John A. Beachy - ABSTRACT ALGEBRA: A STUDY GUIDE FOR...

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Unformatted text preview: 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 December 29, 2008, 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 . . . . . . . . . . . . . . . . . . . . .Commutative rings; Integral Domains ....
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John A. Beachy - ABSTRACT ALGEBRA: A STUDY GUIDE FOR...

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