abstract algebra: a study guide for beginners - Northern Illinois ...

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10 CHAPTER 1. INTEGERS 46. Find the multiplicative orders of the following elements. (a) [5] and [7] in Z × 16 †(b) [5] and [7] in Z × 17 (c) [5] and [7] in Z × 18 47. Find the multiplicative order of each element of the following sets. (a) Z × 8 †(b) Z × 10 (c) Z × 11 48.†Is Z × 14 cyclic? 49.†Is Z × 16 cyclic? 50.†Is Z × 18 cyclic? 51. Find all idempotent elements in the following sets. †(a) Z14 (b) Z16 52. Find all nilpotent elements in the following sets. (a) Z14 (b) Z16 53.†Find {n ∈ Z + | ϕ(n) = 2} and {n ∈ Z + | ϕ(n) = 4}. 54. Prove that if m, n are positive integers with m | n, then ϕ(m) | ϕ(n). 55. Show that n = 7, 9, 14, 18 are the only positive integers n such that ϕ(n) = 6. 56. Use Fermat’s “little” theorem (Corollary 1.4.12) to prove that n 5 − n is divisible by 30, for all integers n. Chapter 1 Review Problems 1. Prove that if a, b, c are integers for which b | a and b | (a − c), then b | c. 2. Find gcd(7605, 5733), and express it as a linear combination of 7605 and 5733. 3. Find the prime factorizations of 1275 and 495 and use them to find gcd(1275, 495). 4. Find ϕ(1275) and ϕ(495).
1.4. INTEGERS MODULO N 11 5. Solve the congruence 24x ≡ 168 (mod 200). 6. Find the additive order of 168 modulo 200. 7. Solve the system of congruences 2x ≡ 9 (mod 15) x ≡ 8 (mod 11) . 8. Find [50] −1 501 and [51]−1 501 , if possible (in Z× 501 ). 9. List the elements of Z × 15 . For each element, find its multiplicative inverse, and find its multiplicative order. 10. Show that 3 n + 4 n − 1 is divisible by 6, for any positive integer n.
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: 1.4. INTEGERS MODULO N 9 SOLVED PRO
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
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
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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
Page 93 and 94:
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
Page 133 and 134:
3.5. CYCLIC GROUPS 127 32. Find all
Page 135 and 136:
3.6. PERMUTATION GROUPS 129 Exercis
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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:
4.4. POLYNOMIALS OVER Z, Q, R, AND
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
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5.4. QUOTIENT FIELDS 165 In the gen
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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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