Solution Manual for A First Course in Abstract Algebra 8th Edition by John B. Fraleigh (All Chapters)

SOLUTION MANUAL l




FirstCourseinAbstractAlgebraA
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8th E
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ditionbyJohnB.Fraleigh
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ll All C
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haptersFullComplete
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, CONTENTS
1. Setsl andl Relations 1

I. Groups and Subgroups
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2. Introductionl andl Examples 4
3. Binaryl Operations 7
4. Isomorphicl Binaryl Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclicll Groups 21
8. Generatorsl andl Cayleyl Digraphs 24

II. Permutations, Cosets, and Direct Products
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9. Groupsl oflPermutations 26
10. Orbits,lCycles,landlthelAlternatinglGroups
30
11. Cosetsl andlthel Theoreml ofl Lagrange 34
12. Directl Productsl andl Finitelyl Generatedl Abelianl Groups 37
13. Planel Isometries 42

III. Homomorphisms and Factor Groups l l l




14. Homomorphisms 44
15. Factorl Groups 49
16. Factor-Groupl Computationsl andl Simplel Groups 53
17. GrouplActionlonlalSet 58
18. ApplicationsloflG-SetsltolCounting 61

IV. Rings and Fields
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19. RingslandlFields 63
20. Integrall Domains 68
21. Fermat’sl andl Euler’sl Theorems 72
22. Thel Fieldl ofl Quotientsl ofl anl Integrall Domain 74
23. Ringsl ofl Polynomials 76
24. FactorizationloflPolynomialsloverlalField 79
25. NoncommutativelExamples 85
26. Orderedl Ringsl andl Fields 87

V. Ideals and Factor Rings
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27. HomomorphismslandlFactorlRings 89

,28. PrimelandlMaximallIdeals 94
29. h
VI. Extension Fields l




30. IntroductionltolExtensionlFields 103
31. Vectorl Spaces 107
32. Algebraicl Extensions 111
33. GeometriclConstructions 115
34. Finitel Fields 116

VII. Advanced Group Theoryl l




35. IsomorphismlTheorems 117
36. SeriesloflGroups 119
37. Sylowl Theorems 122
38. Applicationsl ofl thel Sylowl Theory 124
39. Freel Abelianl Groups 128
40. FreelGroups 130
41. Groupl Presentations 133

VIII. Groups in Topology l l




42. Simpliciall Complexesl andl Homologyl Groups 136
43. Computationsl ofl HomologylGroups 138
44. MorelHomologylComputationslandlApplications 140
45. HomologicallAlgebra 144

IX. Factorization
46. Uniquel Factorizationl Domains 148
47. Euclideanl Domains 151
48. Gaussianl Integersl andl Multiplicativel Norms 154

X. Automorphisms and Galois Theory
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49. AutomorphismsloflFields 159
50. Thel Isomorphisml Extensionl Theorem 164
51. Splittingl Fields 165
52. SeparablelExtensions 167
53. TotallylInseparablelExtensions 171
54. Galoisl Theory 173
55. IllustrationsloflGaloislTheory 176
56. CyclotomiclExtensions 183
57. Insolvabilityl ofl thel Quintic 185

APPENDIXll Matrixll Algebra 187

, 0.l SetslandlRelations 1

1. Sets and Relations
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√ √lll
1.ll { 3,l − 3} 2.l Thel setl isl empty.
3.l {1,l−1,l2,l−2,l3,l−3,l4,l−4,l5,l−5,l6,l−6,l10,l−10,l12,l−12,l15,l−15,l20,l−20,l30,l−30,
60,l−60}

4.l {−10,l−9,l−8,l−7,l−6,l−5,l−4,l−3,l−2,l−1,l0,l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,l11}
5. Itl isl notl al well-
definedl set.l(Somel mayl arguel thatl nol elementl ofl Z+l isl large,l becausel everyl elementl exceedslonlylalfiniteln
umberloflotherlelementslbutlislexceededlbylanlinfinitelnumberloflotherlelements.l Suchlpeoplelmightlclaimlt
helanswerlshouldlbel∅.)

6. ∅ 7.l Thel setl isl ∅l becausel 33l =l27l andl 43l =l64.

8.l Itl isl notl al well-definedl set. 9.l Q

10.l Thel setl containingl alll numbersl thatl arel (positive,l negative,l orl zero)l integerl multiplesl ofl 1,l 1/2,l orl 1/3.

11.lll{(a,l1),l(a,l 2),l (a,l c),l(b,l1),l (b,l 2),l(b,lc),l (c,l1),l(c,l 2),l(c,lc)}

12. a.l Itlislalfunction.l Itlisl notlone-to-onelsinceltherelareltwol pairslwithlsecondlmemberl4.l Itlislnotlonto
Bl becauseltherelislnol pairlwithlsecondlmemberl2.
b. (Samel answerl asl Part(a).)
c. Itlisl notl al functionl becauseltherelarel twol pairsl withl firstlmemberl 1.
d. Itl isl al function.llItl isl one-to-
one.llItl isl ontol Bl becausel everyl elementl ofl Bl appearsl asl secondl memberloflsomelpair.
e. Itlislalfunction.lItlislnotlone-to-
onelbecauseltherelareltwolpairslwithlsecondlmemberl6.l Itlislnotl ontolBlbecauseltherelislnolpairlwithlse
condlmemberl2.
f. Itlisl notl al functionl becauseltherelarel twol pairsl withl firstlmemberl 2.

13. Drawl thel linel throughl Pl andl x,l andl letl yl bel itsl pointl ofl intersectionl withl thel linel segmentl CD.

14.ll a.l φl:l [0,l1]l→l [0,l2]l wherel φ(x)l=l2x b.l φl:l [1,l3]l →l [5,l25]l wherel φ(x)l=l5l+l10(xl−l1)
c.l φl :l [a,lb]l→l [c,ld]l wherel φ(x)l =l cl+l d−cl(xl −lla)
b−ll a

15. Letl φl:lSl →lRl bel definedl byl φ(x)l=ltan(π(xl− 1
)2).
16. a.l ∅;l cardinalityl 1 b.l ∅,l{a};l cardinalityl 2 c.l ∅,l{a},l{b},l{a,lb};l cardinalityl 4
d.l ∅,l{a},l{b},l{c},l{a,lb},l{a,lc},l{b,lc},l{a,lb,lc};l cardinalityl 8

17. Conjecture:l |P(A)|l=l2sl =l2|A|.
Proofl Thel numberl ofl subsetsl ofl al setl Al dependsl onlyl onl thel cardinalityl ofl A,l notl onl whatl thel elem
entslofl Al actuallyl are.l Supposel Bl =l{1,l2,l3,l·l·l·l,lsl−l1}l andl Al=l{1,l2,l3,
,ls}.l Thenl Al hasl alll t
helelementslofl Blplusltheloneladditionallelementls.l AlllsubsetsloflBlarelalsolsubsetsloflA;lthesel arelpr
eciselylthelsubsetsloflAlthatldolnotlcontainls,lsolthelnumberloflsubsetsloflAlnotlcontainingl slisl|P(B)|.l
AnylotherlsubsetloflAlmustlcontainls,landlremovalloflthelslwouldlproducelalsubsetlof