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vi PREFACE PREFACE The changes in the third edition of our book Abstract Algebra have dictated a few minor changes in the study guide. In addition to these, I have added a few new problems and done some editing of the solutions of old ones. I hope this edition will continue to be a help to students who are beginning their study of abstract algebra. DeKalb, Illinois John A. Beachy October 2006 This study guide now contains well over 500 problems, and more than half have detailed solutions, while about a quarter have either an answer or a hint. The ideal way to use the study guide is to work on a solved problem, and if you get stuck, just peek at the solution long enough to get started again. But if the number of problems looks daunting, and you already have a lot of other homework problems, I hope that you can also learn a lot by just reading some solutions. In any case, and however you decide to use the study guide, I hope that you will find the subject as interesting and as challenging as I have. DeKalb, Illinois John A. Beachy January 2012 PREFACE TO THE 2ND ED I first taught an abstract algebra course in 1968, using Herstein’s Topics in Algebra. It’s hard to improve on his book; the subject may have become broader, with applications to computing and other areas, but Topics contains the core of any course. Unfortunately, the subject hasn’t become any easier, so students meeting abstract algebra still struggle to learn the new concepts, especially since they are probably still learning how to write their own proofs. This “study guide” is intended to help students who are beginning to learn about abstract algebra. Instead of just expanding the material that is already written down in our textbook, I decided to try to teach by example, by writing out solutions to problems. I’ve tried to choose problems that would be instructive, and in quite a few cases I’ve included comments to help the reader see what is really going on. Of course, this study guide isn’t a substitute for a good teacher, or for the chance to work together with other students on some hard problems. Finally, I would like to gratefully acknowledge the support of Northern Illinois University while writing this study guide. As part of the recognition as a “Presidential Teaching Professor,” I was given leave in Spring 2000 to work on projects related to teaching. DeKalb, Illinois John A. Beachy October 2000
Chapter 1 INTEGERS Chapter 1 of the text introduces the basic ideas from number theory that are a prerequisite to studying abstract algebra. Many of the concepts introduced there can be abstracted to much more general situations. For example, in Chapter 3 of the text you will be introduced to the concept of a group. One of the first broad classes of groups that you will meet depends on the definition of a cyclic group, one that is obtained by considering all powers of a particular element. The examples in Section 1.4, constructed using congruence classes of integers, actually tell you everything you will need to know about cyclic groups. In fact, although Chapter 1 is very concrete, it is a significant step forward in the study of abstract algebra. 1.1 Divisors Before working through the solved problems for this section, you need to make sure that you are familiar with all of the definitions and theorems in the section. In many cases, the proofs of the theorems contain important techniques that you need to copy in solving the exercises in the text. Here are several useful approaches you should know how to use: —When working on questions involving divisibility you may find it useful to go back to Definition 1.1.1. If you expand the expression b|a by writing “a = bq for some q ∈ Z”, then you have an equation to work with. This equation involves ordinary integers, and so you can use all of the things you already know (from high school algebra) about working with equations. —To show that b|a, try to write down an expression for a and expand, simplify, or substitute for terms in the expression until you can show how to factor out b. —Another approach to proving that b|a is to use the division algorithm (see Theorem 1.1.3) to write a = bq + r, where 0 ≤ r < b. Then to prove that b|a you only need to find some way to check that r = 0. —Theorem 1.1.6 states that any two nonzero integers a and b have a greatest common divisor, which can be expressed as the smallest positive linear combination of a and b. An integer is a linear combination of a and b if and only if it is a multiple of their greatest 1
Page 1 and 2: ABSTRACT ALGEBRA: A STUDY GUIDE FOR
Page 3 and 4: Contents PREFACE vi 1 INTEGERS 1 1.
Page 5: CONTENTS v 5 Commutative Rings 157
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
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4.4. POLYNOMIALS OVER Z, Q, R, AND
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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
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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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Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
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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Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
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
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