SOLUTION MANUAL
c c
First Course in Abstract Algebra A
c c c c c c
8th Edition by John B. Fraleigh
c c c c c c
All Chapters Full Complete
c c c c
, CONTENTS
0. Sets and Relations
c c 1
I. Groups and Subgroups c c
1. Introduction and Examples 4 c c
2. Binary Operations 7 c
3. Isomorphic Binary Structures 9 c c
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
c
7. Generators and Cayley Digraphs 24 c c c
II. Permutations, Cosets, and Direct Products c c c c
8. Groups of Permutations 26 c c
9. Orbits, Cycles, and the Alternating Groups 30
c c c c c
10. Cosets and the Theorem of Lagrange 34
c c c c c
11. Direct Products and Finitely Generated Abelian Groups 37
c c c c c c
12. Plane Isometries 42
c
III. Homomorphisms and Factor Groups c c c
13. Homomorphisms 44
14. Factor Groups 49 c
15. Factor-Group Computations and Simple Groups 53 c c c c
16. Group Action on a Set 58
c c c c
17. Applications of G-Sets to Counting 61 c c c c
IV. Rings and Fields c c
18. Rings and Fields 63
c c
19. Integral Domains 68 c
20. Fermat’s and Euler’s Theorems 72 c c c
21. The Field of Quotients of an Integral Domain 74
c c c c c c c
22. Rings of Polynomials 76
c c
23. Factorization of Polynomials over a Field 79 c c c c c
24. Noncommutative Examples 85 c
25. Ordered Rings and Fields 87 c c c
V. Ideals and Factor Rings c c c
26. Homomorphisms and Factor Rings c c c 89
27. Prime and Maximal Ideals 94
c c c
28. Gröbner Bases for Ideals 99 c c c
, VI. Extension Fields c
29. Introduction to Extension Fields c c c 103
30. Vector Spaces 107 c
31. Algebraic Extensions 111 c
32. Geometric Constructions 115 c
33. Finite Fields 116
c
VII. Advanced Group Theory c c
34. Isomorphism Theorems 117 c
35. Series of Groups 119
c c
36. Sylow Theorems 122
c
37. Applications of the Sylow Theory 124 c c c c
38. Free Abelian Groups 128
c c
39. Free Groups 130
c
40. Group Presentations 133
c
VIII. Groups in Topology
c c
41. Simplicial Complexes and Homology Groups 136
c c c c
42. Computations of Homology Groups 138 c c c
43. More Homology Computations and Applications 140
c c c c
44. Homological Algebra 144 c
IX. Factorization
45. Unique Factorization Domains 148
c c
46. Euclidean Domains 151 c
47. Gaussian Integers and Multiplicative Norms 154
c c c c
X. Automorphisms and Galois Theory c c c
48. Automorphisms of Fields 159 c c
49. The Isomorphism Extension Theorem 164
c c c
50. Splitting Fields 165 c
51. Separable Extensions 167 c
52. Totally Inseparable Extensions 171
c c
53. Galois Theory 173 c
54. Illustrations of Galois Theory 176 c c c
55. Cyclotomic Extensions 183 c
56. Insolvability of the Quintic 185 c c c
APPENDIX Matrix Algebra 187 c c
iv
, 0. Sets and Relations c c c 1
0. Sets and Relations c c
√ √
1. { 3, − 3} c 2. The set is empty.
c c c c
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
c c c c c c c c c c c c c c c c c c c c c c
60, −60} c
4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
c c c c c c c c c c c c c c c c c c c c c c
5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every element
c c c c c c c c c c c c c c c c c c
exceeds only a finite number of other elements but is exceeded by an infinite number of other elements.
c c c c c c c c c c c c c c c c c c
Such people might claim the answer should be ∅.)
c c c c c c c c c
6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
c c c c c c c c c c c c
8. It is not a well-defined set.
c c c c c c 9. Q c
10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1,
c c c c c c c c c c c c c c
1/2, or 1/3.
c c c
11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
c c c c c c c c c c c c c c c 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
c c c c c c c c c c c c c c c c c c c c
onto c
B because there is no pair with second member 2.
c c c c c c c c c
b. (Same answer as Part(a).) c c c
c. It is not a function because there are two pairs with first member 1.
