SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets n z n z and n z n z Relations 1
I. Groups n z n z and n z n z Subgroups
1. Introduction n z n z and n z n z Examples 4
2. Binary n z n z Operations 7
3. Isomorphic n z n z Binary n z n z Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic n z n z Groups 21
7. Generators n z n z and n z n z Cayley n z n z Digraphs 24
II. Permutations, nznz Cosets, nznz and nznz Direct nznz Products
8. Groups n z n z of n z n z Permutations 26
9. Orbits, nznzCycles, nznzand nznzthe nznzAlternating nznzGroups 30
10. Cosets nznz and nznz the n z nz Theorem nznz of n z nz Lagrange 34
11. Direct n z n z Products n z n z and n z n z Finitely n z n z Generated n z n z Abelian nznz Groups 37
12. Plane n z n z Isometries 42
III. Homomorphisms n z n z and n z n z Factor n z n z Groups
13. Homomorphisms 44
14. Factor n z n z Groups 49
15. Factor-Group n z n z Computations n z n z and n z n z Simple n z n z Groups 53
16. Group nznzAction nznzon nznz a nznzSet 58
17. Applications nznzof nznzG-Sets nznzto nznzCounting 61
IV. Rings n z n z and n z n z Fields
18. Rings n z nz and n z n z Fields 63
19. Integral n z n z Domains 68
20. Fermat’s n z n z and n z n z Euler’s n z n z Theorems 72
21. The n z n z Field n z n z of n z n z Quotients n z n z of n z n z an n z n z Integral n z n z Domain 74
22. Rings n z n z of n z n z Polynomials 76
23. Factorization nznzof nznzPolynomials nznzover nznza nznzField 79
24. Noncommutative n z n z Examples 85
25. Ordered n z n z Rings n z n z and n z n z Fields 87
V. Ideals n z n z and n z n z Factor n z n z Rings
26. Homomorphisms nznz and nznz Factor nznz Rings 89
27. Prime nznzand nznzMaximal nznzIdeals 94
28. Gröbner nznzBases nznzfor nznzIdeals 99
, VI. Extension n z n z Fields
29. Introduction nznz to nznz Extension nznzFields 103
30. Vector n z n z Spaces107
31. Algebraic n z n z Extensions 111
32. Geometric nznzConstructions 115
33. Finite n z n z Fields 116
VII. Advanced n zn z Group n zn z Theory
34. Isomorphism nznzTheorems 117
35. Series nznzof nznzGroups 119
36. Sylow n z n z Theorems 122
37. Applications n z n z of n z n z the n z n z Sylow n z n z Theory 124
38. Free n z n z Abelian n z n z Groups 128
39. Free nznzGroups 130
40. Group n z n z Presentations 133
VIII. Groups n z n z in n z n z Topology
41. Simplicial n z n z Complexes n z n z and n z n z Homology n z n z Groups 136
42. Computations n z n z of n z n z Homology n z n z Groups 138
43. More n zn z Homology n z n z Computations n z n z and n z n z Applications 140
44. Homological nznz Algebra 144
IX. Factorization
45. Unique n z n z Factorization n z n z Domains 148
46. Euclidean n z n z Domains 151
47. Gaussian n z n z Integers n z n z and n z n z Multiplicative n z n z Norms 154
X. Automorphisms n z n z and n z n z Galois n z n z Theory
48. Automorphisms nznz of nznz Fields 159
49. The n z n z Isomorphism n z n z Extension n z n z Theorem 164
50. Splitting n z n z Fields 165
51. Separable nznzExtensions 167
52. Totally nznzInseparable nznzExtensions 171
53. Galois n z n z Theory 173
54. Illustrations nznzof nznzGalois nznzTheory 176
55. Cyclotomic nznzExtensions 183
56. Insolvability n z n z of n z n z the n z n z Quintic 185
APPENDIX n z n z Matrix n z n z Algebra 187
iv
, 0. n z n z Sets nznzand nznzRelations 1
0. Sets n z n z and n z n z Relations
√ √
1. { 3, n z n z − 3} 2. nznz The n z n z set n z n z is n z n z empty.
