Abstract Algebra Manual_ Problems and Solutions - Badawi.pdf

ABSTRACT ALGEBRA MANUAL:

PROBLEMS AND SOLUTIONS

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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’.02—dc21

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

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

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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

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Printed in the United States of America

ISBN: 978-1-62808-368-2 (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