SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets4 1 and4 1 Relations 1
I. Groups4 1 and4 1 Subgroups
1. Introduction4 1 and4 1 Examples 4
2. Binary4 1 Operations 7
3. Isomorphic4 1 Binary4 1 Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic4 1 Groups 21
7. Generators4 1 and4 1 Cayley4 1 Digraphs 24
II. Permutations,41 Cosets,41 and41 Direct41 Products
8. Groups4 1 of4 1 Permutations 26
9. Orbits,41 Cycles,41and41 the41Alternating41Groups 30
10. Cosets41 and41 the41 Theorem41 of41 Lagrange 34
11. Direct4 1 Products4 1 and4 1 Finitely4 1 Generated4 1 Abelian4 1 Groups 37
12. Plane4 1 Isometries 42
III. Homomorphisms4 1 and4 1 Factor4 1 Groups
13. Homomorphisms 44
14. Factor4 1 Groups 49
15. Factor-Group4 1 Computations4 1 and4 1 Simple4 1 Groups 53
16. Group41 Action41 on41 a41 Set 58
17. Applications41of41G-Sets41to41Counting 61
IV. Rings4 1 and4 1 Fields
18. Rings41 and41 Fields 63
19. Integral4 1 Domains 68
20. Fermat’s4 1 and4 1 Euler’s4 1 Theorems 72
21. The4 1 Field4 1 of4 1 Quotients4 1 of4 1 an4 1 Integral4 1 Domain 74
22. Rings4 1 of4 1 Polynomials76
23. Factorization41of41Polynomials41over41a41Field 79
24. Noncommutative4 1 Examples 85
25. Ordered4 1 Rings4 1 and4 1 Fields 87
V. Ideals4 1 and4 1 Factor4 1 Rings
26. Homomorphisms41 and41 Factor41 Rings 89
27. Prime41and41Maximal41Ideals 94
28. Gröbner 41Bases41for41Ideals 99
, VI. Extension4 1 Fields
29. Introduction41 to41 Extension41 Fields 103
30. Vector4 1 Spaces 107
31. Algebraic4 1 Extensions 111
32. Geometric41 Constructions 115
33. Finite4 1 Fields 116
VII. Advanced41 Group41 Theory
34. Isomorphism41Theorems 117
35. Series41of41Groups 119
36. Sylow4 1 Theorems 122
37. Applications4 1 of4 1 the4 1 Sylow4 1 Theory 124
38. Free4 1 Abelian4 1 Groups 128
39. Free41Groups 130
40. Group4 1 Presentations 133
VIII. Groups4 1 in4 1 Topology
41. Simplicial4 1 Complexes4 1 and4 1 Homology4 1 Groups 136
42. Computations4 1 of4 1 Homology41 Groups 138
43. More41 Homology41 Computations41 and41 Applications 140
44. Homological41 Algebra 144
IX. Factorization
45. Unique4 1 Factorization4 1 Domains 148
46. Euclidean4 1 Domains 151
47. Gaussian4 1 Integers4 1 and4 1 Multiplicative4 1 Norms 154
X. Automorphisms4 1 and4 1 Galois4 1 Theory
48. Automorphisms41 of41 Fields 159
49. The4 1 Isomorphism4 1 Extension4 1 Theorem 164
50. Splitting4 1 Fields 165
51. Separable41Extensions 167
52. Totally41Inseparable41Extensions171
53. Galois4 1 Theory 173
54. Illustrations41of41Galois41Theory 176
55. Cyclotomic41Extensions 183
56. Insolvability4 1 of4 1 the4 1 Quintic 185
APPENDIX4 1 Matrix4 1 Algebra 187
iv
, 0.4 1 Sets41and41Relations 1
0. Sets4 1 and4 1 Relations
√ √
1. { 3,4 1 − 3} 2.4 1 The4 1 set4 1 is4 1 empty.
