- Page 1 and 2: ABSTRACT ALGEBRA: A STUDY GUIDE FOR
- Page 3 and 4: Contents PREFACE vi 1 INTEGERS 1 1.
- Page 5 and 6: CONTENTS v 5 Commutative Rings 157
- 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: 1.4. INTEGERS MODULO N 11 5. Solve
- 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
- Page 57 and 58: 4.1. FIELDS; ROOTS OF POLYNOMIALS 5
- Page 59 and 60: 4.2. FACTORS 53 23. Find the greate
- Page 61 and 62: 4.4. POLYNOMIALS OVER Z, Q, R, AND
- Page 63 and 64: 4.4. POLYNOMIALS OVER Z, Q, R, AND
- Page 65 and 66: Chapter 5 COMMUTATIVE RINGS This ch
- Page 67 and 68: 5.2. RING HOMOMORPHISMS 61 34. Let
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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
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1.4. INTEGERS MODULO N 83 Finally,
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1.4. INTEGERS MODULO N 85 41. Solve
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1.4. INTEGERS MODULO N 87 � �
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Chapter 2 Functions 2.1 Functions 2
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2.1. FUNCTIONS 91 28. Let a be a fi
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2.1. FUNCTIONS 93 First, if L has a
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2.2. EQUIVALENCE RELATIONS 95 15. F
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2.2. EQUIVALENCE RELATIONS 97 the l
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2.3. PERMUTATIONS 99 21. Prove that
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2.3. PERMUTATIONS 101 Solution: We
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Chapter 3 Groups 3.1 Definition of
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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
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3.2. SUBGROUPS 111 34. Let G be an
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3.2. SUBGROUPS 113 40. Prove that a
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3.3. CONSTRUCTING EXAMPLES 115 Comm
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3.3. CONSTRUCTING EXAMPLES 117 �
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3.4. ISOMORPHISMS 119 Answer: HK =
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3.4. ISOMORPHISMS 121 and so φ : G
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3.4. ISOMORPHISMS 123 since by assu
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3.5. CYCLIC GROUPS 125 Hint: Define
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3.5. CYCLIC GROUPS 127 32. Find all
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3.6. PERMUTATION GROUPS 129 Exercis
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3.7. HOMOMORPHISMS 131 possibilitie
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3.8. COSETS, NORMAL SUBGROUPS, AND
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3.8. COSETS, NORMAL SUBGROUPS, AND
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3.8. COSETS, NORMAL SUBGROUPS, AND
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3.8. COSETS, NORMAL SUBGROUPS, AND
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3.8. COSETS, NORMAL SUBGROUPS, AND
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Chapter 4 Polynomials 4.1 Fields; R
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4.2. FACTORS 145 This gives us the
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4.2. FACTORS 147 over Z3. Using the
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4.3. EXISTENCE OF ROOTS 149 4.3 Exi
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4.4. POLYNOMIALS OVER Z, Q, R, AND
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4.4. POLYNOMIALS OVER Z, Q, R, AND
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4.4. POLYNOMIALS OVER Z, Q, R, AND
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Chapter 5 Commutative Rings 5.1 Com
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5.1. COMMUTATIVE RINGS; INTEGRAL DO
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5.2. RING HOMOMORPHISMS 161 28. Let
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5.3. IDEALS AND FACTOR RINGS 163 so
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5.4. QUOTIENT FIELDS 165 In the gen
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5.4. QUOTIENT FIELDS 167 Z36. The l
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Chapter 6 Fields 1. Let u be a root
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BIBLIOGRAPHY 171 BIBLIOGRAPHY Allen