SOLUTION MANUAL
First Course in Abstract
Algebra A 8th Edition by
John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Sets and Relations 1
I. Groups and Subgroups
2. Introduction and Examples 4
3. Binary Operations 7
4. Isomorphic Binary Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
8. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
9. Groups of Permutations 26
10. Orbits, Cycles, and the Alternating Groups
30
11. Cosets and the Theorem of Lagrange34
12. Direct Products and Finitely Generated Abelian Groups 37
13. Plane Isometries 42
III. Homomorphisms and Factor Groups
14. Homomorphisms 44
15. Factor Groups 49
16. Factor-Group Computations and Simple Groups 53
17. Group Action on a Set 58
18. Applications of G-Sets to Counting 61
IV. Rings and Fields
19. Rings and Fields 63
20. Integral Domains 68
21. Fermat’s and Euler’s Theorems 72
22. The Field of Quotients of an Integral Domain 74
23. Rings of Polynomials 76
24. Factorization of Polynomials over a Field 79
25. Noncommutative Examples 85
26. Ordered Rings and Fields 87
V. Ideals and Factor Rings
27. Homomorphisms and Factor Rings 89
28. Prime and Maximal Ideals 94Gröbner Bases for Ideals 99
, VI. Extension Fields
29. Introduction to Extension Fields 103
30. Vector Spaces 107
31. Algebraic Extensions 111
32. Geometric Constructions 115
33. Finite Fields 116
VII. Advanced Group Theory
34. Isomorphism Theorems 117
35. Series of Groups 119
36. Sylow Theorems 122
37. Applications of the Sylow Theory 124
38. Free Abelian Groups 128
39. Free Groups 130
40. Group Presentations 133
VIII. Groups in Topology
41. Simplicial Complexes and Homology Groups 136
42. Computations of Homology Groups 138
43. More Homology Computations and Applications 140
44. Homological Algebra 144
IX. Factorization
45. Unique Factorization Domains 148
46. Euclidean Domains 151
47. Gaussian Integers and Multiplicative Norms 154
X. Automorphisms and Galois Theory
48. Automorphisms of Fields 159
49. The Isomorphism Extension Theorem164
50. Splitting Fields 165
51. Separable Extensions 167
52. Totally Inseparable Extensions 171
53. Galois Theory 173
54. Illustrations of Galois Theory 176
55. Cyclotomic Extensions 183
56. Insolvability of the Quintic 185
APPENDIX Matrix Algebra 187
iv
, 0. z Sets zand zRelations 1
1. Sets z and z Relations
√ √ zzz
1. z 3, 3} 2. z The z set z is z empty.
z{ z−
3. z {1, z−1, z2, z−2, z3, z−3, z4, z−4, z5, z−5, z6, z−6, z10, z−10, z12, z−12, z15, z−15, z20, z−20, z30, z−30,
60, z−60}
4. z {−10, z−9, z−8, z−7, z−6, z−5, z−4, z−3, z−2, z−1, z0, z1, z2, z3, z4, z5, z6, z7, z8, z9, z10, z11}
5. It z is z not z a z well-defined z set. z(Some z may z argue z that z no z element z of z Z+ z is z large, z because
z every z element z exceeds zonly za zfinite znumber zof zother zelements zbut zis zexceeded zby zan zinfinite
znumber zof zother zelements. z Such zpeople zmight zclaim zthe zanswer zshould zbe z∅.)
6. ∅ 7. z The z set z is z ∅ z because z 33 z = z27 z and z 43 z = z64.
8. z It z is z not z a z well-defined z set. 9. z Q
10. z The z set z containing z all z numbers z that z are z (positive, z negative, z or z zero) z integer z multiples
z of z 1, z 1/2, z or z 1/3.