3.4 1 {1,41−1,412,41−2,413,41−3,414,41−4,415,41−5,416,41−6,4110,41−10,4112,41−12,4115,41−15,4120,41−20,4130,41−30,
60,41−60}
4.4 1 {−10,41−9,41−8,41−7,41−6,41−5,41−4,41−3,41−2,41−1,410,411,412,413,414,415,416,417,418,419,4110,4111}
5. It41is41not41a41well-
defined41set.4 1 (Some41may41argue41that41no41element41of41Z+41is41large,41because41every41elemen
t41exceeds41only41a41finite41number41of41other41elements41but41is41exceeded41by41an41infinite41number
41of41other41elements.41Such41people41might41claim41the41answer41should41be41∅.)
6. ∅ 7.4 1 The41 set41 is41 ∅41 because41 3341=412741 and41 4341=4164.
8.4 1 It41 is41 not41 a41 well-defined41 set. 9.4 1 Q
10. The4 1 set4 1 containing4 1 all4 1 numbers4 1 that4 1 are4 1 (positive,4 1 negative,4 1 or4 1 zero)4 1 integer4 1 mul
tiples4 1 of4 1 1,4 1 1/2,4 1 or411/3.
11. {(a,411),41 (a,41 2),41 (a,41 c),41 (b,411),41 (b,412),41(b,41 c),41 (c,411),41 (c,412),41(c,41 c)}
12. a.4 1 It41 is41 a41 function.4 1 It41 is41 not41 one-to-
one41 since41there41 are41 two41 pairs41 with41 second41 member41 4.4 1 It41 is41 not41 onto
B41 because41 there41 is41 no41 pair41 with41 second41 member41 2.
b. (Same4 1 answer4 1 as4 1 Part(a).)
c. It41 is41 not41 a41 function41 because41 there41 are41 two41 pairs41 with41 first41 member41 1.
d. It41 is41 a41 function.4 1 It41 is41 one-to-
one.4 1 It4 1 is41 onto41 B4 1 because41 every41 element4 1 of41 B4 1 appears4 1 as41 second41member41of
41some41pair.
e. It41is41a41function.41 It41is41not41one-to-
one41because41there41are41two41pairs41with41second41member416.4 1 It41is41not41onto41B41becaus
e41there41is41no41pair41with41second41member412.
f. It41 is41 not41 a41 function41 because41 there41 are41 two41 pairs41 with41 first41 member41 2.
13. Draw41 the4 1 line41 through41 P4 1 and41 x,4 1 and4 1 let41 y4 1 be4 1 its4 1 point4 1 of4 1 intersection4 1 with4 1 the41
line4 1 segment4 1 CD.
14. a.4 1 φ41:41 [0,411]41→41 [0,412]4 1 where4 1 φ(x)41=412x
b.4 1 φ41:41 [1,413]41 →41 [5,4125]4 1 where4 1 φ(x)41=41541+4110(
x41−411)
c.4 1 φ41:41[a,
→41b]
d c
− [c,41d]4 1 where4 1 φ(x)41=41c41+4 1 − 41(x a)
b−a
15. Let41 φ41:41S41 →41R41 be41 defined4 1 by41 φ(x)41
2
=41tan(π(x41−41 141)).
16. a.4 1 ∅;4 1 cardinality4 1 1 b.4 1 ∅,41{a};4 1 cardinality4 1 2c.4 1 ∅,41{a},41{b},41{a,41b};4 1 cardinality4 1 4
d.4 1 ∅,41{a},41{b},41{c},41{a,41b},41{a,41c},41{b,41c},41{a,41b,41c};4 1 cardinality4 1 8
