SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. SetsH andH Relations 1
I. Groups and Subgroups
H H
1. IntroductionH andH Examples 4
2. BinaryH Operations 7
3. IsomorphicH BinaryH Structures 9
4. Groups 13
5. Subgroups 17
6. CyclicH Groups 21
7. GeneratorsH andH CayleyH Digraphs 24
II. Permutations, Cosets, and Direct Products
H H H H
8. GroupsH ofH Permutations 26
9. Orbits,HCycles,HandHtheHAlternatingHGroups 30
10. CosetsH andH theH TheoremH ofH Lagrange 34
11. DirectH ProductsH andH FinitelyH GeneratedH AbelianH Groups 37
12. PlaneH Isometries 42
III. Homomorphisms and Factor Groups H H H
13. Homomorphisms 44
14. FactorH Groups 49
15. Factor-GroupH ComputationsH andH SimpleH Groups 53
16. GroupH ActionHonHaHSet 58
17. ApplicationsHofHG-SetsHtoHCounting 61
IV. Rings and Fields H H
18. RingsH andH Fields 63
19. IntegralH Domains 68
20. Fermat’sH andH Euler’sH Theorems 72
21. TheH FieldH ofH QuotientsH ofH anH IntegralH Domain 74
22. RingsH ofH Polynomials 76
23. FactorizationHofHPolynomialsHoverHaHField 79
24. NoncommutativeH Examples 85
25. OrderedH RingsH andH Fields 87
V. Ideals and Factor Rings
H H H
26. HomomorphismsH andH FactorH Rings 89
27. PrimeHandHMaximalHIdeals 94
28. Gröbner HBasesHforHIdeals 99
, VI. Extension Fields
H
29. IntroductionHtoHExtensionHFields 103
30. VectorH Spaces 107
31. AlgebraicH Extensions 111
32. GeometricHConstructions 115
33. FiniteH Fields 116
VII. Advanced Group Theory
H H
34. IsomorphismHTheorems 117
35. SeriesHofHGroups 119
36. SylowH Theorems 122
37. ApplicationsH ofH theH SylowH Theory 124
38. FreeH AbelianH Groups 128
39. FreeHGroups 130
40. GroupH Presentations 133
VIII. Groups in Topology
H H
41. SimplicialH ComplexesH andH HomologyH Groups 136
42. ComputationsH ofH HomologyH Groups 138
43. MoreH HomologyH ComputationsH andH Applications 140
44. HomologicalH Algebra 144
IX. Factorization
45. UniqueH FactorizationH Domains 148
46. EuclideanH Domains 151
47. GaussianH IntegersH andH MultiplicativeH Norms 154
X. Automorphisms and Galois Theory
H H H
48. AutomorphismsH ofH Fields 159
49. TheH IsomorphismH ExtensionH Theorem 164
50. SplittingH Fields 165
51. SeparableHExtensions 167
52. TotallyHInseparableHExtensions 171
53. GaloisH Theory 173
54. IllustrationsHofHGaloisHTheory 176
55. CyclotomicHExtensions 183
56. InsolvabilityH ofH theH Quintic 185
APPENDIXH MatrixH Algebra 187
iv
, 0.H SetsHandHRelations 1
0. Sets and Relations
H H
√ √
1. { 3,H − 3} 2.H TheH setH isH empty.
3.H {1,H−1,H2,H−2,H3,H−3,H4,H−4,H5,H−5,H6,H−6,H10,H−10,H12,H−12,H15,H−15,H20,H−20,H30,H−30,
60,H−60}
4.H {−10,H−9,H−8,H−7,H−6,H−5,H−4,H−3,H−2,H−1,H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11}
5. ItHisHnotHaHwell-
definedHset.H (SomeHmayHargueHthatHnoHelementHofHZ+HisHlarge,HbecauseHeveryHelementHexceedsHonlyHaH
finiteHnumberHofHotherHelementsHbutHisHexceededHbyHanHinfiniteHnumberHofHotherHelements.HSuchHpeopl
eHmightHclaimHtheHanswerHshouldHbeH∅.)
6. ∅ 7.H TheH setH isH ∅H becauseH 33H=H27H andH 43H=H64.
8.H ItH isH notH aH well-definedH set. 9.H Q
10. TheH setH containingH allH numbersH thatH areH (positive,H negative,H orH zero)H integerH multiplesH ofH 1,H 1/2,H
orH1/3.
11. {(a,H1),H (a,H 2),H (a,H c),H (b,H1),H (b,H2),H(b,H c),H (c,H1),H (c,H2),H(c,H c)}
12. a.H ItHisHaHfunction.H ItHisH notHone-to-
oneHsinceHthereHareHtwoHpairsHwithHsecondHmemberH4.H ItHisHnotHonto
BH becauseHthereHisH noHpairHwithHsecondHmemberH2.
b. (SameH answerH asH Part(a).)
c. ItH isH notH aH functionH becauseH thereH areH twoH pairsH withH firstH memberH 1.
d. ItH isH aH function.H ItH isH one-to-
one.H ItH isH ontoH BH becauseH everyH elementH ofH BH appearsH asH secondHmemberHofHsomeHpair.
e. ItHisHaHfunction.H ItHisHnotHone-to-
oneHbecauseHthereHareHtwoHpairsHwithHsecondHmemberH6.H ItHisHnotHontoHBHbecauseHthereHisHnoHpai
rHwithHsecondHmemberH2.
f. ItHisH notH aH functionH becauseH thereH areH twoH pairsH withH firstH memberH 2.
