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. Gröbner Bases for Ideals 99
, VI. Extension Fields f
30. IntroductionftofExtensionfFields 103
31. Vectorf Spaces 107
32. Algebraicf Extensions 111
33. GeometricfConstructions 115
34. Finitef Fields 116
VII. Advanced Group Theoryf f
35. IsomorphismfTheorems 117
36. SeriesfoffGroups 119
37. Sylowf Theorems 122
38. Applicationsf off thef Sylowf Theory 124
39. Freef Abelianf Groups 128
40. FreefGroups 130
41. Groupf Presentations 133
VIII. Groups in Topology
f f
42. Simplicialf Complexesf andf Homologyf Groups 136
43. Computationsfoff HomologyfGroups 138
44. MorefHomologyfComputationsfandfApplications 140
45. HomologicalfAlgebra 144
IX. Factorization
46. Uniquef Factorizationf Domains 148
47. Euclideanf Domains 151
48. Gaussianf Integersf andf Multiplicativef Norms 154
X. Automorphisms and Galois Theory
f f f
49. AutomorphismsfoffFields 159
50. Thef Isomorphismf Extensionf Theorem 164
51. Splittingf Fields 165
52. SeparablefExtensions 167
53. TotallyfInseparablefExtensions 171
54. Galoisf Theory 173
55. IllustrationsfoffGaloisfTheory 176
56. CyclotomicfExtensions 183
57. Insolvabilityf off thef Quintic 185
APPENDIXff Matrixff Algebra 187
iv
, 0.f SetsfandfRelations 1
1. Sets and Relations
f f
√ √fff
1.ff { 3,f − 3} 2.f Thef setf isf empty.
3.f {1,f−1,f2,f−2,f3,f−3,f4,f−4,f5,f−5,f6,f−6,f10,f−10,f12,f−12,f15,f−15,f20,f−20,f30,f−30,
60,f−60}
4.f {−10,f−9,f−8,f−7,f−6,f−5,f−4,f−3,f−2,f−1,f0,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11}
5. Itf isf notf af well-
definedf set.f(Somef mayf arguef thatf nof elementf off Z+f isf large,f becausef everyf elementf exceedsfonlyfaffinitef
numberfoffotherfelementsfbutfisfexceededfbyfanfinfinitefnumberfoffotherfelements.f Suchfpeoplefmightfclai
mfthefanswerfshouldfbef∅.)
6. ∅ 7.f Thef setf isf ∅f becausef 33f =f27f andf 43f =f64.
8.f Itf isf notf af well-definedf set. 9.f Q
10.f Thef setf containingf allf numbersf thatf aref (positive,f negative,f orf zero)f integerf multiplesf off 1,f 1/2,f orf 1/3.
11.fff{(a,f1),f(a,f2),f (a,fc),f (b,f1),f (b,f 2),f(b,f c),f(c,f1),f (c,f 2),f(c,fc)}
12. a.f Itfisfaffunction.f Itfisf notfone-to-onefsinceftherefareftwof pairsfwithfsecondfmemberf4.f Itfisfnotfonto
Bf becauseftherefisfnof pairfwithfsecondfmemberf2.
b. (Samef answerf asf Part(a).)
c. Itfisf notf af functionf becausef therefaref twof pairsfwithf firstf memberf 1.
d. Itf isf af function.ffItf isf one-to-
one.ffItf isf ontof Bf becausef everyf elementf off Bf appearsf asf secondf memberfoffsomefpair.
e. Itfisfaffunction.f Itfisfnotfone-to-
onefbecauseftherefareftwofpairsfwithfsecondfmemberf6.f Itfisfnotf ontofBfbecauseftherefisfnofpairfwithfs
econdfmemberf2.
f. Itfisf notf af functionf becausef therefaref twof pairsfwithf firstf memberf 2.
13. Drawf thef linef throughf Pf andf x,f andf letf yf bef itsf pointf off intersectionf withf thef linef segmentf CD.
14.ff a.f φf:f [0,f1]f→f [0,f2]f wheref φ(x)f=f2x b.f φf:f [1,f3]f →f [5,f25]f wheref φ(x)f=f5f+f10(xf−f1)
c.f φf :f [a,fb]f→f [c,fd]f wheref φ(x)f =f cf+f d−cf(xf −ffa)
b−
ff a
15. Letfφf:fSf →fRf bef definedf byf φ(x)f=ftan(π(xf− 1
)2).
16. a.f ∅;f cardinalityf 1 b.f ∅,f{a};f cardinalityf 2 c.f ∅,f{a},f{b},f{a,fb};f cardinalityf 4
