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