SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets d and d Relations 1
I. Groups and Subgroups
d d
1. Introduction d and d Examples 4
2. Binary d Operations 7
3. Isomorphic d Binary d Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic d Groups 21
7. Generators d and d Cayley d Digraphs 24
II. Permutations, Cosets, and Direct Products
d d d d
8. Groups d of d Permutations 26
9. Orbits, d Cycles, d and d the d Alternating dGroups 30
10. Cosets d and d the d Theorem d of d Lagrange 34
11. Direct d Products d and d Finitely d Generated d Abelian d Groups 37
12. Plane d Isometries 42
III. Homomorphisms and Factor Groups d d d
13. Homomorphisms 44
14. Factor d Groups 49
15. Factor-Group d Computations d and d Simple d Groups 53
16. Group d Action d on d a dSet 58
17. Applications dof dG-Sets dto dCounting 61
IV. Rings and Fields d d
18. Rings d and d Fields 63
19. Integral d Domains 68
20. Fermat’s d and d Euler’s d Theorems 72
21. The d Field d of d Quotients d of d an d Integral d Domain 74
22. Rings d of d Polynomials 76
23. Factorization dof dPolynomials dover da dField 79
24. Noncommutative d Examples 85
25. Ordered d Rings d and d Fields 87
V. Ideals and Factor Rings
d d d
26. Homomorphisms d and d Factor d Rings 89
27. Prime dand dMaximal dIdeals 94
28. Gröbner dBases dfor dIdeals 99
, VI. Extension Fields
d
29. Introduction d to d Extension d Fields 103
30. Vector d Spaces 107
31. Algebraic d Extensions 111
32. Geometric d Constructions 115
33. Finite d Fields 116
VII. Advanced Group Theory
d d
34. Isomorphism dTheorems 117
35. Series dof dGroups 119
36. Sylow d Theorems 122
37. Applications d of d the d Sylow d Theory 124
38. Free d Abelian d Groups 128
39. Free dGroups 130
40. Group d Presentations 133
VIII. Groups in Topology
d d
41. Simplicial d Complexes d and d Homology d Groups 136
42. Computations d of d Homology d Groups 138
43. More d Homology d Computations d and d Applications 140
44. Homological d Algebra 144
IX. Factorization
45. Unique d Factorization d Domains 148
46. Euclidean d Domains 151
47. Gaussian d Integers d and d Multiplicative d Norms 154
X. Automorphisms and Galois Theory
d d d
48. Automorphisms d of d Fields 159
49. The d Isomorphism d Extension d Theorem 164
50. Splitting d Fields 165
51. Separable dExtensions 167
52. Totally d Inseparable d Extensions 171
53. Galois d Theory 173
54. Illustrations dof dGalois dTheory 176
55. Cyclotomic dExtensions 183
56. Insolvability d of d the d Quintic 185
APPENDIX d Matrix d Algebra 187
iv
, 0. d Sets dand dRelations 1
0. Sets and Relations
d d
√ √
1. { 3, d − 3} 2. d The d set d is d empty.
3. d {1,d−1,d2,d−2,d3,d−3,d4,d−4,d5,d−5,d6,d−6,d10,d−10,d12,d−12,d15,d−15,d20,d−20,d30,d−30,
60, d−60}
4. d {−10,d−9,d−8,d−7,d−6,d−5,d−4,d−3,d−2,d−1,d0,d1,d2,d3,d4,d5,d6,d7,d8,d9,d10,d11}
5. It dis dnot da dwell-defined dset. d (Some dmay dargue dthat dno delement dof dZ+ dis dlarge, dbecause devery
delement dexceeds donly da dfinite dnumber dof dother delements dbut dis dexceeded dby dan dinfinite dnumber dof
dother delements. dSuch dpeople dmight dclaim dthe danswer dshould dbe d∅.)
6. ∅ 7. d The d set d is d ∅ d because d 33 d= d27 d and d 43 d= d64.
8. d It d is d not d a d well-defined d set. 9. d Q
10. The d set d containing d all d numbers d that d are d (positive, d negative, d or d zero) d integer d multiples d of d 1,
d 1/2, d or d1/3.
