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ii This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair ISBN 1–57766–434–4, Copyright 2005 Waveland Press, Inc. 4180 IL Route 83, Suite 101 Long Grove, Illinois 60047 (847) 634-0081 www.waveland.com c○2000, 2006, 2012 by John A. Beachy Permission is granted to copy this document in electronic form, or to print it for personal use, under these conditions: it must be reproduced in whole; it must not be modified in any way; it must not be used as part of another publication. Formatted January 6, 2012, at which time the original was available at: http://www.math.niu.edu/ ∼ beachy/abstract algebra/
Contents PREFACE vi 1 INTEGERS 1 1.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 FUNCTIONS 13 2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 GROUPS 23 3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Constructing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . . . . . . 44 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 POLYNOMIALS 49 4.1 Fields; Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Polynomials over Z, Q, R, and C . . . . . . . . . . . . . . . . . . . . . . . . 55 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 iii
Page 1: ABSTRACT ALGEBRA: A STUDY GUIDE FOR
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 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
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:
Page 65 and 66:
Chapter 5 COMMUTATIVE RINGS This ch
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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
Page 79 and 80:
1.1. DIVISORS 73 where the remainde
Page 81 and 82:
1.2. PRIMES 75 250 � ❅ 50 125
Page 83 and 84:
1.2. PRIMES 77 35. Prove that gcd(2
Page 85 and 86:
1.3. CONGRUENCES 79 (b) Find all so
Page 87 and 88:
1.3. CONGRUENCES 81 40. Prove that
Page 89 and 90:
1.4. INTEGERS MODULO N 83 Finally,
Page 91 and 92:
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
Page 99 and 100:
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
Page 115 and 116:
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
Page 123 and 124:
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:
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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