SOLUTION MANUAL First Course in Abstract Algebra A 8th Edition by John B. Fraleigh All Chapters Full Complete

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 Lagrange 34
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 94
29. Gröbner Bases for Ideals 99

VI. Extension Fields

30. Introduction to Extension Fields 103
31. Vector Spaces 107
32. Algebraic Extensions 111
33. Geometric Constructions 115
34. Finite Fields 116

VII. Advanced Group Theory

35. Isomorphism Theorems 117
36. Series of Groups 119
37. Sylow Theorems 122
38. Applications of the Sylow Theory 124
39. Free Abelian Groups 128
40. Free Groups 130
41. Group Presentations 133

VIII. Groups in Topology

42. Simplicial Complexes and Homology Groups 136
43. Computations of Homology Groups 138
44. More Homology Computations and Applications 140
45. Homological Algebra 144

IX. Factorization
46. Unique Factorization Domains 148
47. Euclidean Domains 151
48. Gaussian Integers and Multiplicative Norms 154

X. Automorphisms and Galois Theory
49. Automorphisms of Fields 159
50. The Isomorphism Extension Theorem 164
51. Splitting Fields 165
52. Separable Extensions 167
53. Totally Inseparable Extensions 171
54. Galois Theory 173
55. Illustrations of Galois Theory 176
56. Cyclotomic Extensions 183
57. Insolvability of the Quintic 185

APPENDIX Matrix Algebra 187

, iv
0. Sets and Relations 1

1. Sets and Relations
√ √
1. { 3, − 3} 2. The set is empty.
3. {1, −1, 2, −2, 3, −3, 4, −4, 5, −5, 6, −6, 10, −10, 12, −12, 15, −15, 20, −20, 30, −30,
60, −60}

4. {−10, −9, −8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

5. It is not a a a well-defined a set. a(Some a may a argue a that a no a element a of a Z+ a is a large, a because
a every a element a exceeds aonly aa afinite anumber aof aother aelements abut ais aexceeded aby aan ainfinite
anumber aof aother aelements. a Such apeople amight aclaim athe aanswer ashould abe a∅.)

6. ∅ 7. a The a set a is a ∅ a because a 33 a = a27 a and a 43 a = a64.

8. a It a is a not a a a well-defined a set. 9. a Q

10. a The a set a containing a all a numbers a that a are a (positive, a negative, a or a zero) a integer a multiples
a of a 1, a 1/2, a or a 1/3.

11. a a a{(a, a1), a(a, a2), a(a, ac), a(b, a1), a(b, a2), a(b, ac), a(c, a1), a(c, a2), a(c, ac)}

12. a. a It ais aa afunction. a It ais a not aone-to-one asince athere aare atwo a pairs awith asecond amember a4. a It
ais anot aonto
B a because athere ais ano apair awith asecond amember a2.
b. (Same a answer a as a Part(a).)
c. It ais a not a a a function a because a there aare a two a pairs a with a first a member a 1.
d. It a is a a a function. a aIt a is a one-to-one. a aIt a is a onto a B a because a every a element a of a B
a appears a as a second a member aof asome apair.

e. It ais aa afunction. a It ais anot aone-to-one abecause athere aare atwo apairs awith asecond amember
a6. a It ais anot a onto aB abecause athere ais ano apair awith asecond amember a2.

f. It ais a not a a a function a because a there aare a two a pairs a with a first a member a 2.

13. Draw a the a line a through a P a and a x, a and a let a y a be a its a point a of a intersection a with a the a line
a segment a CD.

14. a a a. a φ a: a [0, a1] a→ a [0, a2] a where a φ(x) a= a2x b. a φ a: a [1, a3] a → a [5, a25] a where a φ(x) a= a5 a+ a10(x a−
a1)
c. a φ a : a [a, ab] a→ a [c, ad] a where a φ(x) a = a c a+ a d−c a(x a − a aa)
b a−aa

15. Let a φ a: aS a → aR a be a defined a by a φ(x) a= 1
)2).
atan(π(x a−