TEST BANK k
First Course in Abstract Algebra A
k k k k k k
8th Edition by John B. Fraleigh
k k k k k k
All Chapters Full Complete
k k k k
, CONTENTS
0. Setsk andk Relations 1
I. Groups and Subgroups
k k
1. Introductionk andk Examples 4
2. Binaryk Operations 7
3. Isomorphick Binaryk Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclick Groups 21
7. Generatorsk andk Cayleyk Digraphs 24
II. Permutations, Cosets, and Direct Products
k k k k
8. Groupsk ofk Permutations 26
9. Orbits,k Cycles,kandkthek AlternatingkGroups 30
10. Cosetsk andk thek Theoremk ofk Lagrange 34
11. Directk Productsk andk Finitelyk Generatedk Abeliank Groups 37
12. Planek Isometries 42
III. Homomorphisms and Factor Groups k k k
13. Homomorphisms 44
14. Factork Groups 49
15. Factor-Groupk Computationsk andk Simplek Groups 53
16. Groupk Actionk onk akSet 58
17. ApplicationskofkG-SetsktokCounting 61
IV. Rings and Fields k k
18. Ringsk andk Fields 63
19. Integralk Domains 68
20. Fermat’sk andk Euler’sk Theorems 72
21. Thek Fieldk ofk Quotientsk ofk ank Integralk Domain 74
22. Ringsk ofk Polynomials 76
23. FactorizationkofkPolynomialskoverkakField 79
24. Noncommutativek Examples 85
25. Orderedk Ringsk andk Fields 87
V. Ideals and Factor Rings
k k k
26. Homomorphismsk andk Factork Rings 89
27. PrimekandkMaximalkIdeals 94
28. Gröbner kBaseskforkIdeals 99
, VI. Extension Fields k
29. Introductionk tok Extensionk Fields 103
30. Vectork Spaces 107
31. Algebraick Extensions 111
32. Geometrick Constructions 115
33. Finitek Fields 116
VII. Advanced Group Theory
k k
34. IsomorphismkTheorems 117
35. SerieskofkGroups 119
36. Sylowk Theorems 122
37. Applicationsk ofk thek Sylowk Theory 124
38. Freek Abeliank Groups 128
39. FreekGroups 130
40. Groupk Presentations 133
VIII. Groups in Topology k k
41. Simplicialk Complexesk andk Homologyk Groups 136
42. Computationsk ofk Homologyk Groups 138
43. Morek Homologyk Computationsk andk Applications 140
44. Homologicalk Algebra 144
IX. Factorization
45. Uniquek Factorizationk Domains 148
46. Euclideank Domains 151
47. Gaussiank Integersk andk Multiplicativek Norms 154
X. Automorphisms and Galois Theory
k k k
48. Automorphismsk ofk Fields 159
49. Thek Isomorphismk Extensionk Theorem 164
50. Splittingk Fields 165
51. SeparablekExtensions 167
52. TotallykInseparablek Extensions 171
53. Galoisk Theory 173
54. IllustrationskofkGaloiskTheory 176
55. CyclotomickExtensions 183
56. Insolvabilityk ofk thek Quintic 185
APPENDIXk Matrixk Algebra 187
iv
, 0.k SetskandkRelations 1
0. Sets and Relations
k k
√ √
1. { 3,k − 3} 2.k Thek setk isk empty.
3.k {1,k−1,k2,k−2,k3,k−3,k4,k−4,k5,k−5,k6,k−6,k10,k−10,k12,k−12,k15,k−15,k20,k−20,k30,k−30,
60,k−60}
4.k {−10,k−9,k−8,k−7,k−6,k−5,k−4,k−3,k−2,k−1,k0,k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11}
5. Itkisknotkakwell-
definedkset.k (SomekmaykarguekthatknokelementkofkZ+kisklarge,kbecausekeverykelementkexceedskonlykakfini
teknumberkofkotherkelementskbutkiskexceededkbykankinfiniteknumberkofkotherkelements.kSuchkpeoplekmigh
tkclaimkthekanswerkshouldkbek∅.)
6. ∅ 7.k Thek setk isk ∅k becausek 33k=k27k andk 43k=k64.
8.k Itk isk notk ak well-definedk set. 9.k Q
10. Thek setk containingk allk numbersk thatk arek (positive,k negative,k ork zero)k integerk multiplesk ofk 1,k 1/2,k ork
1/3.
11. {(a,k1),k (a,k 2),k (a,k c),k (b,k 1),k (b,k 2),k(b,k c),k (c,k 1),k (c,k 2),k (c,kc)}
12. a.k Itk isk ak function.k Itk isk notk one-to-onek sincektherek arek twok pairsk withk secondkmemberk 4.k Itk isk notk onto
Bk becausek therek isk nok pairk withk secondk memberk 2.
b. (Samek answerk ask Part(a).)
c. Itk isk notk ak functionk becausek therek arek twok pairsk withk firstk memberk 1.
d. Itk isk ak function.k Itk isk one-to-
one.k Itk isk ontok Bk becausek everyk elementk ofk Bk appearsk ask secondkmemberkofksomekpair.
e. Itkiskakfunction.k Itkisknotkone-to-
onekbecausektherekarektwokpairskwithksecondkmemberk6.k ItkisknotkontokBkbecausektherekisknokpairkwi
thksecondkmemberk2.
f. Itk isk notk ak functionk becausek therek arek twok pairsk withk firstk memberk 2.
