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 94
29.
, VI. Extension Fields
30. Introduction to Extension Fields 103
31. Vector Spaces 107
32. Algebraic Extensions 111
33. Geometric Constructions 115
34. Finite Fields 116
VII. Advanced Group Theory
35. Isomorphism Theorems 117
36. Series of Groups 119
37. Sylow Theorems 122
38. Applications of the Sylow Theory 124
39. Free Abelian Groups 128
40. Free Groups 130
41. Group Presentations 133
VIII. Groups in Topology
42. Simplicial Complexes and Homology Groups 136
43. Computations of Homology Groups 138
44. More Homology Computations and Applications 140
45. Homological Algebra 144
IX. Factorization
46. Unique Factorization Domains 148
47. Euclidean Domains 151
48. Gaussian Integers and Multiplicative Norms 154
X. Automorphisms and Galois Theory
49. Automorphisms of Fields 159
50. The Isomorphism Extension Theorem 164
51. Splitting Fields 165
52. Separable Extensions 167
53. Totally Inseparable Extensions 171
54. Galois Theory 173
55. Illustrations of Galois Theory 176
56. Cyclotomic Extensions 183
57. 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 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)
d−c
c. φ : [a, b] → [c, d] where φ(x) = c + (x − a)
b−a
15. Let φ : S → R be defined by φ(x) = tan(π(x − 1
)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)| = 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)|.
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 94
29.
, VI. Extension Fields
30. Introduction to Extension Fields 103
31. Vector Spaces 107
32. Algebraic Extensions 111
33. Geometric Constructions 115
34. Finite Fields 116
VII. Advanced Group Theory
35. Isomorphism Theorems 117
36. Series of Groups 119
37. Sylow Theorems 122
38. Applications of the Sylow Theory 124
39. Free Abelian Groups 128
40. Free Groups 130
41. Group Presentations 133
VIII. Groups in Topology
42. Simplicial Complexes and Homology Groups 136
43. Computations of Homology Groups 138
44. More Homology Computations and Applications 140
45. Homological Algebra 144
IX. Factorization
46. Unique Factorization Domains 148
47. Euclidean Domains 151
48. Gaussian Integers and Multiplicative Norms 154
X. Automorphisms and Galois Theory
49. Automorphisms of Fields 159
50. The Isomorphism Extension Theorem 164
51. Splitting Fields 165
52. Separable Extensions 167
53. Totally Inseparable Extensions 171
54. Galois Theory 173
55. Illustrations of Galois Theory 176
56. Cyclotomic Extensions 183
57. 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 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)
d−c
c. φ : [a, b] → [c, d] where φ(x) = c + (x − a)
b−a
15. Let φ : S → R be defined by φ(x) = tan(π(x − 1
)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)| = 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)|.
