SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Setsj o andj o Relations 1
I. Groupsj o andj o Subgroups
1. Introductionj o andj o Examples 4
2. Binaryj o Operations 7
3. Isomorphicj o Binaryj o Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclicj o Groups 21
7. Generatorsj o andj o Cayleyj o Digraphs 24
II. Permutations,jo Cosets,jo andjo Directjo Products
8. Groupsj o ofj o Permutations 26
9. Orbits,joCycles,joandjothejoAlternatingjoGroups 30
10. Cosetsjo andjo thejo Theoremjo ofjo Lagrange 34
11. Directj o Productsj o andj o Finitelyj o Generatedj o Abelianj o Groups 37
12. Planej o Isometries 42
III. Homomorphismsj o andj o Factorj o Groups
13. Homomorphisms 44
14. Factorj o Groups 49
15. Factor-Groupj o Computationsj o andj o Simplej o Groups 53
16. GroupjoActionjoonjo ajoSet 58
17. ApplicationsjoofjoG-SetsjotojoCounting 61
IV. Ringsj o andj o Fields
18. Ringsjo andjo Fields 63
19. Integralj o Domains 68
20. Fermat’sj o andj o Euler’sj o Theorems 72
21. Thej o Fieldj o ofj o Quotientsj o ofj o anj o Integralj o Domain 74
22. Ringsj o ofj o Polynomials 76
23. FactorizationjoofjoPolynomialsjooverjoajoField 79
24. Noncommutativej o Examples 85
25. Orderedj o Ringsj o andj o Fields 87
V. Idealsj o andj o Factorj o Rings
26. Homomorphismsjo andjo FactorjoRings 89
27. PrimejoandjoMaximaljoIdeals94
28. Gröbner joBasesjoforjoIdeals 99
, VI. Extensionj o Fields
29. Introductionjo tojo Extensionjo Fields 103
30. Vectorj o Spaces 107
31. Algebraicj o Extensions 111
32. GeometricjoConstructions 115
33. Finitej o Fields 116
VII. Advancedjo Groupjo Theory
34. IsomorphismjoTheorems 117
35. SeriesjoofjoGroups 119
36. Sylowj o Theorems 122
37. Applicationsj o ofj o thej o Sylowj o Theory 124
38. Freej o Abelianj o Groups 128
39. FreejoGroups 130
40. Groupj o Presentations 133
VIII. Groupsj o inj o Topology
41. Simplicialj o Complexesj o andj o Homologyj o Groups 136
42. Computationsj o ofj o Homologyjo Groups 138
43. Morejo Homologyjo Computationsjo andjo Applications 140
44. Homologicaljo Algebra 144
IX. Factorization
45. Uniquej o Factorizationj o Domains 148
46. Euclideanj o Domains 151
47. Gaussianj o Integersj o andj o Multiplicativej o Norms 154
X. Automorphismsj o andj o Galoisj o Theory
48. Automorphismsjo ofjo Fields 159
49. Thej o Isomorphismj o Extensionj o Theorem 164
50. Splittingj o Fields 165
51. SeparablejoExtensions 167
52. TotallyjoInseparablejoExtensions 171
53. Galoisj o Theory 173
54. IllustrationsjoofjoGaloisjoTheory 176
55. CyclotomicjoExtensions 183
56. Insolvabilityj o ofj o thej o Quintic 185
APPENDIXj o Matrixj o Algebra 187
iv
, 0.j o SetsjoandjoRelations 1
0. Setsj o andj o Relations
√ √
1. { 3,j o − 3} 2.j o Thej o setj o isj o empty.
