Solution Manual – A First Course in Abstract Algebra, 8th Edition by John B. Fraleigh (All Chapters, Full & Complete, Verified 2024 Update) [Mathematics | Abstract Algebra | Exam Prep]

SOLUTION MANUAL n




First Course inAbstractAlgebra A
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n 8th EditionbyJohnB.Fraleigh
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n ChaptersFullCompleten n

, CONTENTS
1. Sets and Relations
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I. Groups and Subgroups n n




2. Introduction and Examples 4 n n




3. Binary Operations 7 n




4. Isomorphic Binary Structures 9 n n




5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
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8. Generators and Cayley Digraphs 24 n n n




II. Permutations, Cosets, and Direct Products n n n n




9. Groups of Permutations 26 n n




10. Orbits, Cycles, and the Alternating Groups n n n n n




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11. Cosets and the Theorem of Lagrange
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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 n n n




14. Homomorphisms 44
15. Factor Groups 49 n




16. Factor-Group Computations and Simple Groups n n n n 53
17. Group Action on a Set 58
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18. Applications of G-Sets to Counting 61 n n n n




IV. Rings and Fields n n




19. Rings and Fields
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20. Integral Domains 68 n




21. Fermat’s and Euler’s Theorems 72 n n n




22. The Field of Quotients of an Integral Domain
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23. Rings of Polynomials 76
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24. Factorizationof Polynomials over a Field 79 n n n n n




25. Noncommutative Examples 85 n




26. Ordered Rings and Fields 87 n n n




V. Ideals and Factor Rings n n n




27. Homomorphisms and Factor Rings n n n 89
28. Prime and Maximal Ideals
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,29. Gröbner Bases for Ideals
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, VI. Extension Fields n




30. Introduction to Extension Fields n n n 103
31. Vector Spaces 107 n




32. Algebraic Extensions 111 n




33. Geometric Constructions 115 n




34. Finite Fields 116 n




VII. Advanced Group Theory n n




35. IsomorphismTheorems 117 n




36. Series of Groups 119
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37. Sylow Theorems 122 n




38. Applications of the Sylow Theory n n n n 124
39. Free Abelian Groups 128
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40. Free Groups 130
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41. Group Presentations 133 n




VIII. Groups in Topology n n




42. Simplicial Complexes and Homology Groups 136 n n n n




43. Computations of Homology Groups 138 n n n




44. More Homology Computations and Applications
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45. Homological Algebra 144 n




IX. Factorization
46. Unique Factorization Domains 148 n n




47. Euclidean Domains 151 n




48. Gaussian Integers and Multiplicative Norms n n n n 154

X. Automorphisms and Galois Theory n n n




49. Automorphisms of Fields 159 n n




50. The Isomorphism Extension Theorem
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51. Splitting Fields 165 n




52. SeparableExtensions 167 n




53. Totally Inseparable Extensions
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54. Galois Theory 173 n




55. IllustrationsofGalois Theory 176 n n n




56. CyclotomicExtensions 183 n




57. Insolvability of the Quintic 185 n n n




APPENDIX Matrix Algebra n n n n 187


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