SOLUTION MANUAL First Course in Abstract Algebra A 8th Edition by John B. Fraleigh All Chapters Full Complete

SOLUTION MANUAL First Course in Abstract Algebra A 8th Edition by John B. Fraleigh All Chapters Full Complete CONTENTS 1. Sets and Relations 1 I. Groups and Subgroups 2. Introduction and Examples 4 3. Binary Operations 7 4. Isomorphic Binary Structures 9 5. Groups 13 6. Subgroups 17 7. Cyclic Groups 21 8. Generators and Cayley Digraphs 24 II. Permutations, Cosets, and Direct Products 9. Groups of Permutations 26 10. Orbits, Cycles, and the Alternating Groups 30 11. Cosets and the Theorem of Lagrange 34 12. Direct Products and Finitely Generated Abelian Groups 37 13. Plane Isometries 42 III. Homomorphisms and Factor Groups 14. Homomorphisms 44 15. Factor Groups 49 16. Factor-Group Computations and Simple Groups 53 17. Group Action on a Set 58 18. Applications of G-Setsto Counting 61 IV. Rings and Fields 19. Rings and Fields 63 20. Integral Domains 68 21. Fermat’s and Euler’s Theorems 72 22. The Field of Quotients of an Integral Domain 74 23. Rings of Polynomials 76 24. Factorization of Polynomials over a Field 79 25. Noncommutative Examples 85 26. Ordered Rings and Fields 87 V. Ideals and Factor Rings 27. Homomorphisms and Factor Rings 89 28. Prime and Maximal Ideals 94 VI. Extension Fields 29. Introduction to Extension Fields 103 30. Vector Spaces 107 31. Algebraic Extensions 111 32. Geometric Constructions 115 33. Finite Fields 116 VII. Advanced Group Theory 34. IsomorphismTheorems 117 35. Series of Groups 119 36. Sylow Theorems 122 37. Applications of the Sylow Theory 124 38. Free Abelian Groups 128 39. Free Groups 130 40. Group Presentations 133 VIII. Groups in Topology 41. Simplicial Complexes and Homology Groups 136 42. Computations of Homology Groups 138 43. More Homology Computations and Applications 140 44. Homological Algebra 144 IX. Factorization 45. Unique Factorization Domains 148 46. Euclidean Domains 151 47. Gaussian Integers and Multiplicative Norms 154 X. Automorphisms and Galois Theory 48. Automorphisms of Fields 159 49. The Isomorphism Extension Theorem 164 50. Splitting Fields 165 51. Separable Extensions 167 52. Totally Inseparable Extensions 171 53. Galois Theory 173 54. Illustrations ofGalois Theory 176 55. CyclotomicExtensions 183 56. Insolvability of the Quintic 185 APPENDIX Matrix Algebra 187 iv − 0. Sets and Relations 1 1. Sets and Relations √ √ 1. { 3, − 3} 2. The set is empty. 3. {1, −1, 2,−2, 3, −3,4,−4, 5, −5, 6, −6,10, −10,12, −12,15, −15,20, −20,30, −30, 60, −60} 4. {−10,−9,−8,−7,−6,−5, −4,−3, −2,−1,0, 1, 2,3, 4,5, 6,7, 8, 9,10,11} 5. It is not a well-defined set. (Some may argue that no element of Z + is large, because every element exceeds only a finite number of other elements but is exceeded by an infinite number of other elements. Such people might claim the answer should be ∅.) 6. ∅ 7. The set is ∅ because 3 3 = 27 and 4 3 = 64. 8. It is not a well-defined set. 9. Q 10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1, 1/2, or 1/3. 11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)} 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 B because there is no pair with second member 2. b. (Same answer as Part(a).) c. It is not a function because there are two pairs with first member 1. d. It is a function. It is one-to-one. It is onto B because every element of B appears as second member ofsome pair. e. Itis a function. It is not one-to-one because there are two pairs with second member 6. Itis not onto B because there is no pair with second member 2. f. It is not a function because there are two pairs with first member 2. 13. Draw the line through P and x, and let y be its point of intersection with the line segment CD. 14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1) c. φ : [a, b] → [c, d] where φ(x) = c + d− c (x − a) b a 15. Let φ : S → R be defined by φ(x) = tan(π(x − ) 2 ). 