Beachy John A. - Abstract Algebra - A Study Guide for Beginners.pdf

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ABSTRACT ALGEBRA:
A STUDY GUIDE
FOR BEGINNERS
John A. Beachy
Northern Illinois University
2000
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 modied 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/
100501158.001.png
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 Denition 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
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
PREFACE
v
PREFACE
I rst 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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