17. Conjecture: |P(A)|41=412s41 =412|A|.
Proof41The41number41of41subsets41of41a41set41A41depends41only41on41the41cardinality41of41A,41
not41on41what41the41elements41of4 1 A4 1 actually4 1 are.4 1 Suppose41B41=41{1,412,413,41·41·41·41,41s41
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets4 1 and4 1 Relations 1
I. Groups4 1 and4 1 Subgroups
1. Introduction4 1 and4 1 Examples 4
2. Binary4 1 Operations 7
3. Isomorphic4 1 Binary4 1 Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic4 1 Groups 21
7. Generators4 1 and4 1 Cayley4 1 Digraphs 24
II. Permutations,41 Cosets,41 and41 Direct41 Products
8. Groups4 1 of4 1 Permutations 26
9. Orbits,41 Cycles,41and41 the41Alternating41Groups 30
10. Cosets41 and41 the41 Theorem41 of41 Lagrange 34
11. Direct4 1 Products4 1 and4 1 Finitely4 1 Generated4 1 Abelian4 1 Groups 37
12. Plane4 1 Isometries 42
III. Homomorphisms4 1 and4 1 Factor4 1 Groups
13. Homomorphisms 44
14. Factor4 1 Groups 49
15. Factor-Group4 1 Computations4 1 and4 1 Simple4 1 Groups 53
16. Group41 Action41 on41 a41 Set 58
17. Applications41of41G-Sets41to41Counting 61
IV. Rings4 1 and4 1 Fields
18. Rings41 and41 Fields 63
19. Integral4 1 Domains 68
20. Fermat’s4 1 and4 1 Euler’s4 1 Theorems 72
21. The4 1 Field4 1 of4 1 Quotients4 1 of4 1 an4 1 Integral4 1 Domain 74
22. Rings4 1 of4 1 Polynomials76
23. Factorization41of41Polynomials41over41a41Field 79
24. Noncommutative4 1 Examples 85
25. Ordered4 1 Rings4 1 and4 1 Fields 87
V. Ideals4 1 and4 1 Factor4 1 Rings
26. Homomorphisms41 and41 Factor41 Rings 89
27. Prime41and41Maximal41Ideals 94
28. Gröbner 41Bases41for41Ideals 99
, VI. Extension4 1 Fields
29. Introduction41 to41 Extension41 Fields 103
30. Vector4 1 Spaces 107
31. Algebraic4 1 Extensions 111
32. Geometric41 Constructions 115
33. Finite4 1 Fields 116
VII. Advanced41 Group41 Theory
34. Isomorphism41Theorems 117
35. Series41of41Groups 119
36. Sylow4 1 Theorems 122
37. Applications4 1 of4 1 the4 1 Sylow4 1 Theory 124
38. Free4 1 Abelian4 1 Groups 128
39. Free41Groups 130
40. Group4 1 Presentations 133
VIII. Groups4 1 in4 1 Topology
41. Simplicial4 1 Complexes4 1 and4 1 Homology4 1 Groups 136
42. Computations4 1 of4 1 Homology41 Groups 138
43. More41 Homology41 Computations41 and41 Applications 140
44. Homological41 Algebra 144
IX. Factorization
45. Unique4 1 Factorization4 1 Domains 148
46. Euclidean4 1 Domains 151
47. Gaussian4 1 Integers4 1 and4 1 Multiplicative4 1 Norms 154
X. Automorphisms4 1 and4 1 Galois4 1 Theory
48. Automorphisms41 of41 Fields 159
49. The4 1 Isomorphism4 1 Extension4 1 Theorem 164
50. Splitting4 1 Fields 165
51. Separable41Extensions 167
52. Totally41Inseparable41Extensions171
53. Galois4 1 Theory 173
54. Illustrations41of41Galois41Theory 176
55. Cyclotomic41Extensions 183
56. Insolvability4 1 of4 1 the4 1 Quintic 185
APPENDIX4 1 Matrix4 1 Algebra 187
iv
, 0.4 1 Sets41and41Relations 1
0. Sets4 1 and4 1 Relations
√ √
1. { 3,4 1 − 3} 2.4 1 The4 1 set4 1 is4 1 empty.
