SOLUTION MANUAL
hj hj
First Course in Abstract Algebra A
hj hj hj hj hj hj
8th Edition by John B. Fraleigh
hj hj hj hj hj hj
All Chapters Full Complete
hj hj hj hj
, CONTENTS
0. Sets and Relations 1
hj hj
I. Groups and Subgroups h j h j
1. Introduction and Examples 4 hj hj
2. Binary Operations 7
h j
3. Isomorphic Binary Structures 9 h j h j
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
h j
7. Generators and Cayley Digraphs 24 hj hj hj
II. Permutations, Cosets, and Direct Products hj hj hj hj
8. Groups of Permutations 26
hj hj
9. Orbits, Cycles, and the Alternating Groups 30
hj hj hj hj hj
10. Cosets and the Theorem of Lagrange 34
hj hj hj hj hj
11. Direct Products and Finitely Generated Abelian Groups
h j h j h j h j h j h j 37
12. Plane Isometries 42
h j
III. Homomorphisms and Factor Groups h j h j h j
13. Homomorphisms 44
14. Factor Groups 49
hj
15. Factor-Group Computations and Simple Groups hj hj hj h j 53
16. Group Action on a Set 58
hj hj hj hj
17. Applications of G-Sets to Counting 61 hj hj hj hj
IV. Rings and Fields h j h j
18. Rings and Fields 63
hj hj
19. Integral Domains 68 hj
20. Fermat’s and Euler’s Theorems 72
hj hj hj
21. The Field of Quotients of an Integral Domain
hj hj hj hj hj hj hj 74
22. Rings of Polynomials 76
hj hj
23. Factorization of Polynomials over a Field 79 hj hj hj hj hj
24. Noncommutative Examples 85 hj
25. Ordered Rings and Fields 87
hj hj hj
V. Ideals and Factor Rings h j h j h j
26. Homomorphisms and Factor Rings 89 hj hj hj
27. Prime and Maximal Ideals 94
hj hj hj
28. Gröbner Bases for Ideals 99 hj hj hj
, VI. Extension Fieldsh j
29. Introduction to Extension Fields 103 hj hj hj
30. Vector Spaces 107
h j
31. Algebraic Extensions 111
h j
32. Geometric Constructions 115 hj
33. Finite Fields 116
h j
VII. Advanced Group Theory hj hj
34. Isomorphism Theorems 117 hj
35. Series of Groups 119
hj hj
36. Sylow Theorems 122
hj
37. Applications of the Sylow Theory124 hj hj hj hj
38. Free Abelian Groups 128
hj hj
39. Free Groups 130
hj
40. Group Presentations 133
h j
VIII. Groups in Topologyh j h j
41. Simplicial Complexes and Homology Groups 136
hj hj hj hj
42. Computations of Homology Groups 138 hj hj hj
43. More Homology Computations and Applications 140
hj hj hj hj
44. Homological Algebra 144 hj
IX. Factorization
45. Unique Factorization Domains 148
hj hj
46. Euclidean Domains 151 h j
47. Gaussian Integers and Multiplicative Norms 154
hj hj hj hj
X. Automorphisms and Galois Theory h j h j h j
48. Automorphisms of Fields 159 hj hj
49. The Isomorphism Extension Theorem 164
hj hj hj
50. Splitting Fields 165 hj
51. Separable Extensions 167 hj
52. Totally Inseparable Extensions 171
hj hj
53. Galois Theory 173
h j
54. Illustrations of Galois Theory 176 hj hj hj
55. Cyclotomic Extensions 183 hj
56. Insolvability of the Quintic 185 hj hj hj
APPENDIX h j Matrix h j Algebra 187
iv
, 0. Sets and Relations
h j hj hj 1
0. Sets and Relations h j h j
√ √
1. { 3, − 3} hj 2. h j The set is empty. hj hj hj
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
60, −60} hj
4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every
hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj
element exceeds only a finite number of other elements but is exceeded by a n infinite number of
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
other elements. Such people might claim the answer should be ∅.)
hj hj hj hj hj hj hj hj hj hj hj
6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
h j hj hj hj hj hj hj hj hj hj hj hj
8. h j It is not a well-defined set.
hj hj hj hj hj 9. h j Q
10. The set containing all numbers that are (positive, negative, or zero) integer multiples
hj hj hj hj hj hj hj h j h j hj hj hj
of 1, 1/2, or 1/3.
hj hj h j h j hj
11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
12. a. It is a function. It is not one-to-one since there are two pairs with second member 4. It is
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
not onto
hj hj
B because there is no pair with second member 2.
hj hj hj hj hj hj hj hj hj
b. (Same answer as Part(a).) h j h j hj
c. It is not a function because there are two pairs with first member 1.