16. a. ∅; cardinality 1 b. ∅, {a}; cardinality 2 c. ∅,{a},{b},{a, b}; cardinality 4 d. ∅,{a},{b},{c},{a, b},{a, c},{b, c},{a, b, c}; cardinality 8 17. Conjecture: |P(A)| = 2 s = 2| A| . Proof The number of subsets of a set A depends only on the cardinality of A, not on what the elements of A actually are. Suppose B = {1, 2, 3, · · · ,s − 1} and A = {1, 2, 3, ,s}. Then A has all the elements of B plus the one additional element s. All subsets of B are also subsets of A; these are precisely the subsets of A that do not contain s, so the number of subsets of A not containing s is |P(B)|. Any other subset of A must contain s, and removal of the s would produce a subset of B. Thus the number of subsets of A containing s is also |P(B)|. Because every subset of A either contains s or does not contain s (but not both), we see that the number of subsets of A is 2|P(B)|. 1 0 (2 ) ) 2 0. Sets and Relations We have shown that if A has one more element that B, then |P(A)| = 2|P(B)|. Now |P(∅)| = 1, so if |A| = s, then |P(A)| = 2 . s 18. We define a one-to-one map φ of B A onto P(A). Let f ∈ B A , and let φ(f ) = {x ∈ A | f (x) = 1}. Suppose φ(f ) = φ(g). Then f (x) = 1 if and only if g(x) = 1. Because the only possible values for f (x) and g(x) are 0 and 1, we see that f (x) = 0 if and only if g(x) = 0. Consequently f (x) = g(x) for all x ∈ A so f = g and φ is one to one. To show that φ is onto P(A), let S ⊆ A, and let h : A → {0, 1} be defined by h(x) = 1 if x ∈ S and h(x) = 0 otherwise. Clearly φ(h) = S, showing that φ is indeed onto P(A). 19. Picking up from the hint, let Z = {x ∈ A | x ∈/ φ(x)}. We claim that for any a ∈ A, φ(a) /= Z. Either a ∈ φ(a), in which case a ∈/ Z, or a ∈/ φ(a), in which case a ∈ Z. Thus Z and φ(a) are certainly differentsubsetsof A; one ofthemcontains a andthe other one does not. Based on what we just showed, we feel that the power set of A has cardinality greater than |A|. Proceeding naively, we can start with the infinite set Z, form its power set, then form the power set of that, and continue this process indefinitely. If there were only a finite number of infinite cardinal numbers, this process would have to terminate after a fixed finite number of steps. Since it doesn’t, it appears that there must be an infinite number of different infinite cardinal numbers. The set of everything is not logically acceptable, because the set of all subsets of the set of everything would be larger than the set of everything, which is a fallacy. 20. a. The set containing precisely the two elements of A and the three (different) elements of B is C = {1, 2, 3, 4, 5} which has 5 elements. i) Let A = {−2, −1, 0} and B = {1, 2, 3, · · ·} = Z + . Then |A| = 3 and |B| = ℵ0, and A and B have no elements in common. The set C containing all elements in either A or B is C = {−2, −1, 0, 1, 2, 3, · · ·}. The map φ : C → B defined by φ(x) = x + 3 is one to one and onto B, so |C| = |B| = ℵ0. Thus we consider 3 + ℵ0 = ℵ0. ii) Let A = {1, 2, 3, · · ·} and B = {1/2, 3/2,5/2,· · ·}. Then |A| = |B| = ℵ0 and A and B have no elements in common. The set C containing all elements in either A of B is C = {1/2, 1, 3/2, 2, 5/2, 3, · · ·}. The map φ : C → A defined by φ(x) = 2x is one to one and onto A, so |C| = |A| = ℵ0. Thus we consider ℵ0 + ℵ0 = ℵ0. b. We leave the plotting of the points in A × B to you. Figure 0.14 in the text, where there are ℵ0 rows each having ℵ0 entries, illustratesthat we would consider that ℵ0 ·ℵ0 = ℵ0. 21. There are 102 = 100 numbers (.00 through .99) of the form .##, and 105 = 100, 000 numbers (.00000 through .99999) of the form .#####. Thus for .##### · · ·, we expect 10ℵ0 sequences representing all numbers x ∈ R such that 0 ≤ x ≤ 1, but a sequence trailing off in 0’s may represent the same x ∈ R as a sequence trailing of in 9’s. At any rate, we should have 10ℵ0 ≥ |[0, 1]| = |R|; see Exercise 15. On the other hand, we can represent numbers in R using any integer base n > 1, and these same 10ℵ0 sequences using digits from 0 to 9 in base n = 12 would not represent all x ∈ [0, 1], so we have 10ℵ0 ≤ |R|. Thus we consider the value of 10ℵ0 to be |R|. We could make the same argument using any other integer base n > 1, and thus consider nℵ0 = |R| for n ∈ Z + , n > 1. In particular, 12ℵ0 = 2ℵ0 = |R|. 22. ℵ , |R|, 2|R| , 2 (2|R| ) |R| , 2 (2 23. 1. There is only one partition {{a}} of a one-elementset {a}. 24. There are two partitions of {a, b}, namely {{a, b}} and {{a},{b}}.