3.4 1 {1,41−1,412,41−2,413,41−3,414,41−4,415,41−5,416,41−6,4110,41−10,4112,41−12,4115,41−15,4120,41−20,4130,41−30,
60,41−60}
4.4 1 {−10,41−9,41−8,41−7,41−6,41−5,41−4,41−3,41−2,41−1,410,411,412,413,414,415,416,417,418,419,4110,4111}
5. It41is41not41a41well-
defined41set.4 1 (Some41may41argue41that41no41element41of41Z+41is41large,41because41every41elemen
t41exceeds41only41a41finite41number41of41other41elements41but41is41exceeded41by41an41infinite41number
41of41other41elements.41Such41people41might41claim41the41answer41should41be41∅.)
6. ∅ 7.4 1 The41 set41 is41 ∅41 because41 3341=412741 and41 4341=4164.
8.4 1 It41 is41 not41 a41 well-defined41 set. 9.4 1 Q
10. The4 1 set4 1 containing4 1 all4 1 numbers4 1 that4 1 are4 1 (positive,4 1 negative,4 1 or4 1 zero)4 1 integer4 1 mul
tiples4 1 of4 1 1,4 1 1/2,4 1 or411/3.
11. {(a,411),41 (a,41 2),41 (a,41 c),41 (b,411),41 (b,412),41(b,41 c),41 (c,411),41 (c,412),41(c,41 c)}
12. a.4 1 It41 is41 a41 function.4 1 It41 is41 not41 one-to-
one41 since41there41 are41 two41 pairs41 with41 second41 member41 4.4 1 It41 is41 not41 onto
B41 because41 there41 is41 no41 pair41 with41 second41 member41 2.
b. (Same4 1 answer4 1 as4 1 Part(a).)
c. It41 is41 not41 a41 function41 because41 there41 are41 two41 pairs41 with41 first41 member41 1.
d. It41 is41 a41 function.4 1 It41 is41 one-to-
one.4 1 It4 1 is41 onto41 B4 1 because41 every41 element4 1 of41 B4 1 appears4 1 as41 second41member41of
41some41pair.
e. It41is41a41function.41 It41is41not41one-to-
one41because41there41are41two41pairs41with41second41member416.4 1 It41is41not41onto41B41becaus
e41there41is41no41pair41with41second41member412.
f. It41 is41 not41 a41 function41 because41 there41 are41 two41 pairs41 with41 first41 member41 2.
13. Draw41 the4 1 line41 through41 P4 1 and41 x,4 1 and4 1 let41 y4 1 be4 1 its4 1 point4 1 of4 1 intersection4 1 with4 1 the41
line4 1 segment4 1 CD.
14. a.4 1 φ41:41 [0,411]41→41 [0,412]4 1 where4 1 φ(x)41=412x
b.4 1 φ41:41 [1,413]41 →41 [5,4125]4 1 where4 1 φ(x)41=41541+4110(
x41−411)
c.4 1 φ41:41[a,
→41b]
d c
− [c,41d]4 1 where4 1 φ(x)41=41c41+4 1 − 41(x a)
b−a
15. Let41 φ41:41S41 →41R41 be41 defined4 1 by41 φ(x)41
2
=41tan(π(x41−41 141)).
16. a.4 1 ∅;4 1 cardinality4 1 1 b.4 1 ∅,41{a};4 1 cardinality4 1 2c.4 1 ∅,41{a},41{b},41{a,41b};4 1 cardinality4 1 4
d.4 1 ∅,41{a},41{b},41{c},41{a,41b},41{a,41c},41{b,41c},41{a,41b,41c};4 1 cardinality4 1 8
17. Conjecture: |P(A)|41=412s41 =412|A|.
Proof41The41number41of41subsets41of41a41set41A41depends41only41on41the41cardinality41of41A,41
not41on41what41the41elements41of4 1 A4 1 actually4 1 are.4 1 Suppose41B41=41{1,412,413,41·41·41·41,41s41
