algebraguide - ABSTRACT ALGEBRA A STUDY GUIDE FOR BEGINNERS...

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ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2006
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2 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 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 March 1, 2010, at which time the original was available at: http://www.math.niu.edu/ beachy/abstract algebra/
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Contents PREFACE 5 1 INTEGERS 7 1.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 FUNCTIONS 17 2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 GROUPS 27 3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Constructing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . . . . . . 47 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 POLYNOMIALS 53 4.1 Fields; Roots of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Existence of Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Polynomials over Z , Q , R , and C . . . . . . . . . . . . . . . . . . . . . . . . 59 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3
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4 CONTENTS 5 COMMUTATIVE RINGS 63 5.1 Commutative rings; Integral Domains . . . . . . . . . . . . . . . . . . . . . 63 5.2 Ring Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Ideals and Factor Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4 Quotient Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6 FIELDS 71 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 SOLUTIONS 71 1 Integers 73 2 Functions 91 3 Groups 103 4 Polynomials 137 5 Commutative Rings 151 6 Fields 163 BIBLIOGRAPHY 166
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PREFACE 5 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 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 ab- stract 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.
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