Solution Manual for First Course in Abstract Algebra A 8th Edition by John B. Fraleigh All Chapters Full Complete CONTENTS 0. Sets and Relations 1 I. Groups and Subgroups 1. Introduction and Examples 4 2. Binary Operations 7 3. Isomorphic Binary Structures 9 4. Groups 13 5. Subgroups 17 6. Cyclic Groups 21 7. Generators and Cayley Digraphs 24 II. Permutations, Cosets, and Direct Products 8. Groups of Permutations 26 9. Orbits, Cycles, and the Alternating Groups 30 10. Cosets and the Theorem of Lagrange 34 11. Direct Products and Finitely Generated Abelian Groups 37 12. Plane Isometries 42 III. Homomorphisms and Factor Groups 13. Homomorphisms 44 14. Factor Groups 49 15. Factor -Group Computations and Simple Groups 53 16. Group Action on a Set 58 17. Applications of G-Sets to Counting 61 IV. Rings and Fields 18. Rings and Fields 63 19. Integral Domains 68 20. Fermat’s and Euler’s Theorems 72 21. The Field of Quotients of an Integral Domain 74 22. Rings of Polynomials 76 23. Factorization of Polynomials over a Field 79 24. Noncommutative Examples 85 25. Ordered Rings and Fields 87 V. Ideals and Factor Rings 26. Homomorphisms and Factor Rings 89 27. Prime and Maximal Ideals 94 28. Gro¨bner Bases for Ideals 99 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. Isomorphism Theorems 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 of Galois Theory 176 55. Cyclotomic Extensions 183 56. Insolvability of the Quintic 185 APPENDIX Matrix Algebra 187 iv 2 → − 0. Sets and Relations 1 0. 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 33 = 27 and 43 = 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 of some pair. e. It is a function. It is not one-to-one because there are two pairs with second member 6. It is 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 − 1 )). 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)| = 2s = 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)|.
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 33 = 27 and 43 = 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 of some pair. e. It is a function. It is not one-to-one because there are two pairs with second member 6. It is 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 − 1 )). 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)| = 2s = 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)|.