3. n z n z {1, nznz−1, nznz2, nznz−2, nznz3, nznz−3, nznz4, nznz−4, nznz5, nznz−5, nznz6, nznz−6, nznz10, nznz−10, nznz12, nznz−12, nznz15, nznz−15, nznz20, nznz−20, nznz30,
nznz−30,
60, nznz−60}
4. nznz {−10, nznz−9, nznz−8, nznz−7, nznz−6, nznz−5, nznz−4, nznz−3, nznz−2, nznz−1, nznz0, nznz1, nznz2, nznz3, nznz4, nznz5, nznz6, nznz7, nznz8, nznz9, nznz10, nznz11}
5. It nznzis nznznot nznza nznzwell-defined nznzset. n z n z (Some nznzmay nznzargue nznzthat nznzno nznzelement nznzof nznzZ+
nznzis nznzlarge, nznzbecause nznzevery nznzelement nznzexceeds nznzonly nznza nznzfinite nznznumber nznzof nznzother
nznzelements nznzbut nznzis nznzexceeded nznzby nznzan nznzinfinite nznznumber nznzof nznzother nznzelements. nznzSuch nznzpeople
nznzmight nznzclaim nznzthe nznzanswer nznzshould nznzbe nznz∅.)
6. ∅ 7. n z n z The nznz set nznz is nznz ∅ n zn z because nznz 33 nznz= nznz27 nznz and nz nz 43 nznz= nznz64.
8. nznz It n zn z is n z n z not n z nz a nz n z well-defined n z n z set. 9. nznz Q
10. The n z n z set n z n z containing n z n z all n z n z numbers n z n z that n z n z are n z n z (positive, n z n z negative, n z n z or
n z n z zero) n z n z integer n z n z multiples n z n z of n z n z 1, n z n z 1/2, n z n z or nznz1/3.
11. {(a, nznz1), nznz (a, nznz 2), nznz (a, nznz c), nznz (b, nznz1), nznz(b, nznz 2), nznz(b, nznz c), nznz (c, nznz 1), nznz (c, nznz2), nznz (c, nznz c)}
12. a. n z n z It nznzis nznza nznzfunction. n z n z It nznzis nznznot nznzone-to-one nznzsince nznzthere nznzare nznztwo nznzpairs
nznz with nznz second nznz member nznz 4. n z n z It nznz is nznz not nznz onto
B n z n z because nznzthere nznzis nznzno nznz pair nznzwith nznzsecond nznzmember nznz2.
b. (Same n z n z answer n z n z as n z n z Part(a).)
c. It nznz is nznz not nznz a nznz function nznz because nznz there nznz are nznz two nznz pairs nznz with nznz first nznz member
nznz 1.
d. It n z n z is n z n z a n z n z function. n z n z It n z n z is n z n z one-to-one. n z n z It n z n z is n z n z onto n z n z B n z n z because
n z n z every n z n z element n z n z of n z n z B n z n z appears n z n z as n z n z second nznzmember nznzof nznzsome nznzpair.
e. It nznzis nznza nznzfunction. n z n z It nznzis nznznot nznzone-to-one nznzbecause nznzthere nznzare nznztwo nznzpairs nznzwith
nznzsecond nznzmember nznz6. n z n z It nznzis nznznot nznzonto nznzB nznzbecause nznzthere nznzis nznzno nznzpair nznzwith
nznzsecond nznzmember nznz2.
f. It nznzis nznz not nznz a nznz function nznz because nznz there nznz are nznz two nznz pairs nznz with nznz first nznz member
nznz 2.
13. Draw n z n z the n z n z line n z n z through n z n z P n z n z and n z n z x, n z n z and n z n z let n z n z y n z n z be n z n z its n z n z point n z n z of
n z n z intersection n z n z with n z n z the n z n z line n z n z segment n z n z CD.
14. a. n z n z φ nznz: nznz[0, nznz1] nznz→ nznz [0, nznz2] n z n z where n z n z φ(x) nznz= nznz2x b. n z n z φ nznz: nznz[1, nznz3] nznz → nznz [5,
nznz25] n z n z where n z n z φ(x) nznz= nznz5 nznz+ nznz10(x nznz− nznz1)
c. n z n z φ nznz:→
nznz[a, nznzb]
d c
− [c, nznzd] n z n z where n z n z φ(x) nznz= nznzc nznz+ n z n z − nznz(x a)
b−a
15. Let n z n z φ nznz: nznzS n z n z → nznz R n z n z be n z n z defined n z n2z by n z n z φ(x) nznz= nznztan(π(x nznz− n z n z 1 nznz)).
