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