Instructor’s Soluti o
ons Manual o
too accompany
A First Course in Abstrac
o o o o
t Algebra
o
Seventho Edition
Johno B.o Fraleigh
Universityo ofo Rhodeo Island
,Preface
Thisomanualocontainsosolutionsotooalloexercisesoinotheotext,oexceptothoseoodd-
numberedoexercisesoforowhichofairlyolengthyocompleteosolutionsoareogivenoinotheoanswersoatotheobackoofotheo
text.o Thenoreferenceoisosimplyogivenotootheotextoanswersotoosaveotyping.
Iopreparedotheseosolutionsomyself.o WhileoIotriedotoobeoaccurate,othereoareosureotoobeotheoinevitableomista
kesoandotypos.o Anoauthororeadingoprooforendsotooseeowhatoheoorosheowantsotoosee.o However,otheoinstructoro
shouldofindothisomanualoadequateoforotheopurposeoforowhichoitoisointended.
Morgan,oVermont J.B.F
July,o 2002
i
, CONTENTS
0. Setso ando Relations 1
I. Groups and Subgroups
o o
1. Introductiono ando Examples 4
2. Binaryo Operations 7
3. Isomorphico Binaryo Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclico Groups 21
7. Generatorso ando Cayleyo Digraphs 24
II. Permutations, Cosets, and Direct Products
o o o o
8. Groupso ofo Permutations 26
9. Orbits,o Cycles,o ando theo Alternatingo Groups 30
10. Cosetso ando theo Theoremo ofo Lagrange 34
11. Directo Productso ando Finitelyo Generatedo Abeliano Groups 37
12. Planeo Isometries 42
III. Homomorphisms and Factor Groups o o o
13. Homomorphisms 44
14. Factoro Groups 49
15. Factor-Groupo Computationso ando Simpleo Groups 53
16. Groupo Actiono ono ao Set 58
17. Applicationso ofo G-Setso too Counting 61
IV. Rings and Fields o o
18. Ringso ando Fields 63
19. Integralo Domains 68
20. Fermat’so ando Euler’so Theorems 72
21. Theo Fieldo ofo Quotientso ofo ano Integralo Domain 74
22. Ringso ofo Polynomials 76
23. Factorizationo ofo Polynomialso overo ao Field 79
24. NoncommutativeoExamples 85
25. Orderedo Ringso ando Fields 87
V. Ideals and Factor Rings
o o o
26. HomomorphismsoandoFactoroRings 89
27. Primeo ando Maximalo Ideals 94
28. Gröbner o Baseso foro Ideals 99
iii
, VI. Extension Fields o
29. Introductiono too Extensiono Fields 103
30. Vectoro Spaces 107
31. Algebraico Extensions 111
32. Geometrico Constructions 115
33. Finiteo Fields 116
VII. Advanced Group Theoryo o
34. IsomorphismoTheorems 117
35. SeriesoofoGroups 119
36. Sylowo Theorems 122
37. Applicationso ofo theo Sylowo Theory 124
38. Freeo Abeliano Groups 128
39. FreeoGroups 130
40. Groupo Presentations 133
VIII. Groups in Topology o o
41. Simplicialo Complexeso ando Homologyo Groups 136
42. Computationso ofo Homologyo Groups 138
43. Moreo Homologyo Computationso ando Applications 140
44. HomologicaloAlgebra 144
IX. Factorization
45. Uniqueo Factorizationo Domains 148
46. Euclideano Domains 151
47. Gaussiano Integerso ando Multiplicativeo Norms 154
X. Automorphisms and Galois Theory
o o o
48. Automorphismso ofo Fields 159
49. Theo Isomorphismo Extensiono Theorem 164
50. Splittingo Fields 165
51. SeparableoExtensions 167
52. Totallyo Inseparableo Extensions 171
53. Galoiso Theory 173
54. Illustrationso ofo Galoiso Theory 176
55. Cyclotomico Extensions 183
56. Insolvabilityo ofo theo Quintic 185
APPENDIXo o Matrixo Algebra 187
iv
ons Manual o
too accompany
A First Course in Abstrac
o o o o
t Algebra
o
Seventho Edition
Johno B.o Fraleigh
Universityo ofo Rhodeo Island
,Preface
Thisomanualocontainsosolutionsotooalloexercisesoinotheotext,oexceptothoseoodd-
numberedoexercisesoforowhichofairlyolengthyocompleteosolutionsoareogivenoinotheoanswersoatotheobackoofotheo
text.o Thenoreferenceoisosimplyogivenotootheotextoanswersotoosaveotyping.