c c c c c c c c c c c c c
d. It is a function. It is one-to-one. It is onto B because every element of B appears as
c c c c c c c c c c c c c c c c c
second member of some pair.
c c c c c
e. It is a function. It is not one-to-one because there are two pairs with second member 6. It is not
c c c c c c c c c c c c c c c c c c c
onto B because there is no pair with second member 2.
c c c c c c c c c c c
f. It is not a function because there are two pairs with first member 2.
c c c c c c c c c c c c c
13. Draw the line through P and x, and let y be its point of intersection with the line segment CD.
c c c c c c c c c c c c c c c c c c c
14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x
c c c c c c c c c c c b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1)
c c c c c c c c c c c c c c c
d −c
c. φ : [a, b] → [c, d] where φ(x) = c +
c c c c c c c c c c c (x − a)
c
b−a
15. Let φ : S → R be defined by φ(x) = tan(π(x − 21 )).
c c c c c c c c c c c c c
c
16. a. ∅; cardinality 1
c c c b. ∅, {a}; cardinality 2
c c c c c. ∅, {a}, {b}, {a, b}; cardinality 4
c c c c c c c
d. ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}; cardinality 8
c c c c c c c c c c c c c c c
17. Conjecture: |P(A)| = 2s = 2|A|. c c c c
Proof The number of subsets of a set A depends only on the cardinality of A, not on what the
c c c c c c c c c c c c c c c c c c c
elements of A actually are. Suppose B = {1, 2, 3, · · · , s − 1} and A = {1, 2, 3,
c c , s}. Then A has c c c c c c c c c c c c c c c c c c c c c c c c c c c c
call
the elements of B plus the one additional element s. All subsets of B are also subsets of A; these
c c c c c c c c c c c c c c c c c c c
are precisely the subsets of A that do not contain s, so the number of subsets of A not containing
c c c c c c c c c c c c c c c c c c c c
s is |P(B)|. Any other subset of A must contain s, and removal of the s would produce a subset
c c c c c c c c c c c c c c c c c c c c
cof
B. Thus the number of subsets of A containing s is also |P(B)|. Because every subset of A
c c c c c c c c c c c c c c c c c
c c
First Course in Abstract Algebra A
c c c c c c
8th Edition by John B. Fraleigh
c c c c c c
All Chapters Full Complete
c c c c
, CONTENTS
0. Sets and Relations
c c 1
I. Groups and Subgroups c c
1. Introduction and Examples 4 c c
2. Binary Operations 7 c
3. Isomorphic Binary Structures 9 c c
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
c
7. Generators and Cayley Digraphs 24 c c c
II. Permutations, Cosets, and Direct Products c c c c
8. Groups of Permutations 26 c c
9. Orbits, Cycles, and the Alternating Groups 30
c c c c c
10. Cosets and the Theorem of Lagrange 34
c c c c c
11. Direct Products and Finitely Generated Abelian Groups 37
c c c c c c
12. Plane Isometries 42
c
III. Homomorphisms and Factor Groups c c c
13. Homomorphisms 44
14. Factor Groups 49 c
15. Factor-Group Computations and Simple Groups 53 c c c c
16. Group Action on a Set 58
c c c c
17. Applications of G-Sets to Counting 61 c c c c
IV. Rings and Fields c c
18. Rings and Fields 63
c c
19. Integral Domains 68 c
20. Fermat’s and Euler’s Theorems 72 c c c
21. The Field of Quotients of an Integral Domain 74
c c c c c c c
22. Rings of Polynomials 76
c c
23. Factorization of Polynomials over a Field 79 c c c c c
24. Noncommutative Examples 85 c
25. Ordered Rings and Fields 87 c c c
V. Ideals and Factor Rings c c c
26. Homomorphisms and Factor Rings c c c 89
27. Prime and Maximal Ideals 94
c c c
28. Gröbner Bases for Ideals 99 c c c
, VI. Extension Fields c
29. Introduction to Extension Fields c c c 103
30. Vector Spaces 107 c
31. Algebraic Extensions 111 c
32. Geometric Constructions 115 c
33. Finite Fields 116
c
VII. Advanced Group Theory c c
34. Isomorphism Theorems 117 c
35. Series of Groups 119
c c
36. Sylow Theorems 122
c
37. Applications of the Sylow Theory 124 c c c c
38. Free Abelian Groups 128
c c
39. Free Groups 130
c
40. Group Presentations 133
c
VIII. Groups in Topology
c c
41. Simplicial Complexes and Homology Groups 136
c c c c
42. Computations of Homology Groups 138 c c c
43. More Homology Computations and Applications 140
c c c c
44. Homological Algebra 144 c
IX. Factorization
45. Unique Factorization Domains 148
c c
46. Euclidean Domains 151 c
47. Gaussian Integers and Multiplicative Norms 154
c c c c
X. Automorphisms and Galois Theory c c c
48. Automorphisms of Fields 159 c c
49. The Isomorphism Extension Theorem 164
c c c
50. Splitting Fields 165 c
51. Separable Extensions 167 c
52. Totally Inseparable Extensions 171
c c
53. Galois Theory 173 c
54. Illustrations of Galois Theory 176 c c c
55. Cyclotomic Extensions 183 c
56. Insolvability of the Quintic 185 c c c
APPENDIX Matrix Algebra 187 c c
iv
, 0. Sets and Relations c c c 1
0. Sets and Relations c c
√ √
1. { 3, − 3} c 2. The set is empty.
