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ABSTRACT ALGEBRA MANUAL:

PROBLEMS AND SOLUTIONS

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form orby any means. The publisher has taken reasonable care in the preparation of this digital document, but makes noexpressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. Noliability is assumed for incidental or consequential damages in connection with or arising out of informationcontained herein. This digital document is sold with the clear understanding that the publisher is not engaged inrendering legal, medical or any other professional services.

ABSTRACT ALGEBRA MANUAL:

PROBLEMS AND SOLUTIONS

AYMAN BADAWI

Nova Science Publishers, Inc. Huntington, NY

Senior Editors: Susan Boriotti and Donna Dennis

Editorial Coordinator: Tatiana Shohov

Office Manager: Annette Hellinger

Graphics: Wanda Serrano

Book Production: Matthew Kozlowski, Jonathan Rose and Jennifer Vogt

Circulation: Cathy DeGregory, Ave Maria Gonzalez, Ron Hedges and Andre Tillman

Library of Congress Cataloging-in-Publication Data

Badawi, Ayman.

Abstract algebra manual: problems and solutions / Ayman Badawi

p. cm.

Includes bibliographical references and index.

1. Algebra, Abstract. I. Title.

QA162 .B33 2001

512.02dc21

2001031272

Copyright 2001 by Nova Science Publishers, Inc.

227 Main Street, Suite 100

Huntington, New York 11743

Tele. 631-424-6682 Fax 631-425-5933

E Mail Novascience@earthlink.net

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or

transmitted in any form or by any means: electronic, electrostatic, magnetic, tape,

mechanical photocopying, recording or otherwise without permission from the publishers.

The authors and publisher have taken care in preparation of this book, but make no

expressed or implied warranty of any kind and assume no responsibility for any errors or

omissions. No liability is assumed for incidental or consequential damages in connection

with or arising out of information contained in this book.

This publication is designed to provide accurate and authoritative information with regard to

the subject matter covered herein. It is sold with the clear understanding that the publisher is

not engaged in rendering legal or any other professional services. If legal or any other expert

assistance is required, the services of a competent person should be sought. FROM A

DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF

THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

Printed in the United States of America

ISBN: (eBook)

INDEX

A Abelian, 5

automorphisms, 2

B Binomial Expansion Theorem, 68

C cancelation, left-, 32

cancelation, right-, 32

centralizer, 1

closed, 3

closure, 11

common divisor, greatest, 1

common divisor, least, 1

common entries, 4

complex numbers, all nonzero, 29

contradiction, 11

cosets, 1

cosets, left, 21

cycle, even, 19

cycles, disjoint, 4, 18

cyclic group, infinite, 35

cyclic, 3

D distinct powers, 3

distinct prime numbers, 3

divisor, 3

divisor, positive, 4

domain, 62

domain, integral, 65

domain, principal ideal, 69

E element, 2

element, nilpotent, 62

element, non-identity, 8, 49

elements, distinct, 3, 13

elements, nilpotent, 55

F field, 56

field, extension, 58, 93

field, finite extension, 58

field, finite, 59

field, infinite, 85

field, some extension, 96

field, splitting, 92

finite group, 4

A. Badawi 108

finite index, 29

finite set, 2

finite, order, 3

G generators, 16

group homomorphism, 5, 34

group homomorphism, nontrivial, 37

group isomorphism, 36

group, Abelian, 13

group, cyclic, 3

group, finite Abelian, 36

group, infinite cyclic, 16

group, infinite, 17

group, non-Abelian, 24, 47

group, non-cyclic Abelian, 17

group, non-cyclic, 25

group, unit, 2

group-axioms, 11

H homomorphism, ring, 34

I ideal , maximal, 67

ideal domain, principle, 56

ideal, nonzero prime, 88

ideal, prime, 65

ideal, principal, 67

ideal, proper, 66

ideals, all prime (maximal), 88

ideals, prime (maximal), 81

idempotent, 62

identity matrix, 63

infinite order, 3, 15, 85

integer, even, 70

integer, odd, 70

integer, positive, 4

integers, 27

integral domain, 55, 73

integral domain, finite 73

integer, positive, 2, 31

integers, 2

isomorphic, 2

isomorphisms, 2

L least common multiple, 4

linear factors, 90

M matrix, square, 2

matrix-multiplication, 2

maximal ideal, 56, 57

maximal order, 53

module, multiplication, 2

module, intergers, 2

multiplicative identity, 77

multiplicity, 86

N nilpotent, 63

non-Albelian, 45

nonzero-ring homomorphism, 83

normalizer, 1

number, prime, 2, 23

numbers, all rational, 55

numbers, all real, 55

numbers, complex, 2

numbers, rational, 2

O ONTO, 80,

P permutation of a finite set, 4

permutations, 2

polynomial, 55

polynomial, irreducible, 59, 90

Abstract Algebra Manual: Problems and Solutions 109

polynomial, nonzero, 56

polynomials, monic irreducible, 57, 102

prime number, 5, 8

R rational numbers, all nonzero, 28

rational numbers, all positive, 28

real number, positive, 44

real numbers, all nonzero negative, 29

real numbers, all nonzero positive, 29

relatively prime, 3

ring homomorphism, 56, 80

ring isomorphism, 77

ring, commutative, 55, 65

ring, finite commutative, 73

ring, finite, 74

ring, non-commutative, 61

root, common, 99

root, multiple, 99

S subgroup of finite index, proper, 28

subgroup, 1

subgroup, Abelian normal, 33

subgroup, cyclic, 2

subgroup, finite, 17

subgroup, normal cyclic, 30

subgroup, normal, 1, 5, 26

subgroup, proper, 22

subgroup, smallest, 15

subgroup, Sylow p-, 5

subgroup, unique, 4, 18

subgroups, proper, 17

subring isomorphic, 79

subring, 68

Sylow p-subgroups, 52

Sylow q-subgroups, 48

U unit, 74

Z zero (root), 56

zero divisor, 74

zero divisors, 55

ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONSABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONSLibrary of Congress Cataloging-in-Publication DataContents IntroductionChapter 1: Tools and Major Results of Groups1.1 Notations1.2 Results

Chapter 2: Problems in Group Theory2.1 Elementary Properties of Groups 2.2 Subgroups2.3 Cyclic Groups2.4 Permutation Groups2.5 Cosets and Lagrange's Theorem2.6 Normal Subgroups and Factor Groups2.7 Group Homomorphisms and Direct Product 2.8 Sylow Theorems

Chapter 3: Tools and Major Results of Ring Theory3.1 Notations3.2 Major Results of Ring Theory

Chapter 4: Problems in Ring Theory4.1 Basic Properties of Rings 4.2 Ideals, Subrings, and Factor Rings4.3 Integral Domains, and Zero Divisors 4.4 Ring Homomorphisms and Ideals4.5 Polynomial Rings4.6 Factorization in Polynomial Rings4.7 Extension Fields, and Algebraic Fields 4.8 Finite Fields

BibliographyIndex