13. Drawk thek linek throughk Pk andk x,k andk letk yk bek itsk pointk ofk intersectionk withk thek linek segmentk CD.
14. a.k φk:k[0,k1]k→k [0,k2]k wherek φ(x)k=k2x b.k φk:k [1,k3]k →k [5,k25]k wherek φ(x)k=k5k+k10(xk−k1)
c.k φk:k[a,kb] → [c,kd]k wherek φ(x)k=kck+k d−ck(x − a)
b−a
15. Letk φk:kSk →kRk bek definedk byk φ(x)k=ktan(π(xk−k 1k)).
2
16. a.k ∅;k cardinalityk 1 b.k ∅,k{a};k cardinalityk 2 c.k ∅,k{a},k{b},k{a,kb};k cardinalityk 4
d.k ∅,k{a},k{b},k{c},k{a,kb},k{a,kc},k{b,kc},k{a,kb,kc};k cardinalityk 8
17. Conjecture: |P(A)|k=k2sk=k2|A|.
ProofkTheknumberkofksubsetskofkaksetkAkdependskonlykonkthekcardinalitykofkA,knotkonkwhatkthekelem
entskofk Ak actuallyk are.k SupposekBk=k{1,k2,k3,k·k·k·k,ksk−k1}k andk Ak=k{1,k2,k3,k k ,ks}.k Thenk Ak hask all
thekelementskofkBkpluskthekonekadditionalkelementks.k AllksubsetskofkBkarekalsoksubsetskofkA;kthesekar
ekpreciselyktheksubsetskofkAkthatkdoknotkcontainks,ksoktheknumberkofksubsetskofkAknotkcontainingkskis
k|P(B)|.k Anykotherk subsetkofkAk mustk containk s,k andk removalkofk theksk wouldk producek aksubsetk of
B.k Thusk thek numberk ofk subsetsk ofk Ak containingk sk isk alsok |P(B)|.k Becausek everyk subsetk ofk Ak eitherkc
ontainsk sk ork doesk notk containk sk (butk notk both),k wek seek thatk thek numberk ofk subsetsk ofk Ak isk 2|P(B)|.
First Course in Abstract Algebra A
k k k k k k
8th Edition by John B. Fraleigh
k k k k k k
All Chapters Full Complete
k k k k
, CONTENTS
0. Setsk andk Relations 1
I. Groups and Subgroups
k k
1. Introductionk andk Examples 4
2. Binaryk Operations 7
3. Isomorphick Binaryk Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclick Groups 21
7. Generatorsk andk Cayleyk Digraphs 24
II. Permutations, Cosets, and Direct Products
k k k k
8. Groupsk ofk Permutations 26
9. Orbits,k Cycles,kandkthek AlternatingkGroups 30
10. Cosetsk andk thek Theoremk ofk Lagrange 34
11. Directk Productsk andk Finitelyk Generatedk Abeliank Groups 37
12. Planek Isometries 42
III. Homomorphisms and Factor Groups k k k
13. Homomorphisms 44
14. Factork Groups 49
15. Factor-Groupk Computationsk andk Simplek Groups 53
16. Groupk Actionk onk akSet 58
17. ApplicationskofkG-SetsktokCounting 61
IV. Rings and Fields k k
18. Ringsk andk Fields 63
19. Integralk Domains 68
20. Fermat’sk andk Euler’sk Theorems 72
21. Thek Fieldk ofk Quotientsk ofk ank Integralk Domain 74
22. Ringsk ofk Polynomials 76
23. FactorizationkofkPolynomialskoverkakField 79
24. Noncommutativek Examples 85
25. Orderedk Ringsk andk Fields 87
V. Ideals and Factor Rings
k k k
26. Homomorphismsk andk Factork Rings 89
27. PrimekandkMaximalkIdeals 94
28. Gröbner kBaseskforkIdeals 99
, VI. Extension Fields k
29. Introductionk tok Extensionk Fields 103
30. Vectork Spaces 107
31. Algebraick Extensions 111
32. Geometrick Constructions 115
33. Finitek Fields 116
VII. Advanced Group Theory
k k
34. IsomorphismkTheorems 117
35. SerieskofkGroups 119
36. Sylowk Theorems 122
37. Applicationsk ofk thek Sylowk Theory 124
38. Freek Abeliank Groups 128
39. FreekGroups 130
40. Groupk Presentations 133
VIII. Groups in Topology k k
41. Simplicialk Complexesk andk Homologyk Groups 136
42. Computationsk ofk Homologyk Groups 138
43. Morek Homologyk Computationsk andk Applications 140
44. Homologicalk Algebra 144
IX. Factorization
45. Uniquek Factorizationk Domains 148
46. Euclideank Domains 151
47. Gaussiank Integersk andk Multiplicativek Norms 154
X. Automorphisms and Galois Theory
k k k
48. Automorphismsk ofk Fields 159
49. Thek Isomorphismk Extensionk Theorem 164
50. Splittingk Fields 165
51. SeparablekExtensions 167
52. TotallykInseparablek Extensions 171
53. Galoisk Theory 173
54. IllustrationskofkGaloiskTheory 176
55. CyclotomickExtensions 183
56. Insolvabilityk ofk thek Quintic 185
APPENDIXk Matrixk Algebra 187
iv
, 0.k SetskandkRelations 1
0. Sets and Relations
k k
√ √
1. { 3,k − 3} 2.k Thek setk isk empty.
