A First Course in Abstract Algebra – 8th Edition by John B. Fraleigh – Complete Solution Manual (All Chapters)

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-Sets to 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
o o o 94
29. h
VI. Extension Fields o




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



32. Algebraic Extensions 111 o



33. Geometric Constructions 115 o



34. Finite Fields 116
o




VII. Advanced Group Theory o o




35. Isomorphism Theorems 117 o



36. Series of Groups 119
o o



37. Sylow Theorems 122
o



38. Applications of the Sylow Theory o o o o 124
39. Free Abelian Groups 128
o o



40. Free Groups 130
o



41. Group Presentations 133
o




VIII. Groups in Topology o o




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



43. Computations of Homology Groups 138 o o o



44. More Homology Computations and Applications
o 140 o o o



45. Homological Algebra 144 o




IX. Factorization
46. Unique Factorization Domains 148
o o



47. Euclidean Domains 151 o



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

X. Automorphisms and Galois Theory o o o




49. Automorphisms of Fields 159 o o



50. The Isomorphism Extension Theorem
o o o 164
51. Splitting Fields 165 o



52. Separable Extensions 167 o



53. Totally Inseparable Extensions
o 171 o



54. Galois Theory 173 o



55. Illustrations of Galois Theory 176 o o o



56. Cyclotomic Extensions 183 o



57. Insolvability of the Quintic 185 o o o




APPENDIX Matrix Algebra o o o o 187

, 0. Sets and Relations
o o o 1

1. Sets and Relations o o



√ √ ooo


1. o 3, − 3} o 2. The set is empty.o o o o


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



60, −60} o




4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
o o o o o o o o o o o o o o o o o o o o o o




5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every element
o o o o o o o o o o o o o
o
o o o o



exceeds only a finite number of other elements but is exceeded by an infinite number of other elements.
o o o o o o o o o o o o o o o o o o



Such people might claim the answer should be ∅.)
o o o o o o o o o




6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
o o o o o o
o
o o o
o
o




8. It is not a well-defined set.
o o o o o o 9. Q o




10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1, 1/2,
o o o o o o o o o o o o o o o o



or 1/3. o o




11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
ooo o o o o o o o o o o o o o o o o o




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
o o o o o o o o o o o o o o o o o o o o o



B because there is no pair with second member 2.
o o o o o o o o o




b. (Same answer as Part(a).) o o o




c. It is not a function because there are two pairs with first member 1.
o o o o o o o o o o o o o




d. It is a function. It is one-to-one. It is onto B because every element of B appears as second
o o o oo o o oo o o o o o o o o o o o



omember of some pair. o o o




e. It is a function. It is not one-to-one because there are two pairs with second member 6. It is not
o o o o o o o o o o o o o o o o o o o



oonto B because there is no pair with second member 2.
o o o o o o o o o o




f. It is not a function because there are two pairs with first member 2.
o o o o o o o o o o o o o




13. Draw the line through P and x, and let y be its point of intersection with the line segment CD.
o o o o o o o o o o o o o o o o o o o




14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x
o o o o o o o o o o o o o b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1)
o o o o o o o o o o o o o o o


d− c o
c. φ : [a, b] → [c, d] where φ(x) = c +
o o o o o o o o o o o o (x − a) o

b− a
o oo
oo




15. Let φ : S → R be defined by φ(x) = tan(π(x −
o o o o o o o o o o o o
1 )).
2

16. a. ∅; cardinality 1
o o o b. ∅, {a}; cardinality 2
o o o o c. ∅, {a}, {b}, {a, b}; cardinality 4
o o o o o o o




d. ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}; cardinality 8
o o o o o o o o o o o o o o o




17. Conjecture: |P(A)| = 2s = 2|A|. o o o
o
o




Proof The number of subsets of a set A depends only on the cardinality of A, not on what
o o o o o o o o o o o o o o o o o o



the elements of A actually are. Suppose B = {1, 2, 3, · · · , s − 1} and A = {1, 2, 3, , s}. Then A has all
o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o



othe elements of B plus the one additional element s. All subsets of B are also subsets of A;
o o o o o o o o o o o o o o o o o o



these are precisely the subsets of A that do not contain s, so the number of subsets of A not
o o o o o o o o o o o o o o o o o o o o



containing s is |P(B)|. Any other subset of A must contain s, and removal of the s would produce
o o o o o o o o o o o o o o o o o o o



a subset of
o o o