3.j o {1,jo−1,jo2,jo−2,jo3,jo−3,jo4,jo−4,jo5,jo−5,jo6,jo−6,jo10,jo−10,jo12,jo−12,jo15,jo−15,jo20,jo−20,jo30,jo−30,
60,jo−60}
4.j o {−10,jo−9,jo−8,jo−7,jo−6,jo−5,jo−4,jo−3,jo−2,jo−1,jo0,jo1,jo2,jo3,jo4,jo5,jo6,jo7,jo8,jo9,jo10,jo11}
5. Itjoisjonotjoajowell-
definedjoset.j o (SomejomayjoarguejothatjonojoelementjoofjoZ+joisjolarge,jobecausejoeveryjoelementjoexc
eedsjoonlyjoajofinitejonumberjoofjootherjoelementsjobutjoisjoexceededjobyjoanjoinfinitejonumberjoofjootherjoel
ements.joSuchjopeoplejomightjoclaimjothejoanswerjoshouldjobejo∅.)
6. ∅ 7.j o Thejo setjo isjo ∅jo becausejo 33jo=jo27jo andjo 43jo=jo64.
8.j o Itjo isjo notjo ajo well-definedjo set. 9.j o Q
10. Thej o setj o containingj o allj o numbersj o thatj o arej o (positive,j o negative,j o orj o zero)j o integerj o multiples
j o ofj o 1,j o 1/2,j o orjo1/3.
11. {(a,jo1),jo(a,jo2),jo(a,joc),jo (b,jo1),jo(b,jo2),jo(b,joc),jo(c,jo1),jo(c,jo2),jo(c,joc)}
12. a.j o Itjoisjoajofunction.j o Itjoisjonotjoone-to-
onejosincejotherejoarejo twojopairsjowithjosecondjomemberjo4.j o Itjoisjonotjoonto
Bjo becausejotherejoisjonojopairjowithjosecondjomemberjo2.
b. (Samej o answerj o asj o Part(a).)
c. Itjo isjo notjo ajo functionjo becausejo therejo arejo twojo pairsjo withjo firstjo memberjo 1.
d. Itjo isjo ajo function.j o Itjo isjo one-to-
one.j o Itjo isjo ontojo Bj o becausejo everyjo elementjo ofjo Bj o appearsjo asjo secondjomemberjoofjosome
jopair.
e. Itjoisjoajofunction.jo Itjoisjonotjoone-to-
onejobecausejotherejoarejotwojopairsjowithjosecondjomemberjo6.jo ItjoisjonotjoontojoBjobecausejother
ejoisjonojopairjowithjosecondjomemberjo2.
f. Itjoisjo notjo ajo functionjo becausejo therejo arejo twojo pairsjo withjo firstjo memberjo 2.
13. Drawjo thej o linej o throughjo Pj o andj o x,jo andj o letj o yj o bej o itsj o pointj o ofj o intersectionj o withj o thej o linej o se
gmentj o CD.
14. a.j o φjo:jo[0,jo1]jo→jo[0,jo2]jo wherej o φ(x)jo=jo2x
b.j o φjo:jo [1,jo3]jo→jo[5,jo25]j o wherej o φ(x)jo=jo5jo+jo10(xjo−jo1)
c.j o φjo:jo[a,→
job] d c
− [c,jod]j o wherej o φ(x)jo=jocjo+j o − jo(x a)
b−a
1jo
15. Letjo φjo:joSjo →joRjo bejo definedjo byjo φ(x)jo=jotan(π(x
2
jo−jo )).
16. a.j o ∅;j o cardinalityjo 1 b.j o ∅,jo{a};j o cardinalityj o 2 c.j o ∅,jo{a},jo{b},jo{a,job};j o cardinalityj o 4
d.j o ∅,jo{a},jo{b},jo{c},jo{a,job},jo{a,joc},jo{b,joc},jo{a,job,joc};jo cardinalityj o 8