11. z z z{(a, z 1), z (a, z 2), z (a, z c), z (b, z1), z (b, z 2), z (b, z c), z (c, z 1), z (c, z 2), z (c, zc)}
12. a. z It zis za zfunction. z It zis z not zone-to-one zsince zthere zare ztwo zpairs zwith zsecond zmember z4.
z It zis znot zonto
B z because zthere zis zno zpair zwith zsecond zmember z2.
b. (Same z answer z as z Part(a).)
c. It zis z not za zfunction z because zthere zare z two z pairs zwith zfirst zmember z 1.
d. It z is z a z function. z zIt z is z one-to-one. z zIt z is z onto z B z because z every z element z of z B
z appears z as z second z member zof zsome zpair.
e. It zis za zfunction. zIt zis znot zone-to-one zbecause zthere zare ztwo zpairs zwith zsecond zmember
z6. z It zis znot z onto zB zbecause zthere zis zno zpair zwith zsecond zmember z2.
f. It zis z not z a zfunction z because zthere zare z two z pairs zwith zfirst zmember z 2.
13. Draw z the z line z through z P z and z x, z and z let z y z be z its z point z of z intersection z with z the z line
z segment z CD.
14. z z a. z φ z: z [0, z1] z→ z [0, z2] z where z φ(x) z= z2x b. z φ z: z [1, z3] z → z [5, z25] z where z φ(x) z= z5 z+ z10(x
z− z1)
c. z φ z : z [a, zb] z→ z [c, zd] z where z φ(x) z = z c z+ z d−c z(x z − z za)
b z−z a
15. Let zφ z: zS z → zR z be z defined z by z φ(x) z= 1
)2).
ztan(π(x z−
16. a. z ∅; z cardinality z 1 b. z ∅, z{a}; z cardinality z 2 c. z ∅, z{a}, z{b}, z{a, zb}; z cardinality z 4
d. z ∅, z{a}, z{b}, z{c}, z{a, zb}, z{a, zc}, z{b, zc}, z{a, zb, zc}; z cardinality z 8
17. Conjecture: z |P(A)| z= z2s z = z2|A|.
Proof z The z number z of z subsets z of z a z set z A z depends z only z on z the z cardinality z of z A,
z not z on z what z the z elements zof z A z actually z are. z Suppose z B z= z{1, z2, z3, z· z· z· z, zs z− z1}
First Course in Abstract
Algebra A 8th Edition by
John B. Fraleigh
All Chapters Full Complete
, CONTENTS
1. Sets and Relations 1
I. Groups and Subgroups
2. Introduction and Examples 4
3. Binary Operations 7
4. Isomorphic Binary Structures 9
5. Groups 13
6. Subgroups 17
7. Cyclic Groups 21
8. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
9. Groups of Permutations 26
10. Orbits, Cycles, and the Alternating Groups
30
11. Cosets and the Theorem of Lagrange34
12. Direct Products and Finitely Generated Abelian Groups 37
13. Plane Isometries 42
III. Homomorphisms and Factor Groups
14. Homomorphisms 44
15. Factor Groups 49
16. Factor-Group Computations and Simple Groups 53
17. Group Action on a Set 58
18. Applications of G-Sets to Counting 61
IV. Rings and Fields
19. Rings and Fields 63
20. Integral Domains 68
21. Fermat’s and Euler’s Theorems 72
22. The Field of Quotients of an Integral Domain 74
23. Rings of Polynomials 76
24. Factorization of Polynomials over a Field 79
25. Noncommutative Examples 85
26. Ordered Rings and Fields 87
V. Ideals and Factor Rings
27. Homomorphisms and Factor Rings 89
28. Prime and Maximal Ideals 94Gröbner Bases for Ideals 99
, VI. Extension Fields
29. Introduction to Extension Fields 103
30. Vector Spaces 107
31. Algebraic Extensions 111
32. Geometric Constructions 115
33. Finite Fields 116
VII. Advanced Group Theory
34. Isomorphism Theorems 117
35. Series of Groups 119
36. Sylow Theorems 122
37. Applications of the Sylow Theory 124
38. Free Abelian Groups 128
39. Free Groups 130
40. Group Presentations 133
VIII. Groups in Topology
41. Simplicial Complexes and Homology Groups 136
42. Computations of Homology Groups 138
43. More Homology Computations and Applications 140
44. Homological Algebra 144
IX. Factorization
45. Unique Factorization Domains 148
46. Euclidean Domains 151
47. Gaussian Integers and Multiplicative Norms 154
X. Automorphisms and Galois Theory
48. Automorphisms of Fields 159
49. The Isomorphism Extension Theorem164
50. Splitting Fields 165
51. Separable Extensions 167
52. Totally Inseparable Extensions 171
53. Galois Theory 173
54. Illustrations of Galois Theory 176
55. Cyclotomic Extensions 183
56. Insolvability of the Quintic 185
APPENDIX Matrix Algebra 187
iv
, 0. z Sets zand zRelations 1
1. Sets z and z Relations
√ √ zzz
1. z 3, 3} 2. z The z set z is z empty.