11. {(a, d1), d (a, d2), d (a, d c), d (b, d1), d(b, d 2), d(b, d c), d(c, d 1), d (c, d2), d (c, d c)}
12. a. d It d is d a d function. d It d is dnot d one-to-one d since dthere d are d two d pairs d with dsecond d member d 4. d It d is
d not d onto
B d because d there d is d no d pair d with d second d member d 2.
b. (Same d answer d as d Part(a).)
c. It d is d not d a d function d because d there d are d two d pairs d with d first d member d 1.
d. It d is d a d function. d It d is d one-to-one. d It d is d onto d B d because d every d element d of d B d appears
d as d second dmember dof dsome dpair.
e. It dis da dfunction. d It dis dnot done-to-one dbecause dthere dare dtwo dpairs dwith dsecond dmember d6. d It
dis dnot donto dB dbecause dthere dis dno dpair dwith dsecond dmember d2.
f. It d is d not d a d function d because d there d are d two d pairs d with d first d member d 2.
13. Draw d the d line d through d P d and d x, d and d let d y d be d its d point d of d intersection d with d the d line d segment
d CD.
14. a. d φ d: d[0, d1] d→ d [0,d2] d where d φ(x) d= d2x b. d φ d: d [1,d3] d → d [5, d25] d where d φ(x) d= d5 d+ d10(x d− d1)
c. d φ d: d[a, db]→ [c, dd] d where d φ(x) d= dc d+ d d−cd(x
− a) b−a
15. Let d φ d: dS d → dR d be d defined d by d φ(x) d= dtan(π(x2d− d 1 d)).
16. a. d ∅; d cardinality d 1 b. d ∅, d{a}; d cardinality d 2 c. d ∅, d{a},d{b},d{a,db}; d cardinality d 4
d. d ∅,d{a},d{b}, d{c},d{a,db}, d{a,dc}, d{b, dc},d{a, db, dc}; d cardinality d 8
17. Conjecture: |P(A)| d= d2s d = d2|A|.
Proof dThe dnumber dof dsubsets dof da dset dA ddepends donly don dthe dcardinality dof dA, dnot don dwhat
dthe delements dof d A d actually d are. d Suppose dB d= d{1, d2, d3, d· d· d· d, ds d− d1} d and d A d= d{1, d2, d3, d d , ds}.
d Then d A d has d all
the delements dof dB dplus dthe done dadditional delement ds. d All dsubsets dof dB dare dalso dsubsets dof dA;
dthese dare dprecisely dthe dsubsets dof dA dthat ddo dnot dcontain ds, dso dthe dnumber dof dsubsets dof dA d not
dcontaining ds d is d|P(B)|. d Any dother d subset dof dA dmust d contain ds, d and d removal dof d the d s d would
dproduce d a d subset d of
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets d and d Relations 1
I. Groups and Subgroups
d d
1. Introduction d and d Examples 4
2. Binary d Operations 7
3. Isomorphic d Binary d Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic d Groups 21
7. Generators d and d Cayley d Digraphs 24
II. Permutations, Cosets, and Direct Products
d d d d
8. Groups d of d Permutations 26
9. Orbits, d Cycles, d and d the d Alternating dGroups 30
10. Cosets d and d the d Theorem d of d Lagrange 34
11. Direct d Products d and d Finitely d Generated d Abelian d Groups 37
12. Plane d Isometries 42
III. Homomorphisms and Factor Groups d d d
13. Homomorphisms 44
14. Factor d Groups 49
15. Factor-Group d Computations d and d Simple d Groups 53
16. Group d Action d on d a dSet 58
17. Applications dof dG-Sets dto dCounting 61
IV. Rings and Fields d d
18. Rings d and d Fields 63
19. Integral d Domains 68
20. Fermat’s d and d Euler’s d Theorems 72
21. The d Field d of d Quotients d of d an d Integral d Domain 74
22. Rings d of d Polynomials 76
23. Factorization dof dPolynomials dover da dField 79
24. Noncommutative d Examples 85
25. Ordered d Rings d and d Fields 87
V. Ideals and Factor Rings
d d d
26. Homomorphisms d and d Factor d Rings 89
27. Prime dand dMaximal dIdeals 94
28. Gröbner dBases dfor dIdeals 99
, VI. Extension Fields
d
29. Introduction d to d Extension d Fields 103
30. Vector d Spaces 107
31. Algebraic d Extensions 111
32. Geometric d Constructions 115
33. Finite d Fields 116
VII. Advanced Group Theory
d d
34. Isomorphism dTheorems 117
35. Series dof dGroups 119
36. Sylow d Theorems 122
37. Applications d of d the d Sylow d Theory 124
38. Free d Abelian d Groups 128
39. Free dGroups 130
40. Group d Presentations 133
VIII. Groups in Topology
d d
41. Simplicial d Complexes d and d Homology d Groups 136
42. Computations d of d Homology d Groups 138
43. More d Homology d Computations d and d Applications 140
44. Homological d Algebra 144
IX. Factorization
45. Unique d Factorization d Domains 148
46. Euclidean d Domains 151
47. Gaussian d Integers d and d Multiplicative d Norms 154
X. Automorphisms and Galois Theory
d d d
48. Automorphisms d of d Fields 159
49. The d Isomorphism d Extension d Theorem 164
50. Splitting d Fields 165
51. Separable dExtensions 167
52. Totally d Inseparable d Extensions 171
53. Galois d Theory 173
54. Illustrations dof dGalois dTheory 176
55. Cyclotomic dExtensions 183
56. Insolvability d of d the d Quintic 185
APPENDIX d Matrix d Algebra 187
iv
, 0. d Sets dand dRelations 1
0. Sets and Relations
d d
√ √
1. { 3, d − 3} 2. d The d set d is d empty.
