First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh,

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
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I. Groups and Subgroups b b




1. Introduction and Examples 4 b b



2. Binary Operations 7 b



3. Isomorphic Binary Structures 9 b b



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




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




8. Groups of Permutations 26 b b



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 37
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12. Plane Isometries 42
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III. Homomorphisms and Factor Groups b b b




13. Homomorphisms 44
14. Factor Groups 49 b



15. Factor-Group Computations and Simple Groups 53 b b b b



16. Group Action on a Set 58
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17. Applications of G-Sets to Counting 61 b b b b




IV. Rings and Fields b b




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



20. Fermat’s and Euler’s Theorems 72 b b b



21. The Field of Quotients of an Integral Domain 74
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22. Rings of Polynomials 76
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23. Factorization of Polynomials over a Field 79 b b b b b



24. Noncommutative Examples 85 b



25. Ordered Rings and Fields 87 b b b




V. Ideals and Factor Rings b b b




26. Homomorphisms and Factor Rings b b b 89
27. Prime and Maximal Ideals 94
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28. Gröbner Bases for Ideals 99 b b b

, VI. Extension Fields b




29. Introduction to Extension Fields b b b 103
30. Vector Spaces 107 b



31. Algebraic Extensions 111 b



32. Geometric Constructions 115 b



33. Finite Fields 116
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VII. Advanced Group Theoryb b




34. Isomorphism Theorems 117 b



35. Series of Groups 119
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36. Sylow Theorems 122
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37. Applications of the Sylow Theory b b b b 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
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41. Simplicial Complexes and Homology Groups 136
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42. Computations of Homology Groups 138 b b b



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




IX. Factorization
45. Unique Factorization Domains 148
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46. Euclidean Domains 151 b



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




48. Automorphisms of Fields 159 b b



49. The Isomorphism Extension Theorem 164
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50. Splitting Fields 165 b



51. Separable Extensions 167 b



52. Totally Inseparable Extensions 171
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53. Galois Theory 173 b



54. Illustrations of Galois Theory 176 b b b



55. Cyclotomic Extensions 183 b



56. Insolvability of the Quintic 185 b b b




APPENDIX Matrix Algebra 187 b b




iv

, 0. Sets and Relations b b b 1

0. Sets and Relations b b




√ √
1. { 3, − 3} b 2. The set is empty.b b b b




3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
b b b b b b b b b b b b b b b b b b b b b b



60, −60} b




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 element
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exceeds only a finite number of other elements but is exceeded by an infinite number of other elements.
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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. It is not a well-defined set.
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10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1,
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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 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).) b b b




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 appears as
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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 is not
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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 segment CD.
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14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x
b b b b b b b b b b b b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1)
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d− c
c. φ : [a, b] → [c, d] where φ(x) = c +
b b b b b b b b b b b (x − a)
b

b−a

15. Let φ : S → R be defined by φ(x) = tan(π(x − 12)).
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b




16. a. ∅; cardinality 1
b b b b. ∅, {a}; cardinality 2b b b b 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|. b b b b




Proof The number of subsets of a set A depends only on the cardinality of A, not on what the
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elements of A actually are. Suppose B = {1, 2, 3, · · · , s − 1} and A = {1, 2, 3,
b , 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 A; these
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are precisely the subsets of A that do not contain s, so the number of subsets of A not containing s
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is |P(B)|. Any other subset of A must contain s, and removal of the s would produce a subset of
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B. Thus the number of subsets of A containing s is also |P(B)|. Because every subset of A
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beither contains s or does not contain s (but not both), we see that the number of subsets of A is
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b2|P(B)|.