hj hj hj hj hj hj hj hj hj hj hj hj hj
d. It is a function. It is one-to-one. It is onto B because every element of B
hj hj hj h j hj hj h j hj hj hj hj hj hj hj hj
appears as second member of some pair.
hj hj hj hj hj hj hj
e. It is a function. It is not one-to-one because there are two pairs with second member 6. It
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
is not onto B because there is no pair with second member 2.
hj hj hj hj hj hj hj hj hj hj hj hj hj
f. It is not a function because there are two pairs with first member 2.
hj hj hj hj hj hj hj hj hj hj hj hj hj
13. Draw the line through P and x, and let y be its point of intersection with the line
hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj
segment CD.
hj hj
14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 1 0(x − 1)
h j hj hj hj hj hj hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj
] [c, d ] where φ(x) = c + d−c−(x
c. φ : [a, b→
h j hj hj a) hj hj hj hj hj hj hj hj hj
b−a
1 hj
15. Let φ : S → R be defined by φ(x) = tan(π(x
hj hj
2
− hj hj hj hj hj hj hj hj hj hj hj )).
16. a. ∅; cardinality 1
h j hj hj b. h j ∅, {a}; cardinality 2
hj hj hj c. h j ∅, {a}, { b}, { a, b}; cardinality 4
hj hj hj hj hj hj
d. ∅, { a}, { b}, {c}, { a, b}, { a, c}, { b, c}, { a, b, c}; cardinality 8
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj
17. Conjecture: |P(A)| = 2s = 2|A|. hj hj hj hj
Proof The number of subsets of a set A depends only on the cardinality of A, not on what
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
the elements of A actually are. Suppose B = {1, 2, 3 , · · · , s − 1} and A = {1, 2, 3,
hj hj hj , s}. hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj h j h j hj
Then A has all
h j hj hj hj
the elements of B plus the one additional element s. All subsets of B are also subsets of A;
hj hj hj hj hj hj hj hj hj h j hj hj hj hj hj hj hj hj
these are precisely the subsets of A that do not contain s, so the number of subsets of A
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
not containing s is |P(B)|. Any other subset of A must contain s, and removal of the s
hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj
would produce a subset of
hj hj hj hj hj
hj hj
First Course in Abstract Algebra A
hj hj hj hj hj hj
8th Edition by John B. Fraleigh
hj hj hj hj hj hj
All Chapters Full Complete
hj hj hj hj
, CONTENTS
0. Sets and Relations 1
hj hj
I. Groups and Subgroups h j h j
1. Introduction and Examples 4 hj hj
2. Binary Operations 7
h j
3. Isomorphic Binary Structures 9 h j h j
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
h j
7. Generators and Cayley Digraphs 24 hj hj hj
II. Permutations, Cosets, and Direct Products hj hj hj hj
8. Groups of Permutations 26
hj hj
9. Orbits, Cycles, and the Alternating Groups 30
hj hj hj hj hj
10. Cosets and the Theorem of Lagrange 34
hj hj hj hj hj
11. Direct Products and Finitely Generated Abelian Groups
h j h j h j h j h j h j 37
12. Plane Isometries 42
h j
III. Homomorphisms and Factor Groups h j h j h j
13. Homomorphisms 44
14. Factor Groups 49
hj
15. Factor-Group Computations and Simple Groups hj hj hj h j 53
16. Group Action on a Set 58
hj hj hj hj
17. Applications of G-Sets to Counting 61 hj hj hj hj
IV. Rings and Fields h j h j
18. Rings and Fields 63
hj hj
19. Integral Domains 68 hj
20. Fermat’s and Euler’s Theorems 72
hj hj hj
21. The Field of Quotients of an Integral Domain
hj hj hj hj hj hj hj 74
22. Rings of Polynomials 76
hj hj
23. Factorization of Polynomials over a Field 79 hj hj hj hj hj
24. Noncommutative Examples 85 hj
25. Ordered Rings and Fields 87
hj hj hj
V. Ideals and Factor Rings h j h j h j
26. Homomorphisms and Factor Rings 89 hj hj hj
27. Prime and Maximal Ideals 94
hj hj hj
28. Gröbner Bases for Ideals 99 hj hj hj
, VI. Extension Fieldsh j
29. Introduction to Extension Fields 103 hj hj hj
30. Vector Spaces 107
h j
31. Algebraic Extensions 111
h j
32. Geometric Constructions 115 hj
33. Finite Fields 116
h j
VII. Advanced Group Theory hj hj
34. Isomorphism Theorems 117 hj
35. Series of Groups 119
hj hj
36. Sylow Theorems 122
hj
37. Applications of the Sylow Theory124 hj hj hj hj
38. Free Abelian Groups 128
hj hj
39. Free Groups 130
hj
40. Group Presentations 133
h j
VIII. Groups in Topologyh j h j
41. Simplicial Complexes and Homology Groups 136
hj hj hj hj
42. Computations of Homology Groups 138 hj hj hj
43. More Homology Computations and Applications 140
hj hj hj hj
44. Homological Algebra 144 hj
IX. Factorization
45. Unique Factorization Domains 148
hj hj
46. Euclidean Domains 151 h j
47. Gaussian Integers and Multiplicative Norms 154
hj hj hj hj
X. Automorphisms and Galois Theory h j h j h j
48. Automorphisms of Fields 159 hj hj
49. The Isomorphism Extension Theorem 164
hj hj hj
50. Splitting Fields 165 hj
51. Separable Extensions 167 hj
52. Totally Inseparable Extensions 171
hj hj
53. Galois Theory 173
h j
54. Illustrations of Galois Theory 176 hj hj hj
55. Cyclotomic Extensions 183 hj
56. Insolvability of the Quintic 185 hj hj hj
APPENDIX h j Matrix h j Algebra 187
iv
, 0. Sets and Relations
h j hj hj 1
0. Sets and Relations h j h j
√ √
1. { 3, − 3} hj 2. h j The set is empty. hj hj hj
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
60, −60} hj
4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
5. It is not a well-defined set. (Some may argue that no element of Z+ is large, because every
hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj
element exceeds only a finite number of other elements but is exceeded by a n infinite number of
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
other elements. Such people might claim the answer should be ∅.)
