SOLUTION MANUAL j
First Course in Abstract
j j j
Algebra A 8th Edition by John
j j j j j j
B. Fraleigh
j j j j
j All Chapters Full Complete
j j j
, CONTENTS
1. Sets and Relations 1
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I. Groups and Subgroups
j j
2. Introduction and Examples 4 j j
3. Binary Operations 7 j
4. Isomorphic Binary Structures 9 j j
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
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8. Generators and Cayley Digraphs 24 j j j
II. Permutations, Cosets, and Direct Products j j j j
9. Groups of Permutations 26 j j
10. Orbits, Cycles, and the Alternating Groups j j j j j
30
11. Cosets and the Theorem of Lagrange 34
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12. Direct Products and Finitely Generated Abelian Groups 37
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13. Plane Isometries 42
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III. Homomorphisms and Factor Groups j j j
14. Homomorphisms 44
15. Factor Groups 49 j
16. Factor-Group Computations and Simple Groups 53 j j j j
17. Group Action on a Set 58j j j j
18. Applications of G-Sets to Counting 61 j j j j
IV. Rings and Fieldsj j
19. Rings and Fields 63
j j
20. Integral Domains 68 j
21. Fermat’s and Euler’s Theorems 72 j j j
22. The Field of Quotients of an Integral Domain
j j j j j j j 74
23. Rings of Polynomials 76
j j
24. Factorization of Polynomials over a Field 79 j j j j j
25. Noncommutative Examples 85 j
26. Ordered Rings and Fields 87 j j j
V. Ideals and Factor Rings j j j
27. Homomorphisms and Factor Rings 89 j j j
28. Prime and Maximal Ideals 94
j j j
,29. Gröbner Bases for Ideals 99
j j j
, VI. Extension Fields j
30. Introduction to Extension Fields 103 j j j
31. Vector Spaces 107 j
32. Algebraic Extensions 111 j
33. Geometric Constructions 115 j
34. Finite Fields 116 j
VII. Advanced Group Theory j j
35. Isomorphism Theorems 117 j
36. Series of Groups 119j j
37. Sylow Theorems 122 j
38. Applications of the Sylow Theory 124 j j j j
39. Free Abelian Groups 128
j j
40. Free Groups 130
j
41. Group Presentations 133 j
VIII. Groups in Topology j j
42. Simplicial Complexes and Homology Groups 136 j j j j
43. Computations of Homology Groups 138 j j j
44. More Homology Computations and Applications 140
j j j j
45. Homological Algebra 144 j
IX. Factorization
46. Unique Factorization Domains 148 j j
47. Euclidean Domains 151 j
48. Gaussian Integers and Multiplicative Norms 154 j j j j
X. Automorphisms and Galois Theory j j j
49. Automorphisms of Fields 159 j j
50. The Isomorphism Extension Theorem 164
j j j
51. Splitting Fields 165 j
52. Separable Extensions 167 j
53. Totally Inseparable Extensions 171
j j
54. Galois Theory 173 j
55. Illustrations of Galois Theory 176 j j j
56. CyclotomicExtensions 183 j
57. Insolvability of the Quintic 185 j j j
APPENDIX Matrix Algebra j j j j 187
iv
First Course in Abstract
j j j
Algebra A 8th Edition by John
j j j j j j
B. Fraleigh
j j j j
j All Chapters Full Complete
j j j
, CONTENTS
1. Sets and Relations 1
j j
I. Groups and Subgroups
j j
2. Introduction and Examples 4 j j
3. Binary Operations 7 j
4. Isomorphic Binary Structures 9 j j
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
j j
8. Generators and Cayley Digraphs 24 j j j
II. Permutations, Cosets, and Direct Products j j j j
9. Groups of Permutations 26 j j
10. Orbits, Cycles, and the Alternating Groups j j j j j
30
11. Cosets and the Theorem of Lagrange 34
j j j j j
12. Direct Products and Finitely Generated Abelian Groups 37
j j j j j j
13. Plane Isometries 42
j
III. Homomorphisms and Factor Groups j j j
14. Homomorphisms 44
15. Factor Groups 49 j
16. Factor-Group Computations and Simple Groups 53 j j j j
17. Group Action on a Set 58j j j j
18. Applications of G-Sets to Counting 61 j j j j
IV. Rings and Fieldsj j
19. Rings and Fields 63
j j
20. Integral Domains 68 j
21. Fermat’s and Euler’s Theorems 72 j j j
22. The Field of Quotients of an Integral Domain
j j j j j j j 74
23. Rings of Polynomials 76
j j
24. Factorization of Polynomials over a Field 79 j j j j j
25. Noncommutative Examples 85 j
26. Ordered Rings and Fields 87 j j j
V. Ideals and Factor Rings j j j
27. Homomorphisms and Factor Rings 89 j j j
28. Prime and Maximal Ideals 94
j j j
,29. Gröbner Bases for Ideals 99
j j j
, VI. Extension Fields j
30. Introduction to Extension Fields 103 j j j
31. Vector Spaces 107 j
32. Algebraic Extensions 111 j
33. Geometric Constructions 115 j
34. Finite Fields 116 j
VII. Advanced Group Theory j j
35. Isomorphism Theorems 117 j
36. Series of Groups 119j j
37. Sylow Theorems 122 j
38. Applications of the Sylow Theory 124 j j j j
39. Free Abelian Groups 128
j j
40. Free Groups 130
j
41. Group Presentations 133 j
VIII. Groups in Topology j j
42. Simplicial Complexes and Homology Groups 136 j j j j
43. Computations of Homology Groups 138 j j j
44. More Homology Computations and Applications 140
j j j j
45. Homological Algebra 144 j
IX. Factorization
46. Unique Factorization Domains 148 j j
47. Euclidean Domains 151 j
48. Gaussian Integers and Multiplicative Norms 154 j j j j
X. Automorphisms and Galois Theory j j j
49. Automorphisms of Fields 159 j j
50. The Isomorphism Extension Theorem 164
j j j
51. Splitting Fields 165 j
52. Separable Extensions 167 j
53. Totally Inseparable Extensions 171
j j
54. Galois Theory 173 j
55. Illustrations of Galois Theory 176 j j j
56. CyclotomicExtensions 183 j
57. Insolvability of the Quintic 185 j j j
APPENDIX Matrix Algebra j j j j 187
iv
