Complete Solution Manual First Course in Abstract Algebra A, 8th Edition Questions & Answers with rationales

SOLUTION MANUAL
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First Course in Abstract Algebra A
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8th Edition by John B. Fraleigh
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All Chapters Full Complete
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, CONTENTS
0. Sets and Relations 1
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I. Groups and Subgroups d f d f




1. Introduction and Examples 4 df df


2. Binary Operations 7
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3. Isomorphic Binary Structures 9 d f d f


4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
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7. Generators and Cayley Digraphs 24 df df df




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




8. Groups of Permutations 26
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9. Orbits, Cycles, and the Alternating Groups 30
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10. Cosets and the Theorem of Lagrange 34
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11. Direct Products and Finitely Generated Abelian Groups
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12. Plane Isometries 42
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III. Homomorphisms and Factor Groups d f d f d f




13. Homomorphisms 44
14. Factor Groups 49
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15. Factor-Group Computations and Simple Groups 53
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16. Group Action on a Set 58
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17. Applications of G-Sets to Counting 61 df df df df




IV. Rings and Fields d f d f




18. Rings and Fields 63
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19. Integral Domains 68df


20. Fermat’s and Euler’s Theorems72
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21. The Field of Quotients of an Integral Domain
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22. Rings of Polynomials 76
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23. Factorization of Polynomials over a Field 79
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24. Noncommutative Examples 85 df


25. Ordered Rings and Fields 87
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V. Ideals and Factor Rings d f d f d f




26. Homomorphisms and Factor Rings 89 df df df


27. Prime and Maximal Ideals 94
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28. Gröbner Bases for Ideals 99
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, VI. Extension Fields d f




29. Introduction to Extension Fields 103 df df df


30. Vector Spaces 107
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31. Algebraic Extensions 111
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32. Geometric Constructions 115df


33. Finite Fields 116
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VII. Advanced Group Theory df df




34. Isomorphism Theorems 117 df


35. Series of Groups 119
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36. Sylow Theorems 122
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37. Applications of the Sylow Theory df df df df 124
38. Free Abelian Groups 128
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39. Free Groups 130
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40. Group Presentations 133
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VIII. Groups in Topology d f d f




41. Simplicial Complexes and Homology Groups
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42. Computations of Homology Groups 138 df df df


43. More Homology Computations and Applications 140
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44. Homological Algebra 144 df




IX. Factorization
45. Unique Factorization Domains 148
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46. Euclidean Domains 151 d f


47. Gaussian Integers and Multiplicative Norms 154
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X. Automorphisms and Galois Theory d f d f d f




48. Automorphisms of Fields 159 df df


49. The Isomorphism Extension Theorem
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50. Splitting Fields 165 df


51. Separable Extensions 167 df


52. Totally Inseparable Extensions 171
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53. Galois Theory 173
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54. Illustrations of Galois Theory 176 df df df


55. Cyclotomic Extensions 183 df


56. Insolvability of the Quintic 185 df df df




APPENDIX d f Matrix d f Algebra 187


iv

, 0. Sets and Relations
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0. Sets and Relations d f d f


√ √
1. { 3, − 3} df 2. d f The set is empty. df df df




3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
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4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
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5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every
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element exceeds only a finite number of other elements but is exceeded by an infinite number of
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other elements. Such people might claim the answer should be ∅.)
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6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
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8. d f It is not a well-defined set. 9.
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10. The set containing all numbers that are (positive, negative, or zero) integer multiples
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of 1, 1/2, or 1/3.
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11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
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12. a. It is a function. It is not one-to-one since there are two pairs with second member 4. It
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is not onto
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B because there is no pair with second member 2.
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b. (Same answer as Part(a).) d f d f d f



c. It is not a function because there are two pairs with first member 1.
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d. It is a function. It is one-to-one. It is onto B because every element of B
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appears as second member of some pair.
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e. It is a function. It is not one-to-one because there are two pairs with second member 6. It
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is not onto B because there is no pair with second member 2.
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f. It is not a function because there are two pairs with first member 2.
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13. Draw the line through P and x, and let y be its point of intersection with the line
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segment CD.
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14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1)
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c. φ : [a, b]
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→ [c, d] where φ(x) = cb−a df a) df df df df df df df df df




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15. Let φ : S → R be defined by φ(x) = tan(π(x
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2
− df df df df df df df df df df df )).

16. a. ∅; cardinality 1
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df df df c. ∅, { a}, { b}, {a, b }; cardinality 4
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d. ∅, {a}, {b}, {c}, {a, b }, {a, c}, { b, c}, {a, b , c}; cardinality 8
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17. Conjecture: |P(A)| = 2s = 2|A|. df df df df



Proof The number of subsets of a set A depends only on the cardinality of A, not on what
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the elements of A actually are. Suppose B = {1, 2, 3, · · · , s − 1} and A = {1, 2, 3,
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s}. Then A has all
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the elements of B plus the one additional element s. All subsets of B are also subsets of
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A; these are precisely the subsets of A that do not contain s, so the number of subsets of
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A not containing s is |P(B)|. Any other subset of A must contain s, and removal of the s
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would produce a subset of
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