SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets, and, Relations 1
I. Groups, and, Subgroups
1. Introduction, and, Examples 4
2. Binary, Operations 7
3. Isomorphic, Binary, Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic, Groups 21
7. Generators, and, Cayley, Digraphs 24
II. Permutations,, Cosets,, and, Direct, Products
8. Groups, of, Permutations 26
9. Orbits,,Cycles,,and,the,Alternating,Groups 30
10. Cosets, and, the, Theorem, of, Lagrange 34
11. Direct, Products, and, Finitely, Generated, Abelian, Groups 37
12. Plane, Isometries 42
III. Homomorphisms, and, Factor, Groups
13. Homomorphisms 44
14. Factor, Groups 49
15. Factor-Group, Computations, and, Simple, Groups 53
16. Group,Action,on, a,Set 58
17. Applications,of,G-Sets,to,Counting 61
IV. Rings, and, Fields
18. Rings, and, Fields 63
19. Integral, Domains 68
20. Fermat’s, and, Euler’s, Theorems 72
21. The, Field, of, Quotients, of, an, Integral, Domain 74
22. Rings, of, Polynomials 76
23. Factorization,of,Polynomials,over,a,Field 79
24. Noncommutative, Examples 85
25. Ordered, Rings, and, Fields 87
V. Ideals, and, Factor, Rings
26. Homomorphisms, and, Factor, Rings 89
27. Prime,and,Maximal,Ideals 94
28. Gröbner ,Bases,for,Ideals 99
, VI. Extension, Fields
29. Introduction, to,Extension,Fields 103
30. Vector, Spaces 107
31. Algebraic, Extensions 111
32. Geometric, Constructions 115
33. Finite, Fields 116
VII. Advanced, Group, Theory
34. Isomorphism,Theorems 117
35. Series,of,Groups 119
36. Sylow, Theorems 122
37. Applications, of, the, Sylow, Theory 124
38. Free, Abelian, Groups 128
39. Free,Groups 130
40. Group, Presentations 133
VIII. Groups, in, Topology
41. Simplicial, Complexes, and, Homology, Groups 136
42. Computations, of, Homology, Groups 138
43. More, Homology, Computations, and, Applications 140
44. Homological, Algebra 144
IX. Factorization
45. Unique, Factorization, Domains 148
46. Euclidean, Domains 151
47. Gaussian, Integers, and, Multiplicative, Norms 154
X. Automorphisms, and, Galois, Theory
48. Automorphisms, of, Fields 159
49. The, Isomorphism, Extension, Theorem 164
50. Splitting, Fields 165
51. Separable,Extensions 167
52. Totally,Inseparable,Extensions 171
53. Galois, Theory 173
54. Illustrations,of,Galois,Theory 176
55. Cyclotomic,Extensions 183
56. Insolvability, of, the, Quintic 185
APPENDIX, Matrix, Algebra 187
iv
, 0., Sets,and,Relations 1
0. 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,clai
m,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,s
econd,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)
c., φ,:,[a,,b] → [c,,d], where, φ(x),=,c,+, d−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,element
s,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,p
recisely,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,cont
ains, 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
0. Sets, and, Relations 1
I. Groups, and, Subgroups
1. Introduction, and, Examples 4
2. Binary, Operations 7
3. Isomorphic, Binary, Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic, Groups 21
7. Generators, and, Cayley, Digraphs 24
II. Permutations,, Cosets,, and, Direct, Products
8. Groups, of, Permutations 26
9. Orbits,,Cycles,,and,the,Alternating,Groups 30
10. Cosets, and, the, Theorem, of, Lagrange 34
11. Direct, Products, and, Finitely, Generated, Abelian, Groups 37
12. Plane, Isometries 42
III. Homomorphisms, and, Factor, Groups
13. Homomorphisms 44
14. Factor, Groups 49
15. Factor-Group, Computations, and, Simple, Groups 53
16. Group,Action,on, a,Set 58
17. Applications,of,G-Sets,to,Counting 61
IV. Rings, and, Fields
18. Rings, and, Fields 63
19. Integral, Domains 68
20. Fermat’s, and, Euler’s, Theorems 72
21. The, Field, of, Quotients, of, an, Integral, Domain 74
22. Rings, of, Polynomials 76
23. Factorization,of,Polynomials,over,a,Field 79
24. Noncommutative, Examples 85
25. Ordered, Rings, and, Fields 87
V. Ideals, and, Factor, Rings
26. Homomorphisms, and, Factor, Rings 89
27. Prime,and,Maximal,Ideals 94
28. Gröbner ,Bases,for,Ideals 99
, VI. Extension, Fields
29. Introduction, to,Extension,Fields 103
30. Vector, Spaces 107
31. Algebraic, Extensions 111
32. Geometric, Constructions 115
33. Finite, Fields 116
VII. Advanced, Group, Theory
34. Isomorphism,Theorems 117
35. Series,of,Groups 119
36. Sylow, Theorems 122
37. Applications, of, the, Sylow, Theory 124
38. Free, Abelian, Groups 128
39. Free,Groups 130
40. Group, Presentations 133
VIII. Groups, in, Topology
41. Simplicial, Complexes, and, Homology, Groups 136
42. Computations, of, Homology, Groups 138
43. More, Homology, Computations, and, Applications 140
44. Homological, Algebra 144
IX. Factorization
45. Unique, Factorization, Domains 148
46. Euclidean, Domains 151
47. Gaussian, Integers, and, Multiplicative, Norms 154
X. Automorphisms, and, Galois, Theory
48. Automorphisms, of, Fields 159
49. The, Isomorphism, Extension, Theorem 164
50. Splitting, Fields 165
51. Separable,Extensions 167
52. Totally,Inseparable,Extensions 171
53. Galois, Theory 173
54. Illustrations,of,Galois,Theory 176
55. Cyclotomic,Extensions 183
56. Insolvability, of, the, Quintic 185
APPENDIX, Matrix, Algebra 187
iv
, 0., Sets,and,Relations 1
0. 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,clai
m,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,s
econd,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)
c., φ,:,[a,,b] → [c,,d], where, φ(x),=,c,+, d−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,element
s,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,p
recisely,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,cont
ains, s, or, does, not, contain, s, (but, not, both),, we, see, that, the, number, of, subsets, of, A, is, 2|P(B)|.
