SOLUTION MANUAL
t
First Course in Abstract
t t t
tAlgebra A 8th Edition by
t t t t
John B. Fraleigh
t t t t t
t All Chapters Full Complete
t t t
, CONTENTS
1. Sets t and tRelations 1
I. Groups and Subgroups
t t
2. Introduction t and t Examples 4
3. Binary t Operations 7
4. Isomorphic t Binary t Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic t t Groups 21
8. Generators t and t Cayley t Digraphs 24
II. Permutations, Cosets, and Direct Products
t t t t
9. Groups t of tPermutations 26
10. Orbits, tCycles, tand tthe tAlternating tGroups
30
11. Cosets tand tthe tTheorem t of tLagrange 34
12. Direct t Products t and t Finitely t Generated t Abelian t Groups 37
13. Plane t Isometries 42
III. Homomorphisms and Factor Groups t t t
14. Homomorphisms 44
15. Factor t Groups 49
16. Factor-Group t Computations t and t Simple t Groups 53
17. Group tAction ton ta tSet 58
18. Applications tof tG-Sets tto tCounting 61
IV. Rings and Fieldst t
19. Rings tand tFields 63
20. Integral t Domains 68
21. Fermat’s t and t Euler’s t Theorems 72
22. The t Field t of t Quotients t of t an t Integral t Domain 74
23. Rings t of t Polynomials 76
24. Factorization tof tPolynomials tover ta tField 79
25. Noncommutative tExamples 85
26. Ordered t Rings t and t Fields 87
V. Ideals and Factor Rings
t t t
27. Homomorphisms tand tFactor tRings 89
28. Prime tand tMaximal tIdeals 94
,29. Gröbner tBases tfor tIdeals 99
, VI. Extension Fields t
30. Introduction tto tExtension tFields 103
31. Vector t Spaces 107
32. Algebraic t Extensions 111
33. Geometric tConstructions 115
34. Finite t Fields 116
VII. Advanced Group Theory
t t
35. Isomorphism tTheorems 117
36. Series tof tGroups 119
37. Sylow t Theorems 122
38. Applications t of t the t Sylow t Theory 124
39. Free t Abelian t Groups 128
40. Free tGroups 130
41. Group t Presentations 133
VIII. Groups in Topologyt t
42. Simplicial t Complexes t and t Homology t Groups 136
43. Computations tof tHomology tGroups 138
44. More tHomology tComputations tand tApplications 140
45. Homological tAlgebra 144
IX. Factorization
46. Unique t Factorization t Domains148
47. Euclidean t Domains 151
48. Gaussian t Integers t and t Multiplicative t Norms 154
X. Automorphisms and Galois Theory
t t t
49. Automorphisms tof tFields 159
50. The t Isomorphism t Extension t Theorem 164
51. Splitting t Fields 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 t of t the t Quintic 185
APPENDIX t t Matrix t t Algebra 187
iv
t
First Course in Abstract
t t t
tAlgebra A 8th Edition by
t t t t
John B. Fraleigh
t t t t t
t All Chapters Full Complete
t t t
, CONTENTS
1. Sets t and tRelations 1
I. Groups and Subgroups
t t
2. Introduction t and t Examples 4
3. Binary t Operations 7
4. Isomorphic t Binary t Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic t t Groups 21
8. Generators t and t Cayley t Digraphs 24
II. Permutations, Cosets, and Direct Products
t t t t
9. Groups t of tPermutations 26
10. Orbits, tCycles, tand tthe tAlternating tGroups
30
11. Cosets tand tthe tTheorem t of tLagrange 34
12. Direct t Products t and t Finitely t Generated t Abelian t Groups 37
13. Plane t Isometries 42
III. Homomorphisms and Factor Groups t t t
14. Homomorphisms 44
15. Factor t Groups 49
16. Factor-Group t Computations t and t Simple t Groups 53
17. Group tAction ton ta tSet 58
18. Applications tof tG-Sets tto tCounting 61
IV. Rings and Fieldst t
19. Rings tand tFields 63
20. Integral t Domains 68
21. Fermat’s t and t Euler’s t Theorems 72
22. The t Field t of t Quotients t of t an t Integral t Domain 74
23. Rings t of t Polynomials 76
24. Factorization tof tPolynomials tover ta tField 79
25. Noncommutative tExamples 85
26. Ordered t Rings t and t Fields 87
V. Ideals and Factor Rings
t t t
27. Homomorphisms tand tFactor tRings 89
28. Prime tand tMaximal tIdeals 94
,29. Gröbner tBases tfor tIdeals 99
, VI. Extension Fields t
30. Introduction tto tExtension tFields 103
31. Vector t Spaces 107
32. Algebraic t Extensions 111
33. Geometric tConstructions 115
34. Finite t Fields 116
VII. Advanced Group Theory
t t
35. Isomorphism tTheorems 117
36. Series tof tGroups 119
37. Sylow t Theorems 122
38. Applications t of t the t Sylow t Theory 124
39. Free t Abelian t Groups 128
40. Free tGroups 130
41. Group t Presentations 133
VIII. Groups in Topologyt t
42. Simplicial t Complexes t and t Homology t Groups 136
43. Computations tof tHomology tGroups 138
44. More tHomology tComputations tand tApplications 140
45. Homological tAlgebra 144
IX. Factorization
46. Unique t Factorization t Domains148
47. Euclidean t Domains 151
48. Gaussian t Integers t and t Multiplicative t Norms 154
X. Automorphisms and Galois Theory
t t t
49. Automorphisms tof tFields 159
50. The t Isomorphism t Extension t Theorem 164
51. Splitting t Fields 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 t of t the t Quintic 185
APPENDIX t t Matrix t t Algebra 187
iv