13. DrawH theH lineH throughH PH andH x,H andH letH yH beH itsH pointH ofH intersectionH withH theH lineH segmentH CD.
14. a.H φH:H[0,H1]H→H [0,H2]H whereH φ(x)H=H2x b.H φH:H[1,H3]H →H [5,H25]H whereH φ(x)H=H5H+H10(xH−H1)
c.H φH:H[a,Hb]→ [c,Hd]H whereH φ(x)H=HcH+H H(x− a)
d−c
b−a
15. LetH φH:HSH →HRH beH definedH byH φ(x)H=Htan(π(xH−H2 1H)).
16. a.H ∅;H cardinalityH 1 b.H ∅,H{a};H cardinalityH 2 c.H ∅,H{a},H{b},H{a,Hb};H cardinalityH 4
d.H ∅,H{a},H{b},H{c},H{a,Hb},H{a,Hc},H{b,Hc},H{a,Hb,Hc};H cardinalityH 8
17. Conjecture: |P(A)|H=H2sH =H2|A|.
ProofHTheHnumberHofHsubsetsHofHaHsetHAHdependsHonlyHonHtheHcardinalityHofHA,HnotHonHwhatHtheH
elementsHofH AH actuallyH are.H SupposeHBH=H{1,H2,H3,H·H·H·H,HsH−H1}H andH AH=H{1,H2,H3,H H ,Hs}.H ThenH AH ha
sH all
theHelementsHofHBHplusHtheHoneHadditionalHelementHs.H AllHsubsetsHofHBHareHalsoHsubsetsHofHA;Hthes
eHareHpreciselyHtheHsubsetsHofHAHthatHdoHnotHcontainHs,HsoHtheHnumberHofHsubsetsHofHAHnotHcontai
ningHsHisH|P(B)|.H AnyHotherHsubsetHofHAHmustHcontainHs,HandHremovalHofHtheHsHwouldHproduceHaHsu
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. SetsH andH Relations 1
I. Groups and Subgroups
H H
1. IntroductionH andH Examples 4
2. BinaryH Operations 7
3. IsomorphicH BinaryH Structures 9
4. Groups 13
5. Subgroups 17
6. CyclicH Groups 21
7. GeneratorsH andH CayleyH Digraphs 24
II. Permutations, Cosets, and Direct Products
H H H H
8. GroupsH ofH Permutations 26
9. Orbits,HCycles,HandHtheHAlternatingHGroups 30
10. CosetsH andH theH TheoremH ofH Lagrange 34
11. DirectH ProductsH andH FinitelyH GeneratedH AbelianH Groups 37
12. PlaneH Isometries 42
III. Homomorphisms and Factor Groups H H H
13. Homomorphisms 44
14. FactorH Groups 49
15. Factor-GroupH ComputationsH andH SimpleH Groups 53
16. GroupH ActionHonHaHSet 58
17. ApplicationsHofHG-SetsHtoHCounting 61
IV. Rings and Fields H H
18. RingsH andH Fields 63
19. IntegralH Domains 68
20. Fermat’sH andH Euler’sH Theorems 72
21. TheH FieldH ofH QuotientsH ofH anH IntegralH Domain 74
22. RingsH ofH Polynomials 76
23. FactorizationHofHPolynomialsHoverHaHField 79
24. NoncommutativeH Examples 85
25. OrderedH RingsH andH Fields 87
V. Ideals and Factor Rings
H H H
26. HomomorphismsH andH FactorH Rings 89
27. PrimeHandHMaximalHIdeals 94
28. Gröbner HBasesHforHIdeals 99
, VI. Extension Fields
H
29. IntroductionHtoHExtensionHFields 103
30. VectorH Spaces 107
31. AlgebraicH Extensions 111
32. GeometricHConstructions 115
33. FiniteH Fields 116
VII. Advanced Group Theory
H H
34. IsomorphismHTheorems 117
35. SeriesHofHGroups 119
36. SylowH Theorems 122
37. ApplicationsH ofH theH SylowH Theory 124
38. FreeH AbelianH Groups 128
39. FreeHGroups 130
40. GroupH Presentations 133
VIII. Groups in Topology
H H
41. SimplicialH ComplexesH andH HomologyH Groups 136
42. ComputationsH ofH HomologyH Groups 138
43. MoreH HomologyH ComputationsH andH Applications 140
44. HomologicalH Algebra 144
IX. Factorization
45. UniqueH FactorizationH Domains 148
46. EuclideanH Domains 151
47. GaussianH IntegersH andH MultiplicativeH Norms 154
X. Automorphisms and Galois Theory
H H H
48. AutomorphismsH ofH Fields 159
49. TheH IsomorphismH ExtensionH Theorem 164
50. SplittingH Fields 165
51. SeparableHExtensions 167
52. TotallyHInseparableHExtensions 171
53. GaloisH Theory 173
54. IllustrationsHofHGaloisHTheory 176
55. CyclotomicHExtensions 183
56. InsolvabilityH ofH theH Quintic 185
APPENDIXH MatrixH Algebra 187
iv
, 0.H SetsHandHRelations 1
0. Sets and Relations
H H
√ √
1. { 3,H − 3} 2.H TheH setH isH empty.
