SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Sets and Relations 1
I. Groups and Subgroups
2. Introduction and Examples 4
3. Binary Operations 7
4. Isomorphic Binary Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
8. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
9. Groups of Permutations 26
10. Orbits, Cycles, and the Alternating Groups
30
11. Cosets and the Theorem of Lagrange 34
12. Direct Products and Finitely Generated Abelian Groups 37
13. Plane Isometries 42
III. Homomorphisms and Factor Groups
14. Homomorphisms 44
15. Factor Groups 49
16. Factor-Group Computations and Simple Groups 53
17. Group Action on a Set 58
18. Applications of G-Sets to Counting 61
IV. Rings and Fields
19. Rings and Fields 63
20. Integral Domains 68
21. Fermat’s and Euler’s Theorems 72
22. The Field of Quotients of an Integral Domain 74
23. Rings of Polynomials 76
24. Factorization of Polynomials over a Field 79
25. Noncommutative Examples 85
26. Ordered Rings and Fields 87
V. Ideals and Factor Rings
27. Homomorphisms and Factor Rings 89
28. Prime and Maximal Ideals 94
,29. Gröbner xBases xfor xIdeals 99
, VI. Extension x Fields
30. Introduction xto xExtension xFields 103
31. Vector x Spaces 107
32. Algebraic x Extensions 111
33. Geometric xConstructions 115
34. Finite xFields 116
VII. Advanced xGroup xTheory
35. Isomorphism xTheorems 117
36. Series xof xGroups 119
37. Sylow xTheorems 122
38. Applications xof xthe xSylow xTheory 124
39. Free xAbelian xGroups 128
40. Free xGroups 130
41. Group xPresentations 133
VIII. Groups x in x Topology
42. Simplicial xComplexes xand xHomology xGroups 136
43. Computations xof xHomology xGroups 138
44. More xHomology xComputations xand xApplications 140
45. Homological xAlgebra 144
IX. Factorization
46. Unique xFactorization xDomains 148
47. Euclidean x Domains 151
48. Gaussian xIntegers xand xMultiplicative xNorms 154
X. Automorphisms x and x Galois x Theory
49. Automorphisms xof xFields 159
50. The xIsomorphism xExtension x Theorem 164
51. Splitting xFields 165
52. Separable xExtensions 167
53. Totally xInseparable xExtensions 171
54. Galois x Theory 173
55. Illustrations xof xGalois xTheory 176
56. Cyclotomic xExtensions 183
57. Insolvability xof xthe xQuintic185
APPENDIX x xMatrix x xAlgebra 187
iv
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Sets and Relations 1
I. Groups and Subgroups
2. Introduction and Examples 4
3. Binary Operations 7
4. Isomorphic Binary Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
8. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
9. Groups of Permutations 26
10. Orbits, Cycles, and the Alternating Groups
30
11. Cosets and the Theorem of Lagrange 34
12. Direct Products and Finitely Generated Abelian Groups 37
13. Plane Isometries 42
III. Homomorphisms and Factor Groups
14. Homomorphisms 44
15. Factor Groups 49
16. Factor-Group Computations and Simple Groups 53
17. Group Action on a Set 58
18. Applications of G-Sets to Counting 61
IV. Rings and Fields
19. Rings and Fields 63
20. Integral Domains 68
21. Fermat’s and Euler’s Theorems 72
22. The Field of Quotients of an Integral Domain 74
23. Rings of Polynomials 76
24. Factorization of Polynomials over a Field 79
25. Noncommutative Examples 85
26. Ordered Rings and Fields 87
V. Ideals and Factor Rings
27. Homomorphisms and Factor Rings 89
28. Prime and Maximal Ideals 94
,29. Gröbner xBases xfor xIdeals 99
, VI. Extension x Fields
30. Introduction xto xExtension xFields 103
31. Vector x Spaces 107
32. Algebraic x Extensions 111
33. Geometric xConstructions 115
34. Finite xFields 116
VII. Advanced xGroup xTheory
35. Isomorphism xTheorems 117
36. Series xof xGroups 119
37. Sylow xTheorems 122
38. Applications xof xthe xSylow xTheory 124
39. Free xAbelian xGroups 128
40. Free xGroups 130
41. Group xPresentations 133
VIII. Groups x in x Topology
42. Simplicial xComplexes xand xHomology xGroups 136
43. Computations xof xHomology xGroups 138
44. More xHomology xComputations xand xApplications 140
45. Homological xAlgebra 144
IX. Factorization
46. Unique xFactorization xDomains 148
47. Euclidean x Domains 151
48. Gaussian xIntegers xand xMultiplicative xNorms 154
X. Automorphisms x and x Galois x Theory
49. Automorphisms xof xFields 159
50. The xIsomorphism xExtension x Theorem 164
51. Splitting xFields 165
52. Separable xExtensions 167
53. Totally xInseparable xExtensions 171
54. Galois x Theory 173
55. Illustrations xof xGalois xTheory 176
56. Cyclotomic xExtensions 183
57. Insolvability xof xthe xQuintic185
APPENDIX x xMatrix x xAlgebra 187
iv