3. d {1,d−1,d2,d−2,d3,d−3,d4,d−4,d5,d−5,d6,d−6,d10,d−10,d12,d−12,d15,d−15,d20,d−20,d30,d−30,
60, d−60}
4. d {−10,d−9,d−8,d−7,d−6,d−5,d−4,d−3,d−2,d−1,d0,d1,d2,d3,d4,d5,d6,d7,d8,d9,d10,d11}
5. It dis dnot da dwell-defined dset. d (Some dmay dargue dthat dno delement dof dZ+ dis dlarge, dbecause devery
delement dexceeds donly da dfinite dnumber dof dother delements dbut dis dexceeded dby dan dinfinite dnumber dof
dother delements. dSuch dpeople dmight dclaim dthe danswer dshould dbe d∅.)
6. ∅ 7. d The d set d is d ∅ d because d 33 d= d27 d and d 43 d= d64.
8. d It d is d not d a d well-defined d set. 9. d Q
10. The d set d containing d all d numbers d that d are d (positive, d negative, d or d zero) d integer d multiples d of d 1,
d 1/2, d or d1/3.
11. {(a, d1), d (a, d2), d (a, d c), d (b, d1), d(b, d 2), d(b, d c), d(c, d 1), d (c, d2), d (c, d c)}
12. a. d It d is d a d function. d It d is dnot d one-to-one d since dthere d are d two d pairs d with dsecond d member d 4. d It d is
d not d onto
B d because d there d is d no d pair d with d second d member d 2.
b. (Same d answer d as d Part(a).)
c. It d is d not d a d function d because d there d are d two d pairs d with d first d member d 1.
d. It d is d a d function. d It d is d one-to-one. d It d is d onto d B d because d every d element d of d B d appears
d as d second dmember dof dsome dpair.
e. It dis da dfunction. d It dis dnot done-to-one dbecause dthere dare dtwo dpairs dwith dsecond dmember d6. d It
dis dnot donto dB dbecause dthere dis dno dpair dwith dsecond dmember d2.
f. It d is d not d a d function d because d there d are d two d pairs d with d first d member d 2.
13. Draw d the d line d through d P d and d x, d and d let d y d be d its d point d of d intersection d with d the d line d segment
d CD.
14. a. d φ d: d[0, d1] d→ d [0,d2] d where d φ(x) d= d2x b. d φ d: d [1,d3] d → d [5, d25] d where d φ(x) d= d5 d+ d10(x d− d1)
c. d φ d: d[a, db]→ [c, dd] d where d φ(x) d= dc d+ d d−cd(x
− a) b−a
15. Let d φ d: dS d → dR d be d defined d by d φ(x) d= dtan(π(x2d− d 1 d)).
16. a. d ∅; d cardinality d 1 b. d ∅, d{a}; d cardinality d 2 c. d ∅, d{a},d{b},d{a,db}; d cardinality d 4
d. d ∅,d{a},d{b}, d{c},d{a,db}, d{a,dc}, d{b, dc},d{a, db, dc}; d cardinality d 8
17. Conjecture: |P(A)| d= d2s d = d2|A|.
Proof dThe dnumber dof dsubsets dof da dset dA ddepends donly don dthe dcardinality dof dA, dnot don dwhat
dthe delements dof d A d actually d are. d Suppose dB d= d{1, d2, d3, d· d· d· d, ds d− d1} d and d A d= d{1, d2, d3, d d , ds}.
d Then d A d has d all
the delements dof dB dplus dthe done dadditional delement ds. d All dsubsets dof dB dare dalso dsubsets dof dA;
dthese dare dprecisely dthe dsubsets dof dA dthat ddo dnot dcontain ds, dso dthe dnumber dof dsubsets dof dA d not
dcontaining ds d is d|P(B)|. d Any dother d subset dof dA dmust d contain ds, d and d removal dof d the d s d would
dproduce d a d subset d of