3.H {1,H−1,H2,H−2,H3,H−3,H4,H−4,H5,H−5,H6,H−6,H10,H−10,H12,H−12,H15,H−15,H20,H−20,H30,H−30,
60,H−60}
4.H {−10,H−9,H−8,H−7,H−6,H−5,H−4,H−3,H−2,H−1,H0,H1,H2,H3,H4,H5,H6,H7,H8,H9,H10,H11}
5. ItHisHnotHaHwell-
definedHset.H (SomeHmayHargueHthatHnoHelementHofHZ+HisHlarge,HbecauseHeveryHelementHexceedsHonlyHaH
finiteHnumberHofHotherHelementsHbutHisHexceededHbyHanHinfiniteHnumberHofHotherHelements.HSuchHpeopl
eHmightHclaimHtheHanswerHshouldHbeH∅.)
6. ∅ 7.H TheH setH isH ∅H becauseH 33H=H27H andH 43H=H64.
8.H ItH isH notH aH well-definedH set. 9.H Q
10. TheH setH containingH allH numbersH thatH areH (positive,H negative,H orH zero)H integerH multiplesH ofH 1,H 1/2,H
orH1/3.
11. {(a,H1),H (a,H 2),H (a,H c),H (b,H1),H (b,H2),H(b,H c),H (c,H1),H (c,H2),H(c,H c)}
12. a.H ItHisHaHfunction.H ItHisH notHone-to-
oneHsinceHthereHareHtwoHpairsHwithHsecondHmemberH4.H ItHisHnotHonto
BH becauseHthereHisH noHpairHwithHsecondHmemberH2.
b. (SameH answerH asH Part(a).)
c. ItH isH notH aH functionH becauseH thereH areH twoH pairsH withH firstH memberH 1.
d. ItH isH aH function.H ItH isH one-to-
one.H ItH isH ontoH BH becauseH everyH elementH ofH BH appearsH asH secondHmemberHofHsomeHpair.
e. ItHisHaHfunction.H ItHisHnotHone-to-
oneHbecauseHthereHareHtwoHpairsHwithHsecondHmemberH6.H ItHisHnotHontoHBHbecauseHthereHisHnoHpai
rHwithHsecondHmemberH2.
f. ItHisH notH aH functionH becauseH thereH areH twoH pairsH withH firstH memberH 2.
13. DrawH theH lineH throughH PH andH x,H andH letH yH beH itsH pointH ofH intersectionH withH theH lineH segmentH CD.
14. a.H φH:H[0,H1]H→H [0,H2]H whereH φ(x)H=H2x b.H φH:H[1,H3]H →H [5,H25]H whereH φ(x)H=H5H+H10(xH−H1)
c.H φH:H[a,Hb]→ [c,Hd]H whereH φ(x)H=HcH+H H(x− a)
d−c
b−a
15. LetH φH:HSH →HRH beH definedH byH φ(x)H=Htan(π(xH−H2 1H)).
16. a.H ∅;H cardinalityH 1 b.H ∅,H{a};H cardinalityH 2 c.H ∅,H{a},H{b},H{a,Hb};H cardinalityH 4
d.H ∅,H{a},H{b},H{c},H{a,Hb},H{a,Hc},H{b,Hc},H{a,Hb,Hc};H cardinalityH 8
17. Conjecture: |P(A)|H=H2sH =H2|A|.
ProofHTheHnumberHofHsubsetsHofHaHsetHAHdependsHonlyHonHtheHcardinalityHofHA,HnotHonHwhatHtheH
elementsHofH AH actuallyH are.H SupposeHBH=H{1,H2,H3,H·H·H·H,HsH−H1}H andH AH=H{1,H2,H3,H H ,Hs}.H ThenH AH ha
sH all
theHelementsHofHBHplusHtheHoneHadditionalHelementHs.H AllHsubsetsHofHBHareHalsoHsubsetsHofHA;Hthes
eHareHpreciselyHtheHsubsetsHofHAHthatHdoHnotHcontainHs,HsoHtheHnumberHofHsubsetsHofHAHnotHcontai
ningHsHisH|P(B)|.H AnyHotherHsubsetHofHAHmustHcontainHs,HandHremovalHofHtheHsHwouldHproduceHaHsu
