SOLUTION MANUAL for First Course In Abstract Algebra A 8th Edition By John B. Fraleigh All Chapters Full Complete PDF

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

, CONTENTS
0. Setsg andg Relations 1

I. Groups and Subgroups
g g




1. Introductiong andg Examples 4
2. Binaryg Operations 7
3. Isomorphicg Binaryg Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclicg Groups 21
7. Generatorsg andg Cayleyg Digraphs 24

II. Permutations, Cosets, and Direct Products
g g g g




8. Groupsg ofg Permutations 26
9. Orbits,g Cycles,g andg theg Alternatingg Groups 30
10. Cosetsg andg theg Theoremg ofg Lagrange 34
11. Directg Productsg andg Finitelyg Generatedg Abeliang Groups 37
12. Planeg Isometries 42

III. Homomorphisms and Factor Groups g g g




13. Homomorphisms 44
14. Factorg Groups 49
15. Factor-Groupg Computationsg andg Simpleg Groups53
16. Groupg Actiong ong ag Set 58
17. ApplicationsgofgG-SetsgtogCounting 61

IV. Rings and Fields
g g




18. Ringsg andg Fields 63
19. Integralg Domains 68
20. Fermat’sg andg Euler’sg Theorems 72
21. Theg Fieldg ofg Quotientsg ofg ang Integralg Domain 74
22. Ringsg ofg Polynomials 76
23. FactorizationgofgPolynomialsgovergagField79
24. Noncommutativeg Examples 85
25. Orderedg Ringsg andg Fields 87

V. Ideals and Factor Rings
g g g




26. Homomorphismsg andg Factorg Rings 89
27. PrimegandgMaximalg Ideals 94
28. Grö bner gBasesgforgIdeals 99

, VI. Extension Fields g




29. Introductiongtog Extensiong Fields 103
30. Vectorg Spaces 107
31. Algebraicg Extensions 111
32. Geometricg Constructions 115
33. Finiteg Fields 116

VII. Advanced Group Theory
g g




34. IsomorphismgTheorems 117
35. Seriesg ofgGroups 119
36. Sylowg Theorems 122
37. Applicationsg ofg theg Sylowg Theory 124
38. Freeg Abeliang Groups 128
39. FreegGroups 130
40. Groupg Presentations 133

VIII. Groups in Topology
g g




41. Simplicialg Complexesg andg Homologyg Groups 136
42. Computationsg ofg Homologyg Groups 138
43. Moreg Homologyg Computationsg andg Applications 140
44. Homologicalg Algebra 144

IX. Factorization
45. Uniqueg Factorizationg Domains 148
46. Euclideang Domains 151
47. Gaussiang Integersg andg Multiplicativeg Norms 154

X. Automorphisms and Galois Theory
g g g




48. Automorphismsg ofg Fields 159
49. Theg Isomorphismg Extensiong Theorem 164
50. Splittingg Fields 165
51. SeparablegExtensions 167
52. TotallygInseparableg Extensions 171
53. Galoisg Theory 173
54. IllustrationsgofgGaloisgTheory 176
55. CyclotomicgExtensions 183
56. Insolvabilityg ofg theg Quintic 185

APPENDIXg Matrixg Algebra 187


iv

, 0.g SetsgandgRelations 1

0. Sets and Relations
g g


√ √
1. { 3,g − 3} 2.g Theg setg isg empty.

3.g {1,g−1,g2,g−2,g3,g−3,g4,g−4,g5,g−5,g6,g−6,g10,g−10,g12,g−12,g15,g−15,g20,g−20,g30,g−30,
60,g−60}

4.g {−10,g−9,g−8,g−7,g−6,g−5,g−4,g−3,g−2,g−1,g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11}

5. Itgisgnotgagwell-
definedgset.g (SomegmaygarguegthatgnogelementgofgZ+gisglarge,gbecausegeverygelementgexceedsgonlyg
agfinitegnumbergofgothergelementsgbutgisgexceededgbyganginfinitegnumbergofgothergelements.gSuchgpe
oplegmightgclaimgtheganswergshouldgbeg∅.)

6. ∅ 7.g Theg setg isg ∅g becauseg 33g=g27g andg 43g=g64.

8.g Itg isg notg ag well-definedg set. 9.g Q

10. Theg setg containingg allg numbersg thatg areg (positive,g negative,g org zero)g integerg multiplesg ofg 1,g 1/2
,g org1/3.

11. {(a,g 1),g (a,g 2),g (a,g c),g (b,g 1),g (b,g 2),g (b,gc),g (c,g 1),g (c,g 2),g (c,gc)}
12. a.g Itg isg ag function.g Itgisg notg one-to-
oneg sinceg thereg areg twog pairsg withg secondg memberg 4.g Itg isg notg onto
Bg becauseg theregisg nog pairg withg secondg memberg 2.
b. (Sameg answerg asg Part(a).)
c. Itg isg notg ag functiong becauseg thereg areg twog pairsg withg firstg memberg 1.
d. Itg isg ag function.g Itg isg one-to-
one.g Itg isg ontog Bg becauseg everyg elementg ofg Bg appearsg asg secondgmembergofgsomegpair.
e. Itgisgagfunction.g Itgisgnotgone-to-
onegbecausegtheregaregtwogpairsgwithgsecondgmemberg6.g ItgisgnotgontogBgbecausegtheregisgnogpa
irgwithgsecondgmemberg2.
f. Itg isg notg ag functiong becauseg thereg areg twog pairsg withg firstg memberg 2.

13. Drawg theg lineg throughg Pg andg x,g andg letg yg beg itsg pointg ofg intersectiong withg theg lineg segmentg CD.

14. a.g φg :g [0,g1]g →g [0,g2]g whereg φ(x)g=g2x b.g φg :g [1,g3]g →g [5,g25]g whereg φ(x)g=g5g+g10(xg−g1)
c.g φg:g[a,gb]→ [c,gd]g whereg φ(x)g=gcg+g d−cg−(x a)
b−a

15. Letg φg :gSg →g Rg beg definedg byg φ(x)g=gtan(π(xg2−g 1g)).

16. a.g ∅;g cardinalityg 1 b.g ∅,g{a};g cardinalityg 2 c.g ∅,g{a},g{b},g{a,gb};g cardinalityg 4
d.g ∅,g{a},g{b},g{c},g{a,gb},g{a,gc},g{b,gc},g{a,gb,gc};g cardinalityg 8

17. Conjecture: |P(A)|g=g2sg =g2|A|.
ProofgThegnumbergofgsubsetsgofgagsetgAgdependsgonlygongthegcardinalitygofgA,gnotgongwhatgthe
gelements gofg Ag actuallyg are.g SupposegBg=g{1,g2,g3,g·g·g·g, gs g−g1}g and g Ag=g{1,g2,g3,g g ,gs }.g Then g Ag h

asg all
thegelementsgofgBgplusgthegonegadditionalgelementgs.g AllgsubsetsgofgBgaregalsogsubsetsgofgA;gthes
egaregpreciselygthegsubsetsgofgAgthatgdognotgcontaings,gsogthegnumbergofgsubsetsgofgAgnotgcontai
ninggsg isg |P(B)|.g Anyg otherg subsetgofgAgmustg contaings,g andgremovalg ofg thegsg wouldgproduceg a
g subsetgof