17. Conjecture: |P(A)|jo=jo2sjo =jo2|A|.
ProofjoThejonumberjoofjosubsetsjoofjoajosetjoAjodependsjoonlyjoonjothejocardinalityjoofjoA,jonotjoon
jowhatjothejoelementsjoofj o Ajo actuallyjo are.j o SupposejoBjo=jo{1,jo2,jo3,jo·jo·jo·jo,josjo−jo1}j o andj o Ajo=jo
{1,jo2,jo3,j o j o ,jos}.j o Thenjo Ajo hasjo all
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Setsj o andj o Relations 1
I. Groupsj o andj o Subgroups
1. Introductionj o andj o Examples 4
2. Binaryj o Operations 7
3. Isomorphicj o Binaryj o Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclicj o Groups 21
7. Generatorsj o andj o Cayleyj o Digraphs 24
II. Permutations,jo Cosets,jo andjo Directjo Products
8. Groupsj o ofj o Permutations 26
9. Orbits,joCycles,joandjothejoAlternatingjoGroups 30
10. Cosetsjo andjo thejo Theoremjo ofjo Lagrange 34
11. Directj o Productsj o andj o Finitelyj o Generatedj o Abelianj o Groups 37
12. Planej o Isometries 42
III. Homomorphismsj o andj o Factorj o Groups
13. Homomorphisms 44
14. Factorj o Groups 49
15. Factor-Groupj o Computationsj o andj o Simplej o Groups 53
16. GroupjoActionjoonjo ajoSet 58
17. ApplicationsjoofjoG-SetsjotojoCounting 61
IV. Ringsj o andj o Fields
18. Ringsjo andjo Fields 63
19. Integralj o Domains 68
20. Fermat’sj o andj o Euler’sj o Theorems 72
21. Thej o Fieldj o ofj o Quotientsj o ofj o anj o Integralj o Domain 74
22. Ringsj o ofj o Polynomials 76
23. FactorizationjoofjoPolynomialsjooverjoajoField 79
24. Noncommutativej o Examples 85
25. Orderedj o Ringsj o andj o Fields 87
V. Idealsj o andj o Factorj o Rings
26. Homomorphismsjo andjo FactorjoRings 89
27. PrimejoandjoMaximaljoIdeals94
28. Gröbner joBasesjoforjoIdeals 99
, VI. Extensionj o Fields
29. Introductionjo tojo Extensionjo Fields 103
30. Vectorj o Spaces 107
31. Algebraicj o Extensions 111
32. GeometricjoConstructions 115
33. Finitej o Fields 116
VII. Advancedjo Groupjo Theory
34. IsomorphismjoTheorems 117
35. SeriesjoofjoGroups 119
36. Sylowj o Theorems 122
37. Applicationsj o ofj o thej o Sylowj o Theory 124
38. Freej o Abelianj o Groups 128
39. FreejoGroups 130
40. Groupj o Presentations 133
VIII. Groupsj o inj o Topology
41. Simplicialj o Complexesj o andj o Homologyj o Groups 136
42. Computationsj o ofj o Homologyjo Groups 138
43. Morejo Homologyjo Computationsjo andjo Applications 140
44. Homologicaljo Algebra 144
IX. Factorization
45. Uniquej o Factorizationj o Domains 148
46. Euclideanj o Domains 151
47. Gaussianj o Integersj o andj o Multiplicativej o Norms 154
X. Automorphismsj o andj o Galoisj o Theory
48. Automorphismsjo ofjo Fields 159
49. Thej o Isomorphismj o Extensionj o Theorem 164
50. Splittingj o Fields 165
51. SeparablejoExtensions 167
52. TotallyjoInseparablejoExtensions 171
53. Galoisj o Theory 173
54. IllustrationsjoofjoGaloisjoTheory 176
55. CyclotomicjoExtensions 183
56. Insolvabilityj o ofj o thej o Quintic 185
APPENDIXj o Matrixj o Algebra 187
iv
, 0.j o SetsjoandjoRelations 1
0. Setsj o andj o Relations
√ √
1. { 3,j o − 3} 2.j o Thej o setj o isj o empty.
