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 Bases for Ideals 99
, VI. Extension Fields
30. Introduction to Extension Fields 103
31. Vector Spaces 107
32. Algebraic Extensions 111
33. Geometric Constructions 115
34. Finite Fields 116
VII. Advanced Group Theory
35. Isomorphism Theorems 117
36. Series of Groups 119
37. Sylow Theorems 122
38. Applications of the Sylow Theory 124
39. Free Abelian Groups 128
40. Free Groups 130
41. Group Presentations 133
VIII. Groups in Topology
42. Simplicial Complexes and Homology Groups 136
43. Computations of Homology Groups 138
44. More Homology Computations and Applications 140
45. Homological Algebra 144
IX. Factorization
46. Unique Factorization Domains 148
47. Euclidean Domains 151
48. Gaussianx Integersx andx Multiplicativex Norms 154
X. Automorphismsx andx Galoisx Theory
49. AutomorphismsxofxFields 159
50. Thex Isomorphismx Extensionx Theorem 164
51. Splittingx Fields 165
52. SeparablexExtensions 167
53. TotallyxInseparablexExtensions 171
54. Galoisx Theory 173
55. IllustrationsxofxGaloisxTheory 176
56. CyclotomicxExtensions 183
57. Insolvabilityx ofx thex Quintic 185
APPENDIXxx Matrixxx Algebra 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 Bases for Ideals 99
, VI. Extension Fields
30. Introduction to Extension Fields 103
31. Vector Spaces 107
32. Algebraic Extensions 111
33. Geometric Constructions 115
34. Finite Fields 116
VII. Advanced Group Theory
35. Isomorphism Theorems 117
36. Series of Groups 119
37. Sylow Theorems 122
38. Applications of the Sylow Theory 124
39. Free Abelian Groups 128
40. Free Groups 130
41. Group Presentations 133
VIII. Groups in Topology
42. Simplicial Complexes and Homology Groups 136
43. Computations of Homology Groups 138
44. More Homology Computations and Applications 140
45. Homological Algebra 144
IX. Factorization
46. Unique Factorization Domains 148
47. Euclidean Domains 151
48. Gaussianx Integersx andx Multiplicativex Norms 154
X. Automorphismsx andx Galoisx Theory
49. AutomorphismsxofxFields 159
50. Thex Isomorphismx Extensionx Theorem 164
51. Splittingx Fields 165
52. SeparablexExtensions 167
53. TotallyxInseparablexExtensions 171
54. Galoisx Theory 173
55. IllustrationsxofxGaloisxTheory 176
56. CyclotomicxExtensions 183
57. Insolvabilityx ofx thex Quintic 185
APPENDIXxx Matrixxx Algebra 187
iv