d.f ∅,f{a},f{b},f{c},f{a,fb},f{a,fc},f{b,fc},f{a,fb,fc};f cardinalityf 8
17. Conjecture:f |P(A)|f=f2sf =f2|A|.
Prooff Thef numberf off subsetsf off af setf Af dependsf onlyf onf thef cardinalityf off A,f notf onf whatf thef ele
mentsfoff Af actuallyf are.f Supposef Bf =f{1,f2,f3,f·f·f·f,fsf−f1}f andf Af=f{1,f2,f3,
,fs}.f Thenf Af hasf allf t
hefelementsfoff Bfplusfthef onefadditionalfelementf s.f AllfsubsetsfoffBfarefalsofsubsetsfoff A;fthesef aref
preciselyfthefsubsetsfoffAfthatfdofnotfcontainfs,fsofthefnumberfoffsubsetsfoffAfnotfcontainingf sfisf|P(
B)|.f AnyfotherfsubsetfoffAfmustfcontainfs,fandfremovalfoffthefsfwouldfproducefafsubsetfof
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. Gröbner Bases for Ideals 99
, VI. Extension Fields f
30. IntroductionftofExtensionfFields 103
31. Vectorf Spaces 107
32. Algebraicf Extensions 111
33. GeometricfConstructions 115
34. Finitef Fields 116
VII. Advanced Group Theoryf f
35. IsomorphismfTheorems 117
36. SeriesfoffGroups 119
37. Sylowf Theorems 122
38. Applicationsf off thef Sylowf Theory 124
39. Freef Abelianf Groups 128
40. FreefGroups 130
41. Groupf Presentations 133
VIII. Groups in Topology
f f
42. Simplicialf Complexesf andf Homologyf Groups 136
43. Computationsfoff HomologyfGroups 138
44. MorefHomologyfComputationsfandfApplications 140
45. HomologicalfAlgebra 144
IX. Factorization
46. Uniquef Factorizationf Domains 148
47. Euclideanf Domains 151
48. Gaussianf Integersf andf Multiplicativef Norms 154
X. Automorphisms and Galois Theory
f f f
49. AutomorphismsfoffFields 159
50. Thef Isomorphismf Extensionf Theorem 164
51. Splittingf Fields 165
52. SeparablefExtensions 167
53. TotallyfInseparablefExtensions 171
54. Galoisf Theory 173
55. IllustrationsfoffGaloisfTheory 176
56. CyclotomicfExtensions 183
57. Insolvabilityf off thef Quintic 185
APPENDIXff Matrixff Algebra 187
iv
, 0.f SetsfandfRelations 1
1. Sets and Relations
f f
√ √fff
1.ff { 3,f − 3} 2.f Thef setf isf empty.
3.f {1,f−1,f2,f−2,f3,f−3,f4,f−4,f5,f−5,f6,f−6,f10,f−10,f12,f−12,f15,f−15,f20,f−20,f30,f−30,
60,f−60}
4.f {−10,f−9,f−8,f−7,f−6,f−5,f−4,f−3,f−2,f−1,f0,f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11}
5. Itf isf notf af well-
definedf set.f(Somef mayf arguef thatf nof elementf off Z+f isf large,f becausef everyf elementf exceedsfonlyfaffinitef
numberfoffotherfelementsfbutfisfexceededfbyfanfinfinitefnumberfoffotherfelements.f Suchfpeoplefmightfclai
mfthefanswerfshouldfbef∅.)
6. ∅ 7.f Thef setf isf ∅f becausef 33f =f27f andf 43f =f64.
8.f Itf isf notf af well-definedf set. 9.f Q
10.f Thef setf containingf allf numbersf thatf aref (positive,f negative,f orf zero)f integerf multiplesf off 1,f 1/2,f orf 1/3.
11.fff{(a,f1),f(a,f2),f (a,fc),f (b,f1),f (b,f 2),f(b,f c),f(c,f1),f (c,f 2),f(c,fc)}
12. a.f Itfisfaffunction.f Itfisf notfone-to-onefsinceftherefareftwof pairsfwithfsecondfmemberf4.f Itfisfnotfonto
Bf becauseftherefisfnof pairfwithfsecondfmemberf2.
b. (Samef answerf asf Part(a).)
c. Itfisf notf af functionf becausef therefaref twof pairsfwithf firstf memberf 1.
d. Itf isf af function.ffItf isf one-to-
one.ffItf isf ontof Bf becausef everyf elementf off Bf appearsf asf secondf memberfoffsomefpair.
e. Itfisfaffunction.f Itfisfnotfone-to-
onefbecauseftherefareftwofpairsfwithfsecondfmemberf6.f Itfisfnotf ontofBfbecauseftherefisfnofpairfwithfs
econdfmemberf2.
f. Itfisf notf af functionf becausef therefaref twof pairsfwithf firstf memberf 2.
13. Drawf thef linef throughf Pf andf x,f andf letf yf bef itsf pointf off intersectionf withf thef linef segmentf CD.
14.ff a.f φf:f [0,f1]f→f [0,f2]f wheref φ(x)f=f2x b.f φf:f [1,f3]f →f [5,f25]f wheref φ(x)f=f5f+f10(xf−f1)
c.f φf :f [a,fb]f→f [c,fd]f wheref φ(x)f =f cf+f d−cf(xf −ffa)
b−
ff a
15. Letfφf:fSf →fRf bef definedf byf φ(x)f=ftan(π(xf− 1
)2).
16. a.f ∅;f cardinalityf 1 b.f ∅,f{a};f cardinalityf 2 c.f ∅,f{a},f{b},f{a,fb};f cardinalityf 4
d.f ∅,f{a},f{b},f{c},f{a,fb},f{a,fc},f{b,fc},f{a,fb,fc};f cardinalityf 8
17. Conjecture:f |P(A)|f=f2sf =f2|A|.
Prooff Thef numberf off subsetsf off af setf Af dependsf onlyf onf thef cardinalityf off A,f notf onf whatf thef ele
mentsfoff Af actuallyf are.f Supposef Bf =f{1,f2,f3,f·f·f·f,fsf−f1}f andf Af=f{1,f2,f3,
,fs}.f Thenf Af hasf allf t
hefelementsfoff Bfplusfthef onefadditionalfelementf s.f AllfsubsetsfoffBfarefalsofsubsetsfoff A;fthesef aref
preciselyfthefsubsetsfoffAfthatfdofnotfcontainfs,fsofthefnumberfoffsubsetsfoffAfnotfcontainingf sfisf|P(
B)|.f AnyfotherfsubsetfoffAfmustfcontainfs,fandfremovalfoffthefsfwouldfproducefafsubsetfof
