand tDirect tProducts
9. Groups tof tPermutations 26
10. Orbits, tCycles, tand tthe tAlternating tGroups
30
11. Cosets tand tthe tTheorem tof tLagrange 34
12. Direct t Products t and t Finitely t Generated t Abelian t Groups 37
13. Plane tIsometries 42
III. Homomorphisms tand tFactor tGroups
14. Homomorphisms 44
15. Factor tGroups 49
16. Factor-Group tComputations t and tSimple tGroups 53
17. Group tAction ton ta tSet 58
18. Applications tof tG-Sets tto tCounting 61
IV. Rings t and t Fields
19. Rings tand tFields 63
20. Integral tDomains 68
21. Fermat’s tand tEuler’s tTheorems 72
22. The tField tof tQuotients tof tan tIntegral tDomain 74
23. Rings tof tPolynomials 76
24. Factorization tof tPolynomials tover ta tField 79
25. Noncommutative tExamples 85
26. Ordered tRings tand tFields 87
V. Ideals t and t Factor t Rings
27. Homomorphisms tand tFactor tRings 89
, 28. Prime tand tMaximal tIdeals 94
29. h
VI. Extension t Fields
30. Introduction tto tExtension tFields 103
31. Vector t Spaces 107
32. Algebraic t Extensions 111
33. Geometric tConstructions 115
34. Finite tFields 116
VII. Advanced tGroup tTheory
35. Isomorphism tTheorems 117
36. Series tof tGroups 119
37. Sylow tTheorems 122
38. Applications tof tthe tSylow tTheory 124
39. Free tAbelian tGroups 128
40. Free tGroups 130
41. Group tPresentations 133
VIII. Groups t in t Topology
42. Simplicial tComplexes tand tHomology tGroups 136
43. Computations tof tHomology tGroups 138
44. More tHomology tComputations tand tApplications 140
45. Homological tAlgebra 144
IX. Factorization
46. Unique tFactorization tDomains 148
47. Euclidean t Domains 151
48. Gaussian tIntegers tand tMultiplicative tNorms 154
X. Automorphisms t and t Galois t Theory
49. Automorphisms tof tFields 159
50. The tIsomorphism tExtension tTheorem 164
51. Splitting tFields 165
52. Separable tExtensions 167
53. Totally tInseparable tExtensions 171
54. Galois t Theory 173
55. Illustrations tof tGalois tTheory 176
56. Cyclotomic tExtensions 183
57. Insolvability tof tthe tQuintic 185
APPENDIX t tMatrix t tAlgebra 187
First tCourse tin tAbstract tAlgebra
tA t t
t8th tEdition tby tJohn tB.
tFraleigh t t
tAll tChapters tFull tComplete
,
, CONTENTS
1. Sets tand tRelations 1
I. Groups t and t Subgroups
2. Introduction tand tExamples 4
3. Binary t Operations 7
4. Isomorphic t Binary t Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic t tGroups 21
8. Generators tand tCayley tDigraphs 24
II. Permutations, tCosets, tand tDirect tProducts
9. Groups tof tPermutations 26
10. Orbits, tCycles, tand tthe tAlternating tGroups
30
11. Cosets tand tthe tTheorem tof tLagrange 34
12. Direct t Products t and t Finitely t Generated t Abelian t Groups 37
13. Plane tIsometries 42
III. Homomorphisms tand tFactor tGroups
14. Homomorphisms 44
15. Factor tGroups 49
16. Factor-Group tComputations t and tSimple tGroups 53
17. Group tAction ton ta tSet 58
18. Applications tof tG-Sets tto tCounting 61
IV. Rings t and t Fields
19. Rings tand tFields 63
20. Integral tDomains 68
21. Fermat’s tand tEuler’s tTheorems 72
22. The tField tof tQuotients tof tan tIntegral tDomain 74
23. Rings tof tPolynomials 76
24. Factorization tof tPolynomials tover ta tField 79
25. Noncommutative tExamples 85
26. Ordered tRings tand tFields 87
V. Ideals t and t Factor t Rings
27. Homomorphisms tand tFactor tRings 89
, 28. Prime tand tMaximal tIdeals 94
29. h
VI. Extension t Fields
30. Introduction tto tExtension tFields 103
31. Vector t Spaces 107
32. Algebraic t Extensions 111
33. Geometric tConstructions 115
34. Finite tFields 116
VII. Advanced tGroup tTheory
35. Isomorphism tTheorems 117
36. Series tof tGroups 119
37. Sylow tTheorems 122
38. Applications tof tthe tSylow tTheory 124
39. Free tAbelian tGroups 128
40. Free tGroups 130
41. Group tPresentations 133
VIII. Groups t in t Topology
42. Simplicial tComplexes tand tHomology tGroups 136
43. Computations tof tHomology tGroups 138
44. More tHomology tComputations tand tApplications 140
45. Homological tAlgebra 144
IX. Factorization
46. Unique tFactorization tDomains 148
47. Euclidean t Domains 151
48. Gaussian tIntegers tand tMultiplicative tNorms 154
X. Automorphisms t and t Galois t Theory
49. Automorphisms tof tFields 159
50. The tIsomorphism tExtension tTheorem 164
51. Splitting tFields 165
52. Separable tExtensions 167
53. Totally tInseparable tExtensions 171
54. Galois t Theory 173
55. Illustrations tof tGalois tTheory 176
56. Cyclotomic tExtensions 183
57. Insolvability tof tthe tQuintic 185
APPENDIX t tMatrix t tAlgebra 187