c c c c
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
c c c c c c c c c c c c c c c c c c c c c c
60, −60} c
4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
c c c c c c c c c c c c c c c c c c c c c c
5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every element
c c c c c c c c c c c c c c c c c c
exceeds only a finite number of other elements but is exceeded by an infinite number of other elements.
c c c c c c c c c c c c c c c c c c
Such people might claim the answer should be ∅.)
c c c c c c c c c
6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
c c c c c c c c c c c c
8. It is not a well-defined set.
c c c c c c 9. Q c
10. The set containing all numbers that are (positive, negative, or zero) integer multiples of 1,
c c c c c c c c c c c c c c
1/2, or 1/3.
c c c
11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
c c c c c c c c c c c c c c c 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
c c c c c c c c c c c c c c c c c c c c
onto c
B because there is no pair with second member 2.
c c c c c c c c c
b. (Same answer as Part(a).) c c c
c. It is not a function because there are two pairs with first member 1.
c c c c c c c c c c c c c
d. It is a function. It is one-to-one. It is onto B because every element of B appears as
c c c c c c c c c c c c c c c c c
second member of some pair.
c c c c c
e. It is a function. It is not one-to-one because there are two pairs with second member 6. It is not
c c c c c c c c c c c c c c c c c c c
onto B because there is no pair with second member 2.
c c c c c c c c c c c
f. It is not a function because there are two pairs with first member 2.
c c c c c c c c c c c c c
13. Draw the line through P and x, and let y be its point of intersection with the line segment CD.
c c c c c c c c c c c c c c c c c c c
14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x
c c c c c c c c c c c b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 10(x − 1)
c c c c c c c c c c c c c c c
d −c
c. φ : [a, b] → [c, d] where φ(x) = c +
c c c c c c c c c c c (x − a)
c
b−a
15. Let φ : S → R be defined by φ(x) = tan(π(x − 21 )).
c c c c c c c c c c c c c
c
16. a. ∅; cardinality 1
c c c b. ∅, {a}; cardinality 2
c c c c c. ∅, {a}, {b}, {a, b}; cardinality 4
c c c c c c c
d. ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}; cardinality 8
c c c c c c c c c c c c c c c
17. Conjecture: |P(A)| = 2s = 2|A|. c c c c
Proof The number of subsets of a set A depends only on the cardinality of A, not on what the
c c c c c c c c c c c c c c c c c c c
elements of A actually are. Suppose B = {1, 2, 3, · · · , s − 1} and A = {1, 2, 3,
c c , s}. Then A has c c c c c c c c c c c c c c c c c c c c c c c c c c c c
call
the elements of B plus the one additional element s. All subsets of B are also subsets of A; these
c c c c c c c c c c c c c c c c c c c
are precisely the subsets of A that do not contain s, so the number of subsets of A not containing
c c c c c c c c c c c c c c c c c c c c
s is |P(B)|. Any other subset of A must contain s, and removal of the s would produce a subset
c c c c c c c c c c c c c c c c c c c c
cof
B. Thus the number of subsets of A containing s is also |P(B)|. Because every subset of A
c c c c c c c c c c c c c c c c c