3.k {1,k−1,k2,k−2,k3,k−3,k4,k−4,k5,k−5,k6,k−6,k10,k−10,k12,k−12,k15,k−15,k20,k−20,k30,k−30,
60,k−60}
4.k {−10,k−9,k−8,k−7,k−6,k−5,k−4,k−3,k−2,k−1,k0,k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11}
5. Itkisknotkakwell-
definedkset.k (SomekmaykarguekthatknokelementkofkZ+kisklarge,kbecausekeverykelementkexceedskonlykakfini
teknumberkofkotherkelementskbutkiskexceededkbykankinfiniteknumberkofkotherkelements.kSuchkpeoplekmigh
tkclaimkthekanswerkshouldkbek∅.)
6. ∅ 7.k Thek setk isk ∅k becausek 33k=k27k andk 43k=k64.
8.k Itk isk notk ak well-definedk set. 9.k Q
10. Thek setk containingk allk numbersk thatk arek (positive,k negative,k ork zero)k integerk multiplesk ofk 1,k 1/2,k ork
1/3.
11. {(a,k1),k (a,k 2),k (a,k c),k (b,k 1),k (b,k 2),k(b,k c),k (c,k 1),k (c,k 2),k (c,kc)}
12. a.k Itk isk ak function.k Itk isk notk one-to-onek sincektherek arek twok pairsk withk secondkmemberk 4.k Itk isk notk onto
Bk becausek therek isk nok pairk withk secondk memberk 2.
b. (Samek answerk ask Part(a).)
c. Itk isk notk ak functionk becausek therek arek twok pairsk withk firstk memberk 1.
d. Itk isk ak function.k Itk isk one-to-
one.k Itk isk ontok Bk becausek everyk elementk ofk Bk appearsk ask secondkmemberkofksomekpair.
e. Itkiskakfunction.k Itkisknotkone-to-
onekbecausektherekarektwokpairskwithksecondkmemberk6.k ItkisknotkontokBkbecausektherekisknokpairkwi
thksecondkmemberk2.
f. Itk isk notk ak functionk becausek therek arek twok pairsk withk firstk memberk 2.
13. Drawk thek linek throughk Pk andk x,k andk letk yk bek itsk pointk ofk intersectionk withk thek linek segmentk CD.
14. a.k φk:k[0,k1]k→k [0,k2]k wherek φ(x)k=k2x b.k φk:k [1,k3]k →k [5,k25]k wherek φ(x)k=k5k+k10(xk−k1)
c.k φk:k[a,kb] → [c,kd]k wherek φ(x)k=kck+k d−ck(x − a)
b−a
15. Letk φk:kSk →kRk bek definedk byk φ(x)k=ktan(π(xk−k 1k)).
2
16. a.k ∅;k cardinalityk 1 b.k ∅,k{a};k cardinalityk 2 c.k ∅,k{a},k{b},k{a,kb};k cardinalityk 4
d.k ∅,k{a},k{b},k{c},k{a,kb},k{a,kc},k{b,kc},k{a,kb,kc};k cardinalityk 8
17. Conjecture: |P(A)|k=k2sk=k2|A|.
ProofkTheknumberkofksubsetskofkaksetkAkdependskonlykonkthekcardinalitykofkA,knotkonkwhatkthekelem
entskofk Ak actuallyk are.k SupposekBk=k{1,k2,k3,k·k·k·k,ksk−k1}k andk Ak=k{1,k2,k3,k k ,ks}.k Thenk Ak hask all
thekelementskofkBkpluskthekonekadditionalkelementks.k AllksubsetskofkBkarekalsoksubsetskofkA;kthesekar
ekpreciselyktheksubsetskofkAkthatkdoknotkcontainks,ksoktheknumberkofksubsetskofkAknotkcontainingkskis
k|P(B)|.k Anykotherk subsetkofkAk mustk containk s,k andk removalkofk theksk wouldk producek aksubsetk of
B.k Thusk thek numberk ofk subsetsk ofk Ak containingk sk isk alsok |P(B)|.k Becausek everyk subsetk ofk Ak eitherkc
ontainsk sk ork doesk notk containk sk (butk notk both),k wek seek thatk thek numberk ofk subsetsk ofk Ak isk 2|P(B)|.
