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7 are at Y 27 i. ? = a o= d 7p are ca t on iJ oc ed en aye cee) al ea Essential VOLUME THREE Student Algebra —— Abstract Algebra ASS Essential VOLUME THREE Student Algebra Abstract Algebra T. S. BLYTH & E. F. ROBERTSON University of StA ndrews London New York CHAPMAN AND HALL First published in 1986 by Chapman and Hall Ltd 11 New Fetter Lane, London EC4P 4EE Published in the USA by Chapman and Hall 29 West 35th Street, New York NY 10001 © 1986 T. S. Blyth and E. F. Robertson Printed in Great Britain by J. W. Arrowsmith Ltd., Bristol ISBN 0 412 27860 X This paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher’s prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. All rights reserved. No part of this book may be reprinted or reproduced, or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Blyth, T.S. Essential student algebra. Vol 3: Abstract algebra 1. Algebra I. Title II. Robertson, E. F. 512 QA155 ISBN 0-412-27860-X Contents Preface Chapter One : Semigroups and groups Chapter Two : Subgroups 19 Chapter Three : Quotient groups 38 Chapter Four : Group morphisms 47 Chapter Five : Isomorphism theorems 59 Chapter Six : Rings, integral domains, and fields 66 Chapter Seven : Quotient rings and ring morphisms va Chapter Eight : Polynomials 88 Chapter Nine : Fields of quotients 108 Index 119 Preface If, as it is often said, mathematics is the queen of science then algebra is surely the jewel in her crown. In the course of its vast development over the last half-century, algebra has emerged as the subject in which one can observe pure mathe- matical reasoning at its best. Its elegance is matched only by the ever-increasing number of its applications to an extraordinarily wide range of topics in areas other than ‘pure’ mathematics. Here our objective is to present, in the form of a series of five concise volumes, the fundamentals of the subject. Broadly speaking, we have covered in all the now traditional syllabus that is found in first and second year university courses, as well as some third year material. Further study would be at the level of ‘honours options’. The reasoning that lies behind this modular presentation is simple, namely to allow the student (be he a mathematician or not) to read the subject in a way that is more appropriate to the length, content, and extent, of the various courses he has to take. Although we have taken great pains to include a wide selec- tion of illustrative examples, we have not included any exer- cises. For a suitable companion collection of worked examples, we would refer the reader to our series Algebra through practice (Cambridge University Press), the first five books of which are appropriate to the material covered here. TS\B., EER. ae a e apie ciepimaremmermenninelant 7 i. _ | . aaitentas Te agen odd, al eemmedten bis ete i - - : ay ce Bta s ai. dices twa tee Iswaj. 912 views of - > aa aeaghe~ W ires led dal od) wevo aasingotevely teas yeas “she fa Mie onyoels exe suo dzidw ai danicue ait 7 7 i} mr ‘S. ‘elaw Seba. a sSaagale ei ased wep agent 4 “isa wise Be Of SoyNanilqgs a to wsdewe gtk tien dames aq’ neds adto anne aie . he aviine. be aredl osls- al jarset, ald sae ie Abresi aeighon. ou to eles renaban) ose #tihe pipeline Le mokeibeys eel Head been. id ed : te a Series hruwins agey Sourps des Ri al bewit a te ed pings ciate. ede. eaten. oad eee > BI batited wel 2 Cake YAO we ai? sige pe tee ; = ml} socohe? nasi. dla at bease is pas rs 7 jadi vere & ai-v orbis wal ba jee E ofa te ya we 5848. (Red iaD: alete S Soins; iie:s g_ ye oapesahtyng ., etiet w bAesta pa.? omtv oamd ecboaudy: ofw.s 2gacar: y - wea at besd on % 708 sallos einer & sau akteere-7 eawrAl«P rpedp ony hod ane =@ "e. ilsa hied dod wil vad oid (ore aise Sang 82,3; 427