Iopreparedotheseosolutionsomyself.o WhileoIotriedotoobeoaccurate,othereoareosureotoobeotheoinevitableomista
kesoandotypos.o Anoauthororeadingoprooforendsotooseeowhatoheoorosheowantsotoosee.o However,otheoinstructoro
shouldofindothisomanualoadequateoforotheopurposeoforowhichoitoisointended.
Morgan,oVermont J.B.F
July,o 2002
i
, CONTENTS
0. Setso ando Relations 1
I. Groups and Subgroups
o o
1. Introductiono ando Examples 4
2. Binaryo Operations 7
3. Isomorphico Binaryo Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclico Groups 21
7. Generatorso ando Cayleyo Digraphs 24
II. Permutations, Cosets, and Direct Products
o o o o
8. Groupso ofo Permutations 26
9. Orbits,o Cycles,o ando theo Alternatingo Groups 30
10. Cosetso ando theo Theoremo ofo Lagrange 34
11. Directo Productso ando Finitelyo Generatedo Abeliano Groups 37
12. Planeo Isometries 42
III. Homomorphisms and Factor Groups o o o
13. Homomorphisms 44
14. Factoro Groups 49
15. Factor-Groupo Computationso ando Simpleo Groups 53
16. Groupo Actiono ono ao Set 58
17. Applicationso ofo G-Setso too Counting 61
IV. Rings and Fields o o
18. Ringso ando Fields 63
19. Integralo Domains 68
20. Fermat’so ando Euler’so Theorems 72
21. Theo Fieldo ofo Quotientso ofo ano Integralo Domain 74
22. Ringso ofo Polynomials 76
23. Factorizationo ofo Polynomialso overo ao Field 79
24. NoncommutativeoExamples 85
25. Orderedo Ringso ando Fields 87
V. Ideals and Factor Rings
o o o
26. HomomorphismsoandoFactoroRings 89
27. Primeo ando Maximalo Ideals 94
28. Gröbner o Baseso foro Ideals 99
iii
, VI. Extension Fields o
29. Introductiono too Extensiono Fields 103
30. Vectoro Spaces 107
31. Algebraico Extensions 111
32. Geometrico Constructions 115
33. Finiteo Fields 116
VII. Advanced Group Theoryo o
34. IsomorphismoTheorems 117
35. SeriesoofoGroups 119
36. Sylowo Theorems 122
37. Applicationso ofo theo Sylowo Theory 124
38. Freeo Abeliano Groups 128
39. FreeoGroups 130
40. Groupo Presentations 133
VIII. Groups in Topology o o
41. Simplicialo Complexeso ando Homologyo Groups 136
42. Computationso ofo Homologyo Groups 138
43. Moreo Homologyo Computationso ando Applications 140
44. HomologicaloAlgebra 144
IX. Factorization
45. Uniqueo Factorizationo Domains 148
46. Euclideano Domains 151
47. Gaussiano Integerso ando Multiplicativeo Norms 154
X. Automorphisms and Galois Theory
o o o
48. Automorphismso ofo Fields 159
49. Theo Isomorphismo Extensiono Theorem 164
50. Splittingo Fields 165
51. SeparableoExtensions 167
52. Totallyo Inseparableo Extensions 171
53. Galoiso Theory 173
54. Illustrationso ofo Galoiso Theory 176
55. Cyclotomico Extensions 183
56. Insolvabilityo ofo theo Quintic 185
APPENDIXo o Matrixo Algebra 187
iv