16. a. n z n z ∅; n z n z cardinality n z n z 1 b. nznz ∅, nznz{a}; n z n z cardinality n z n z 2 c. nznz ∅,
nznz{a}, nznz{b}, nznz{a, nznzb}; n z n z cardinality n z n z 4
d. nznz ∅, nznz{a}, nznz{b}, nznz{c}, nznz{a, nznzb}, nznz{a, nznzc}, nznz{b, nznzc}, nznz{a, nznzb, nznzc}; n z n z cardinality n z n z 8
17. Conjecture: |P(A)| nznz= nznz2s nznz= nznz2|A|.
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets n z n z and n z n z Relations 1
I. Groups n z n z and n z n z Subgroups
1. Introduction n z n z and n z n z Examples 4
2. Binary n z n z Operations 7
3. Isomorphic n z n z Binary n z n z Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic n z n z Groups 21
7. Generators n z n z and n z n z Cayley n z n z Digraphs 24
II. Permutations, nznz Cosets, nznz and nznz Direct nznz Products
8. Groups n z n z of n z n z Permutations 26
9. Orbits, nznzCycles, nznzand nznzthe nznzAlternating nznzGroups 30
10. Cosets nznz and nznz the n z nz Theorem nznz of n z nz Lagrange 34
11. Direct n z n z Products n z n z and n z n z Finitely n z n z Generated n z n z Abelian nznz Groups 37
12. Plane n z n z Isometries 42
III. Homomorphisms n z n z and n z n z Factor n z n z Groups
13. Homomorphisms 44
14. Factor n z n z Groups 49
15. Factor-Group n z n z Computations n z n z and n z n z Simple n z n z Groups 53
16. Group nznzAction nznzon nznz a nznzSet 58
17. Applications nznzof nznzG-Sets nznzto nznzCounting 61
IV. Rings n z n z and n z n z Fields
18. Rings n z nz and n z n z Fields 63
19. Integral n z n z Domains 68
20. Fermat’s n z n z and n z n z Euler’s n z n z Theorems 72
21. The n z n z Field n z n z of n z n z Quotients n z n z of n z n z an n z n z Integral n z n z Domain 74
22. Rings n z n z of n z n z Polynomials 76
23. Factorization nznzof nznzPolynomials nznzover nznza nznzField 79
24. Noncommutative n z n z Examples 85
25. Ordered n z n z Rings n z n z and n z n z Fields 87
V. Ideals n z n z and n z n z Factor n z n z Rings
26. Homomorphisms nznz and nznz Factor nznz Rings 89
27. Prime nznzand nznzMaximal nznzIdeals 94
28. Gröbner nznzBases nznzfor nznzIdeals 99
, VI. Extension n z n z Fields
29. Introduction nznz to nznz Extension nznzFields 103
30. Vector n z n z Spaces107
31. Algebraic n z n z Extensions 111
32. Geometric nznzConstructions 115
33. Finite n z n z Fields 116
VII. Advanced n zn z Group n zn z Theory
34. Isomorphism nznzTheorems 117
35. Series nznzof nznzGroups 119
36. Sylow n z n z Theorems 122
37. Applications n z n z of n z n z the n z n z Sylow n z n z Theory 124
38. Free n z n z Abelian n z n z Groups 128
39. Free nznzGroups 130
40. Group n z n z Presentations 133
VIII. Groups n z n z in n z n z Topology
41. Simplicial n z n z Complexes n z n z and n z n z Homology n z n z Groups 136
42. Computations n z n z of n z n z Homology n z n z Groups 138
43. More n zn z Homology n z n z Computations n z n z and n z n z Applications 140
44. Homological nznz Algebra 144
IX. Factorization
45. Unique n z n z Factorization n z n z Domains 148
46. Euclidean n z n z Domains 151
47. Gaussian n z n z Integers n z n z and n z n z Multiplicative n z n z Norms 154
X. Automorphisms n z n z and n z n z Galois n z n z Theory
48. Automorphisms nznz of nznz Fields 159
49. The n z n z Isomorphism n z n z Extension n z n z Theorem 164
50. Splitting n z n z Fields 165
51. Separable nznzExtensions 167
52. Totally nznzInseparable nznzExtensions 171
53. Galois n z n z Theory 173
54. Illustrations nznzof nznzGalois nznzTheory 176
55. Cyclotomic nznzExtensions 183
56. Insolvability n z n z of n z n z the n z n z Quintic 185
APPENDIX n z n z Matrix n z n z Algebra 187
iv
, 0. n z n z Sets nznzand nznzRelations 1
0. Sets n z n z and n z n z Relations
√ √
1. { 3, n z n z − 3} 2. nznz The n z n z set n z n z is n z n z empty.
