SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Setsj andj Relations 1
I. Groups and Subgroups
j j
2. Introductionj andj Examples 4
3. Binaryj Operations 7
4. Isomorphicj Binaryj Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclicjj Groups 21
8. Generatorsj andj Cayleyj Digraphs 24
II. Permutations, Cosets, and Direct Products
j j j j
9. Groupsj ofjPermutations 26
10. Orbits,jCycles,jandjthejAlternatingjGroups
30
11. Cosetsj andjthej Theoremj ofj Lagrange 34
12. Directj Productsj andj Finitelyj Generatedj Abelianj Groups 37
13. Planej Isometries 42
III. Homomorphisms and Factor Groups j j j
14. Homomorphisms 44
15. Factorj Groups 49
16. Factor-Groupj Computationsj andj Simplej Groups 53
17. GroupjActionjonjajSet 58
18. ApplicationsjofjG-SetsjtojCounting 61
IV. Rings and Fields
j j
19. RingsjandjFields 63
20. Integralj Domains 68
21. Fermat’sj andj Euler’sj Theorems 72
22. Thej Fieldj ofj Quotientsj ofj anj Integralj Domain 74
23. Ringsj ofj Polynomials 76
24. FactorizationjofjPolynomialsjoverjajField 79
25. NoncommutativejExamples 85
26. Orderedj Ringsj andj Fields 87
V. Ideals and Factor Rings
j j j
27. HomomorphismsjandjFactorjRings 89
28. PrimejandjMaximaljIdeals 94
,29. GröbnerjBasesjforjIdeals 99
, VI. Extension Fields j
30. IntroductionjtojExtensionjFields 103
31. Vectorj Spaces 107
32. Algebraicj Extensions 111
33. GeometricjConstructions 115
34. Finitej Fields 116
VII. Advanced Group Theoryj j
35. IsomorphismjTheorems 117
36. SeriesjofjGroups 119
37. Sylowj Theorems 122
38. Applicationsj ofj thej Sylowj Theory 124
39. Freej Abelianj Groups 128
40. FreejGroups 130
41. Groupj Presentations 133
VIII. Groups in Topology j j
42. Simplicialj Complexesj andj Homologyj Groups 136
43. Computationsjofj HomologyjGroups 138
44. MorejHomologyjComputationsjandjApplications 140
45. HomologicaljAlgebra 144
IX. Factorization
46. Uniquej Factorizationj Domains 148
47. Euclideanj Domains 151
48. Gaussianj Integersj andj Multiplicativej Norms 154
X. Automorphisms and Galois Theory
j j j
49. AutomorphismsjofjFields 159
50. Thej Isomorphismj Extensionj Theorem 164
51. Splittingj Fields 165
52. SeparablejExtensions 167
53. TotallyjInseparablejExtensions 171
54. Galoisj Theory 173
55. IllustrationsjofjGaloisjTheory 176
56. CyclotomicjExtensions 183
57. Insolvabilityj ofj thej Quintic 185
APPENDIXjj Matrixjj Algebra 187
iv
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Setsj andj Relations 1
I. Groups and Subgroups
j j
2. Introductionj andj Examples 4
3. Binaryj Operations 7
4. Isomorphicj Binaryj Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclicjj Groups 21
8. Generatorsj andj Cayleyj Digraphs 24
II. Permutations, Cosets, and Direct Products
j j j j
9. Groupsj ofjPermutations 26
10. Orbits,jCycles,jandjthejAlternatingjGroups
30
11. Cosetsj andjthej Theoremj ofj Lagrange 34
12. Directj Productsj andj Finitelyj Generatedj Abelianj Groups 37
13. Planej Isometries 42
III. Homomorphisms and Factor Groups j j j
14. Homomorphisms 44
15. Factorj Groups 49
16. Factor-Groupj Computationsj andj Simplej Groups 53
17. GroupjActionjonjajSet 58
18. ApplicationsjofjG-SetsjtojCounting 61
IV. Rings and Fields
j j
19. RingsjandjFields 63
20. Integralj Domains 68
21. Fermat’sj andj Euler’sj Theorems 72
22. Thej Fieldj ofj Quotientsj ofj anj Integralj Domain 74
23. Ringsj ofj Polynomials 76
24. FactorizationjofjPolynomialsjoverjajField 79
25. NoncommutativejExamples 85
26. Orderedj Ringsj andj Fields 87
V. Ideals and Factor Rings
j j j
27. HomomorphismsjandjFactorjRings 89
28. PrimejandjMaximaljIdeals 94
,29. GröbnerjBasesjforjIdeals 99
, VI. Extension Fields j
30. IntroductionjtojExtensionjFields 103
31. Vectorj Spaces 107
32. Algebraicj Extensions 111
33. GeometricjConstructions 115
34. Finitej Fields 116
VII. Advanced Group Theoryj j
35. IsomorphismjTheorems 117
36. SeriesjofjGroups 119
37. Sylowj Theorems 122
38. Applicationsj ofj thej Sylowj Theory 124
39. Freej Abelianj Groups 128
40. FreejGroups 130
41. Groupj Presentations 133
VIII. Groups in Topology j j
42. Simplicialj Complexesj andj Homologyj Groups 136
43. Computationsjofj HomologyjGroups 138
44. MorejHomologyjComputationsjandjApplications 140
45. HomologicaljAlgebra 144
IX. Factorization
46. Uniquej Factorizationj Domains 148
47. Euclideanj Domains 151
48. Gaussianj Integersj andj Multiplicativej Norms 154
X. Automorphisms and Galois Theory
j j j
49. AutomorphismsjofjFields 159
50. Thej Isomorphismj Extensionj Theorem 164
51. Splittingj Fields 165
52. SeparablejExtensions 167
53. TotallyjInseparablejExtensions 171
54. Galoisj Theory 173
55. IllustrationsjofjGaloisjTheory 176
56. CyclotomicjExtensions 183
57. Insolvabilityj ofj thej Quintic 185
APPENDIXjj Matrixjj Algebra 187
iv
