John A. Beachy2 - ABSTRACT ALGEBRA A STUDY GUIDE FOR...

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ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2000
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ii This is a supplement to Abstract Algebra , Second Edition by John A. Beachy and William D. Blair ISBN 0–88133–866–4, Copyright 1996 Waveland Press, Inc. P.O. Box 400 Prospect Heights, Illinois 60070 847 / 634-0081 www.waveland.com c John A. Beachy 2000 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 February 8, 2002, at which time the original was available at: http://www.math.niu.edu/ beachy/abstract algebra/
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Contents PREFACE v 1 INTEGERS 1 1.1 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Integers Modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 FUNCTIONS 7 2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 GROUPS 13 3.1 Definition of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Constructing Examples . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5 Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.6 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.7 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.8 Cosets, Normal Subgroups, and Factor Groups . . . . . . . . . . . . 24 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 POLYNOMIALS 27 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 COMMUTATIVE RINGS 29 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 iii
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iv CONTENTS 6 FIELDS 33 Review problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 SOLUTIONS 33 1 Integers 35 2 Functions 49 3 Groups 57 4 Polynomials 87 5 Commutative Rings 93 6 Fields 101 BIBLIOGRAPHY 104 INDEX 105
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PREFACE v PREFACE 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 “Presi- dential Teaching Professor,” I was given leave in Spring 2000 to work on projects related to teaching. DeKalb, Illinois John A. Beachy October 2000
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vi PREFACE
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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
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