hj hj hj hj hj hj hj hj hj hj hj
6. ∅ 7. The set is ∅ because 33 = 27 and 43 = 64.
h j hj hj hj hj hj hj hj hj hj hj hj
8. h j It is not a well-defined set.
hj hj hj hj hj 9. h j Q
10. The set containing all numbers that are (positive, negative, or zero) integer multiples
hj hj hj hj hj hj hj h j h j hj hj hj
of 1, 1/2, or 1/3.
hj hj h j h j hj
11. {(a, 1), (a, 2), (a, c), (b, 1), (b, 2), (b, c), (c, 1), (c, 2), (c, c)}
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
12. a. It is a function. It is not one-to-one since there are two pairs with second member 4. It is
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
not onto
hj hj
B because there is no pair with second member 2.
hj hj hj hj hj hj hj hj hj
b. (Same answer as Part(a).) h j h j hj
c. It is not a function because there are two pairs with first member 1.
hj hj hj hj hj hj hj hj hj hj hj hj hj
d. It is a function. It is one-to-one. It is onto B because every element of B
hj hj hj h j hj hj h j hj hj hj hj hj hj hj hj
appears as second member of some pair.
hj hj hj hj hj hj hj
e. It is a function. It is not one-to-one because there are two pairs with second member 6. It
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
is not onto B because there is no pair with second member 2.
hj hj hj hj hj hj hj hj hj hj hj hj hj
f. It is not a function because there are two pairs with first member 2.
hj hj hj hj hj hj hj hj hj hj hj hj hj
13. Draw the line through P and x, and let y be its point of intersection with the line
hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj
segment CD.
hj hj
14. a. φ : [0, 1] → [0, 2] where φ(x) = 2x b. φ : [1, 3] → [5, 25] where φ(x) = 5 + 1 0(x − 1)
h j hj hj hj hj hj hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj
] [c, d ] where φ(x) = c + d−c−(x
c. φ : [a, b→
h j hj hj a) hj hj hj hj hj hj hj hj hj
b−a
1 hj
15. Let φ : S → R be defined by φ(x) = tan(π(x
hj hj
2
− hj hj hj hj hj hj hj hj hj hj hj )).
16. a. ∅; cardinality 1
h j hj hj b. h j ∅, {a}; cardinality 2
hj hj hj c. h j ∅, {a}, { b}, { a, b}; cardinality 4
hj hj hj hj hj hj
d. ∅, { a}, { b}, {c}, { a, b}, { a, c}, { b, c}, { a, b, c}; cardinality 8
h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj
17. Conjecture: |P(A)| = 2s = 2|A|. hj hj hj hj
Proof The number of subsets of a set A depends only on the cardinality of A, not on what
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
the elements of A actually are. Suppose B = {1, 2, 3 , · · · , s − 1} and A = {1, 2, 3,
hj hj hj , s}. hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj h j h j hj
Then A has all
h j hj hj hj
the elements of B plus the one additional element s. All subsets of B are also subsets of A;
hj hj hj hj hj hj hj hj hj h j hj hj hj hj hj hj hj hj
these are precisely the subsets of A that do not contain s, so the number of subsets of A
hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj hj
not containing s is |P(B)|. Any other subset of A must contain s, and removal of the s
hj hj hj hj hj h j hj hj hj hj hj hj hj hj hj hj hj hj
would produce a subset of
hj hj hj hj hj