3. n z n z {1, nznz−1, nznz2, nznz−2, nznz3, nznz−3, nznz4, nznz−4, nznz5, nznz−5, nznz6, nznz−6, nznz10, nznz−10, nznz12, nznz−12, nznz15, nznz−15, nznz20, nznz−20, nznz30,
nznz−30,
60, nznz−60}
4. nznz {−10, nznz−9, nznz−8, nznz−7, nznz−6, nznz−5, nznz−4, nznz−3, nznz−2, nznz−1, nznz0, nznz1, nznz2, nznz3, nznz4, nznz5, nznz6, nznz7, nznz8, nznz9, nznz10, nznz11}
5. It nznzis nznznot nznza nznzwell-defined nznzset. n z n z (Some nznzmay nznzargue nznzthat nznzno nznzelement nznzof nznzZ+
nznzis nznzlarge, nznzbecause nznzevery nznzelement nznzexceeds nznzonly nznza nznzfinite nznznumber nznzof nznzother
nznzelements nznzbut nznzis nznzexceeded nznzby nznzan nznzinfinite nznznumber nznzof nznzother nznzelements. nznzSuch nznzpeople
nznzmight nznzclaim nznzthe nznzanswer nznzshould nznzbe nznz∅.)
6. ∅ 7. n z n z The nznz set nznz is nznz ∅ n zn z because nznz 33 nznz= nznz27 nznz and nz nz 43 nznz= nznz64.
8. nznz It n zn z is n z n z not n z nz a nz n z well-defined n z n z set. 9. nznz Q
10. The n z n z set n z n z containing n z n z all n z n z numbers n z n z that n z n z are n z n z (positive, n z n z negative, n z n z or
n z n z zero) n z n z integer n z n z multiples n z n z of n z n z 1, n z n z 1/2, n z n z or nznz1/3.
11. {(a, nznz1), nznz (a, nznz 2), nznz (a, nznz c), nznz (b, nznz1), nznz(b, nznz 2), nznz(b, nznz c), nznz (c, nznz 1), nznz (c, nznz2), nznz (c, nznz c)}
12. a. n z n z It nznzis nznza nznzfunction. n z n z It nznzis nznznot nznzone-to-one nznzsince nznzthere nznzare nznztwo nznzpairs
nznz with nznz second nznz member nznz 4. n z n z It nznz is nznz not nznz onto
B n z n z because nznzthere nznzis nznzno nznz pair nznzwith nznzsecond nznzmember nznz2.
b. (Same n z n z answer n z n z as n z n z Part(a).)
c. It nznz is nznz not nznz a nznz function nznz because nznz there nznz are nznz two nznz pairs nznz with nznz first nznz member
nznz 1.
d. It n z n z is n z n z a n z n z function. n z n z It n z n z is n z n z one-to-one. n z n z It n z n z is n z n z onto n z n z B n z n z because
n z n z every n z n z element n z n z of n z n z B n z n z appears n z n z as n z n z second nznzmember nznzof nznzsome nznzpair.
e. It nznzis nznza nznzfunction. n z n z It nznzis nznznot nznzone-to-one nznzbecause nznzthere nznzare nznztwo nznzpairs nznzwith
nznzsecond nznzmember nznz6. n z n z It nznzis nznznot nznzonto nznzB nznzbecause nznzthere nznzis nznzno nznzpair nznzwith
nznzsecond nznzmember nznz2.
f. It nznzis nznz not nznz a nznz function nznz because nznz there nznz are nznz two nznz pairs nznz with nznz first nznz member
nznz 2.
13. Draw n z n z the n z n z line n z n z through n z n z P n z n z and n z n z x, n z n z and n z n z let n z n z y n z n z be n z n z its n z n z point n z n z of
n z n z intersection n z n z with n z n z the n z n z line n z n z segment n z n z CD.
14. a. n z n z φ nznz: nznz[0, nznz1] nznz→ nznz [0, nznz2] n z n z where n z n z φ(x) nznz= nznz2x b. n z n z φ nznz: nznz[1, nznz3] nznz → nznz [5,
nznz25] n z n z where n z n z φ(x) nznz= nznz5 nznz+ nznz10(x nznz− nznz1)
c. n z n z φ nznz:→
nznz[a, nznzb]
d c
− [c, nznzd] n z n z where n z n z φ(x) nznz= nznzc nznz+ n z n z − nznz(x a)
b−a
15. Let n z n z φ nznz: nznzS n z n z → nznz R n z n z be n z n z defined n z n2z by n z n z φ(x) nznz= nznztan(π(x nznz− n z n z 1 nznz)).
16. a. n z n z ∅; n z n z cardinality n z n z 1 b. nznz ∅, nznz{a}; n z n z cardinality n z n z 2 c. nznz ∅,
nznz{a}, nznz{b}, nznz{a, nznzb}; n z n z cardinality n z n z 4
d. nznz ∅, nznz{a}, nznz{b}, nznz{c}, nznz{a, nznzb}, nznz{a, nznzc}, nznz{b, nznzc}, nznz{a, nznzb, nznzc}; n z n z cardinality n z n z 8
17. Conjecture: |P(A)| nznz= nznz2s nznz= nznz2|A|.
