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iv CONTENTS 5 COMMUTATIVE RINGS 59 5.1 Commutative rings; Integral Domains . . . . . . . . . . . . . . . . . . . . . 59 5.2 Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Ideals and Factor Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4 Quotient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6 FIELDS 67 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 SOLUTIONS 68 1 Integers 69 1.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.4 Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2 Functions 89 2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3 Groups 103 3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.3 Constructing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . . . . . . 133 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4 Polynomials 143 4.1 Fields; Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.3 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.4 Polynomials over Z, Q, R, and C . . . . . . . . . . . . . . . . . . . . . . . . 151 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
CONTENTS v 5 Commutative Rings 157 5.1 Commutative rings; Integral Domains . . . . . . . . . . . . . . . . . . . . . 157 5.2 Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.3 Ideals and Factor Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4 Quotient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6 Fields 169 BIBLIOGRAPHY 171
Page 1 and 2: ABSTRACT ALGEBRA: A STUDY GUIDE FOR
Page 3: Contents PREFACE vi 1 INTEGERS 1 1.
Page 7 and 8: Chapter 1 INTEGERS Chapter 1 of the
Page 9 and 10: 1.2. PRIMES 3 (d) a = 12345 , b = 9
Page 11 and 12: 1.3. CONGRUENCES 5 1.3 Congruences
Page 13 and 14: 1.3. CONGRUENCES 7 45.†Solve the
Page 15 and 16: 1.4. INTEGERS MODULO N 9 SOLVED PRO
Page 17 and 18: 1.4. INTEGERS MODULO N 11 5. Solve
Page 19 and 20: Chapter 2 FUNCTIONS The first goal
Page 21 and 22: 2.1. FUNCTIONS 15 33. Let A be an n
Page 23 and 24: 2.2. EQUIVALENCE RELATIONS 17 SOLVE
Page 25 and 26: 2.3. PERMUTATIONS 19 30.†Let ∼
Page 27 and 28: 2.3. PERMUTATIONS 21 Chapter 2 Revi
Page 29 and 30: Chapter 3 GROUPS The study of group
Page 31 and 32: 3.1. DEFINITION OF A GROUP 25 Table
Page 33 and 34: 3.1. DEFINITION OF A GROUP 27 36.
Page 35 and 36: 3.2. SUBGROUPS 29 29. Find all cycl
Page 37 and 38: 3.2. SUBGROUPS 31 54.† In each of
Page 39 and 40: 3.3. CONSTRUCTING EXAMPLES 33 25. F
Page 41 and 42: 3.4. ISOMORPHISMS 35 (b) Show that
Page 43 and 44: 3.4. ISOMORPHISMS 37 MORE PROBLEMS:
Page 45 and 46: 3.5. CYCLIC GROUPS 39 27. In Z45 fi
Page 47 and 48: 3.6. PERMUTATION GROUPS 41 29. In t
Page 49 and 50: 3.7. HOMOMORPHISMS 43 26. For the g
Page 51 and 52: 3.8. COSETS, NORMAL SUBGROUPS, AND
Page 53 and 54: 3.8. COSETS, NORMAL SUBGROUPS, AND
Page 55 and 56:
Chapter 4 POLYNOMIALS In this chapt
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4.1. FIELDS; ROOTS OF POLYNOMIALS 5
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4.2. FACTORS 53 23. Find the greate
Page 61 and 62:
4.4. POLYNOMIALS OVER Z, Q, R, AND
Page 63 and 64:
Page 65 and 66:
Chapter 5 COMMUTATIVE RINGS This ch
Page 67 and 68:
5.2. RING HOMOMORPHISMS 61 34. Let
Page 69 and 70:
5.3. IDEALS AND FACTOR RINGS 63 5.3
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5.4. QUOTIENT FIELDS 65 Chapter 5 R
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Chapter 6 FIELDS These review probl
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Chapter 1 Integers 1.1 Divisors 25.
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1.1. DIVISORS 71 � 1 0 3553 0 1 5
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1.1. DIVISORS 73 where the remainde
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1.2. PRIMES 75 250 � ❅ 50 125
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1.2. PRIMES 77 35. Prove that gcd(2
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1.3. CONGRUENCES 79 (b) Find all so
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1.3. CONGRUENCES 81 40. Prove that
Page 89 and 90:
1.4. INTEGERS MODULO N 83 Finally,
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1.4. INTEGERS MODULO N 85 41. Solve
Page 93 and 94:
1.4. INTEGERS MODULO N 87 � �
Page 95 and 96:
Chapter 2 Functions 2.1 Functions 2
Page 97 and 98:
2.1. FUNCTIONS 91 28. Let a be a fi
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2.1. FUNCTIONS 93 First, if L has a
Page 101 and 102:
2.2. EQUIVALENCE RELATIONS 95 15. F
Page 103 and 104:
2.2. EQUIVALENCE RELATIONS 97 the l
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2.3. PERMUTATIONS 99 21. Prove that
Page 107 and 108:
2.3. PERMUTATIONS 101 Solution: We
Page 109 and 110:
Chapter 3 Groups 3.1 Definition of
Page 111 and 112:
3.1. DEFINITION OF A GROUP 105 30.
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3.1. DEFINITION OF A GROUP 107 f1(z
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3.2. SUBGROUPS 109 39. For each bin
Page 117 and 118:
3.2. SUBGROUPS 111 34. Let G be an
Page 119 and 120:
3.2. SUBGROUPS 113 40. Prove that a
Page 121 and 122:
3.3. CONSTRUCTING EXAMPLES 115 Comm
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3.3. CONSTRUCTING EXAMPLES 117 �
Page 125 and 126:
3.4. ISOMORPHISMS 119 Answer: HK =
Page 127 and 128:
3.4. ISOMORPHISMS 121 and so φ : G
Page 129 and 130:
3.4. ISOMORPHISMS 123 since by assu
Page 131 and 132:
3.5. CYCLIC GROUPS 125 Hint: Define
Page 133 and 134:
3.5. CYCLIC GROUPS 127 32. Find all
Page 135 and 136:
3.6. PERMUTATION GROUPS 129 Exercis
Page 137 and 138:
3.7. HOMOMORPHISMS 131 possibilitie
Page 139 and 140:
3.8. COSETS, NORMAL SUBGROUPS, AND
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Chapter 4 Polynomials 4.1 Fields; R
Page 151 and 152:
4.2. FACTORS 145 This gives us the
Page 153 and 154:
4.2. FACTORS 147 over Z3. Using the
Page 155 and 156:
4.3. EXISTENCE OF ROOTS 149 4.3 Exi
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Chapter 5 Commutative Rings 5.1 Com
Page 165 and 166:
5.1. COMMUTATIVE RINGS; INTEGRAL DO
Page 167 and 168:
5.2. RING HOMOMORPHISMS 161 28. Let
Page 169 and 170:
5.3. IDEALS AND FACTOR RINGS 163 so
Page 171 and 172:
5.4. QUOTIENT FIELDS 165 In the gen
Page 173 and 174:
5.4. QUOTIENT FIELDS 167 Z36. The l
Page 175 and 176:
Chapter 6 Fields 1. Let u be a root
Page 177:
BIBLIOGRAPHY 171 BIBLIOGRAPHY Allen
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