Solution Manual First Course in Abstract Algebra A, 8th Edition by John B. Fraleigh All 1-56 Chapters Covered ,Latest Edition

SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh All
Chapters Full Complete

, CONTENTS
1. SetsQ andQRelations 1

I. GroupsQ andQ Subgroups

2. IntroductionQ andQ Examples 4
3. BinaryQ Operations 7
4. IsomorphicQ BinaryQ Structures 9
5. Groups 13
6. Subgroups 17
7. CyclicQQ Groups 21
8. GeneratorsQ andQ CayleyQ Digraphs 24

II. Permutations,QCosets,QandQDirectQProducts

9. GroupsQofQPermutations 26
10. Orbits,QCycles,QandQtheQAlternatingQGroups
30
11. CosetsQandQtheQTheoremQofQLagrange 34
12. DirectQ ProductsQ andQ FinitelyQ GeneratedQ AbelianQ Groups 37
13. PlaneQ Isometries 42

III. HomomorphismsQ andQ FactorQ Groups

14. Homomorphisms 44
15. FactorQ Groups 49
16. Factor-GroupQ ComputationsQ andQ SimpleQ Groups 53
17. GroupQActionQonQaQSet 58
18. ApplicationsQofQG-SetsQtoQCounting 61

IV. RingsQ andQ Fields

19. RingsQandQFields 63
20. IntegralQ Domains 68
21. Fermat’sQ andQ Euler’sQ Theorems 72
22. TheQ FieldQ ofQ QuotientsQ ofQ anQ IntegralQ Domain 74
23. RingsQ ofQ Polynomials 76
24. FactorizationQofQPolynomialsQoverQaQField 79
25. NoncommutativeQExamples 85
26. OrderedQ RingsQ andQ Fields 87

V. IdealsQ andQ FactorQ Rings

,27. HomomorphismsQandQFactorQRings 89
28. PrimeQandQMaximalQIdeals 94
29. GröbnerQBasesQforQIdeals 99

VI. ExtensionQ Fields

30. IntroductionQtoQExtensionQFields 103
31. VectorQ Spaces 107
32. AlgebraicQ Extensions 111
33. GeometricQConstructions 115
34. FiniteQ Fields 116

VII. AdvancedQGroupQTheory

35. IsomorphismQTheorems 117
36. SeriesQofQGroups 119
37. SylowQ Theorems 122
38. ApplicationsQ ofQ theQ SylowQ Theory 124
39. FreeQ AbelianQ Groups 128
40. FreeQGroups 130
41. GroupQ Presentations 133

VIII. GroupsQ inQ Topology

42. SimplicialQ ComplexesQ andQ HomologyQ Groups136
43. ComputationsQofQHomologyQGroups 138
44. MoreQHomologyQComputationsQandQApplications 140
45. HomologicalQAlgebra 144

IX. Factorization
46. UniqueQ FactorizationQ Domains 148
47. EuclideanQ Domains 151
48. GaussianQ IntegersQ andQ MultiplicativeQ Norms 154

X. AutomorphismsQ andQ GaloisQ Theory
49. AutomorphismsQofQFields 159
50. TheQ IsomorphismQ ExtensionQ Theorem 164
51. SplittingQ Fields 165
52. SeparableQExtensions 167
53. TotallyQInseparableQExtensions 171
54. GaloisQ Theory 173
55. IllustrationsQofQGaloisQTheory 176
56. CyclotomicQExtensions 183
57. InsolvabilityQ ofQ theQ Quintic 185

, APPENDIXQQ MatrixQQ Algebra 187


iv
0.Q SetsQandQRelations 1

1. SetsQ andQ Relations
√ √QQQ
1.Q 3,Q 3} 2.Q TheQ setQ isQ empty.
Q{ −
3.Q {1,Q−1,Q2,Q−2,Q3,Q−3,Q4,Q−4,Q5,Q−5,Q6,Q−6,Q10,Q−10,Q12,Q−12,Q15,Q−15,Q20,Q−20,Q30,Q−30,
60,Q−60}

4.Q {−10,Q−9,Q−8,Q−7,Q−6,Q−5,Q−4,Q−3,Q−2,Q−1,Q0,Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,Q9,Q10,Q11}
5. ItQ isQ notQ aQ well-
definedQ set.Q(SomeQ mayQ argueQ thatQ noQ elementQ ofQ Z+Q isQ large,Q becauseQ everyQ elementQ e
xceedsQonlyQaQfiniteQnumberQofQotherQelementsQbutQisQexceededQbyQanQinfiniteQnumberQofQot
herQelements.Q SuchQpeopleQmightQclaimQtheQanswerQshouldQbeQ∅.)

6. ∅ 7.Q TheQ setQ isQ ∅Q becauseQ 33Q =Q27Q andQ 43Q =Q64.

8.Q ItQ isQ notQ aQ well-definedQ set. 9.Q Q

10.Q TheQ setQ containingQ allQ numbersQ thatQ areQ (positive,Q negative,Q orQ zero)Q integerQ multiplesQ of
Q 1,Q 1/2,Q orQ 1/3.

11.QQQ{(a,Q1),Q (a,Q2),Q(a,Q c),Q(b,Q1),Q (b,Q 2),Q(b,Q c),Q (c,Q1),Q (c,Q2),Q (c,Qc)}

12. a.Q ItQisQaQfunction.Q ItQisQnotQone-to-
oneQsinceQthereQareQtwoQpairsQwithQsecondQmemberQ4.Q ItQisQnotQonto
BQ becauseQthereQisQnoQpairQwithQsecondQmemberQ2.
b. (SameQ answerQ asQ Part(a).)
c. ItQisQnotQaQfunctionQbecauseQthereQareQtwoQpairsQwithQfirstQmemberQ1.
d. ItQ isQ aQ function.QQItQ isQ one-to-
one.QQItQ isQ ontoQ BQ becauseQ everyQ elementQ ofQ BQ appearsQ asQ secondQ memberQofQsom
eQpair.
e. ItQisQaQfunction.QItQisQnotQone-to-
oneQbecauseQthereQareQtwoQpairsQwithQsecondQmemberQ6.Q ItQisQnotQ ontoQBQbecauseQt
hereQisQnoQpairQwithQsecondQmemberQ2.
f. ItQisQnotQaQfunctionQbecauseQthereQareQtwoQpairsQwithQfirstQmemberQ2.

13. DrawQ theQ lineQ throughQ PQ andQ x,Q andQ letQ yQ beQ itsQ pointQ ofQ intersectionQ withQ theQ lineQ seg
mentQ CD.

14.QQ a.Q φQ:Q [0,Q1]Q→Q[0,Q2]Q whereQ φ(x)Q=Q2x
b.Q φQ:Q[1,Q3]Q→Q [5,Q25]Q whereQ φ(x)Q=Q5Q+Q10(xQ−Q1
)
c.Q φQ:Q[a,Qb]Q→Q [c,Qd]Q whereQ φ(x)Q =Q cQ+Q d−cQ(xQ −QQa)
− a
bQQ
Qtan(π(xQ−
15. LetQ φQ:QSQ →QRQ beQ definedQ byQ φ(x)Q= 1