Summary College Algebra Exam Survival Guide | Quick Review, Key Formulas & Practice Solutions

ABSTRACT ALGEBRA:


A STUDY GUIDE
FOR BEGINNERS




John A. Beachy

Northern Illinois University

2006

,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/

,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

, 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