3.j o {1,jo−1,jo2,jo−2,jo3,jo−3,jo4,jo−4,jo5,jo−5,jo6,jo−6,jo10,jo−10,jo12,jo−12,jo15,jo−15,jo20,jo−20,jo30,jo−30,
60,jo−60}
4.j o {−10,jo−9,jo−8,jo−7,jo−6,jo−5,jo−4,jo−3,jo−2,jo−1,jo0,jo1,jo2,jo3,jo4,jo5,jo6,jo7,jo8,jo9,jo10,jo11}
5. Itjoisjonotjoajowell-
definedjoset.j o (SomejomayjoarguejothatjonojoelementjoofjoZ+joisjolarge,jobecausejoeveryjoelementjoexc
eedsjoonlyjoajofinitejonumberjoofjootherjoelementsjobutjoisjoexceededjobyjoanjoinfinitejonumberjoofjootherjoel
ements.joSuchjopeoplejomightjoclaimjothejoanswerjoshouldjobejo∅.)
6. ∅ 7.j o Thejo setjo isjo ∅jo becausejo 33jo=jo27jo andjo 43jo=jo64.
8.j o Itjo isjo notjo ajo well-definedjo set. 9.j o Q
10. Thej o setj o containingj o allj o numbersj o thatj o arej o (positive,j o negative,j o orj o zero)j o integerj o multiples
j o ofj o 1,j o 1/2,j o orjo1/3.
11. {(a,jo1),jo(a,jo2),jo(a,joc),jo (b,jo1),jo(b,jo2),jo(b,joc),jo(c,jo1),jo(c,jo2),jo(c,joc)}
12. a.j o Itjoisjoajofunction.j o Itjoisjonotjoone-to-
onejosincejotherejoarejo twojopairsjowithjosecondjomemberjo4.j o Itjoisjonotjoonto
Bjo becausejotherejoisjonojopairjowithjosecondjomemberjo2.
b. (Samej o answerj o asj o Part(a).)
c. Itjo isjo notjo ajo functionjo becausejo therejo arejo twojo pairsjo withjo firstjo memberjo 1.
d. Itjo isjo ajo function.j o Itjo isjo one-to-
one.j o Itjo isjo ontojo Bj o becausejo everyjo elementjo ofjo Bj o appearsjo asjo secondjomemberjoofjosome
jopair.
e. Itjoisjoajofunction.jo Itjoisjonotjoone-to-
onejobecausejotherejoarejotwojopairsjowithjosecondjomemberjo6.jo ItjoisjonotjoontojoBjobecausejother
ejoisjonojopairjowithjosecondjomemberjo2.
f. Itjoisjo notjo ajo functionjo becausejo therejo arejo twojo pairsjo withjo firstjo memberjo 2.
13. Drawjo thej o linej o throughjo Pj o andj o x,jo andj o letj o yj o bej o itsj o pointj o ofj o intersectionj o withj o thej o linej o se
gmentj o CD.
14. a.j o φjo:jo[0,jo1]jo→jo[0,jo2]jo wherej o φ(x)jo=jo2x
b.j o φjo:jo [1,jo3]jo→jo[5,jo25]j o wherej o φ(x)jo=jo5jo+jo10(xjo−jo1)
c.j o φjo:jo[a,→
job] d c
− [c,jod]j o wherej o φ(x)jo=jocjo+j o − jo(x a)
b−a
1jo
15. Letjo φjo:joSjo →joRjo bejo definedjo byjo φ(x)jo=jotan(π(x
2
jo−jo )).
16. a.j o ∅;j o cardinalityjo 1 b.j o ∅,jo{a};j o cardinalityj o 2 c.j o ∅,jo{a},jo{b},jo{a,job};j o cardinalityj o 4
d.j o ∅,jo{a},jo{b},jo{c},jo{a,job},jo{a,joc},jo{b,joc},jo{a,job,joc};jo cardinalityj o 8
17. Conjecture: |P(A)|jo=jo2sjo =jo2|A|.
ProofjoThejonumberjoofjosubsetsjoofjoajosetjoAjodependsjoonlyjoonjothejocardinalityjoofjoA,jonotjoon
jowhatjothejoelementsjoofj o Ajo actuallyjo are.j o SupposejoBjo=jo{1,jo2,jo3,jo·jo·jo·jo,josjo−jo1}j o andj o Ajo=jo
{1,jo2,jo3,j o j o ,jos}.j o Thenjo Ajo hasjo all