z{ z−
3. z {1, z−1, z2, z−2, z3, z−3, z4, z−4, z5, z−5, z6, z−6, z10, z−10, z12, z−12, z15, z−15, z20, z−20, z30, z−30,
60, z−60}
4. z {−10, z−9, z−8, z−7, z−6, z−5, z−4, z−3, z−2, z−1, z0, z1, z2, z3, z4, z5, z6, z7, z8, z9, z10, z11}
5. It z is z not z a z well-defined z set. z(Some z may z argue z that z no z element z of z Z+ z is z large, z because
z every z element z exceeds zonly za zfinite znumber zof zother zelements zbut zis zexceeded zby zan zinfinite
znumber zof zother zelements. z Such zpeople zmight zclaim zthe zanswer zshould zbe z∅.)
6. ∅ 7. z The z set z is z ∅ z because z 33 z = z27 z and z 43 z = z64.
8. z It z is z not z a z well-defined z set. 9. z Q
10. z The z set z containing z all z numbers z that z are z (positive, z negative, z or z zero) z integer z multiples
z of z 1, z 1/2, z or z 1/3.
11. z z z{(a, z 1), z (a, z 2), z (a, z c), z (b, z1), z (b, z 2), z (b, z c), z (c, z 1), z (c, z 2), z (c, zc)}
12. a. z It zis za zfunction. z It zis z not zone-to-one zsince zthere zare ztwo zpairs zwith zsecond zmember z4.
z It zis znot zonto
B z because zthere zis zno zpair zwith zsecond zmember z2.
b. (Same z answer z as z Part(a).)
c. It zis z not za zfunction z because zthere zare z two z pairs zwith zfirst zmember z 1.
d. It z is z a z function. z zIt z is z one-to-one. z zIt z is z onto z B z because z every z element z of z B
z appears z as z second z member zof zsome zpair.
e. It zis za zfunction. zIt zis znot zone-to-one zbecause zthere zare ztwo zpairs zwith zsecond zmember
z6. z It zis znot z onto zB zbecause zthere zis zno zpair zwith zsecond zmember z2.
f. It zis z not z a zfunction z because zthere zare z two z pairs zwith zfirst zmember z 2.
13. Draw z the z line z through z P z and z x, z and z let z y z be z its z point z of z intersection z with z the z line
z segment z CD.
14. z z a. z φ z: z [0, z1] z→ z [0, z2] z where z φ(x) z= z2x b. z φ z: z [1, z3] z → z [5, z25] z where z φ(x) z= z5 z+ z10(x
z− z1)
c. z φ z : z [a, zb] z→ z [c, zd] z where z φ(x) z = z c z+ z d−c z(x z − z za)
b z−z a
15. Let zφ z: zS z → zR z be z defined z by z φ(x) z= 1
)2).
ztan(π(x z−
16. a. z ∅; z cardinality z 1 b. z ∅, z{a}; z cardinality z 2 c. z ∅, z{a}, z{b}, z{a, zb}; z cardinality z 4
d. z ∅, z{a}, z{b}, z{c}, z{a, zb}, z{a, zc}, z{b, zc}, z{a, zb, zc}; z cardinality z 8
17. Conjecture: z |P(A)| z= z2s z = z2|A|.
Proof z The z number z of z subsets z of z a z set z A z depends z only z on z the z cardinality z of z A,
z not z on z what z the z elements zof z A z actually z are. z Suppose z B z= z{1, z2, z3, z· z· z· z, zs z− z1}
