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Sc Undergraduate Algebra A FIRST CO:U RS E C.W. Norman Undergraduate Algebra A First Course C. W. NORMAN Department of Mathematics Royal Holloway and Bedford New College, University of London CLARENDON PRESS OXFORD 1986 Oxford University Press, Walton Street, Oxford OX2 6DP Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Petaling Jaya Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland and associated companies in Beirut Berlin Ibadan Nicosia Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press, New York © C. W. Norman, 1986 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, 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 British Library Cataloguing in Publication Data Norman, C. W. Undergraduate algebra: a first course. 1. Algebra L Title 512 QA 155 ISBN 0-19-853249-0 ISBN 0-19-853248-2 Pbk Library of Congress Cataloging in Publication Data Norman, C. W. Undergraduate algebra. Bibliography: p. Includes index. 1. Algebra. I. Title. QA 154.2. N65 1986 512 85-31057 ISBN 0-19-853249-0 ISBN 0-19-853248-2 (pbk.) Typeset and printed by The Universities Press (Belfast) Ltd To Lucy, Tessa, and Timmy Preface Arithmetic is part of everyone's education: at an early age we learn how to add, multiply, and perform rote calculations such as `long' division; not only are these exercises a help to our understanding- we may even enjoy doing them! The present book aims to make its readers feel equally at home with the basic techniques of contem- porary algebra, especially with matrix manipulation-skills entirely analogous to those of elementary arithmetic and just as useful. Here is a first course in algebra which, though written primarily for mathematics students at college or university, will I hope be useful to aspiring engineers and scientists generally; much of the material is particularly relevant to computer science. The subject lends itself to a virtually self-contained treatment, and the amount of knowledge presupposed is indeed small; however, the reader is expected to have some familiarity with calculus, co-ordinate geometry, and trigonometry. A glance at the list of contents will convey the scope of the book. After a preliminary chapter on sets, Part One introduces the concept of a ring (which is no more than arithmetic in an abstract setting) and leads the reader gently but firmly through the basic theory, including complex numbers, integers, and polynomials. He or she is now well prepared to meet vector spaces (which generalize everyday 3-dimensional space), matrices, and groups in Part Two. Throughout, systematic techniques are given pride of place: the Euclidean algorithm for finding greatest common divisors is at the heart of Chapters 3 and 4; many of the problems which linear algebra sets out to solve are dealt with in a practical way by the row-reduction algorithm in Chapter S. Determinants, matrix diago- nalization, and quadratic and hermitian forms-topics which have wide application are thoroughly discussed, and a final chapter with a geometric flavour treats Euclidean and unitary spaces. In short, this first course is comprehensive and suitable for students with a clear commitment to algebra; for those who want a direct route to the rudiments of linear algebra, Part Two may be tackled as soon as the concepts of ring, field, and polynomial have been grasped. Starred sections are optional. However, each section ends with exercises arranged roughly in order of difficulty; these are ignored at the reader's peril! As it is all too easy to be discouraged by the abstract nature of viii Preface algebra, I have tried to keep in mind the standpoint of a student meeting the subject for the first time; as it is all too easy for a university teacher to forget what this entails, I gratefully acknow- ledge the reminders supplied by my classes at Westfield College, University of London. I must thank an ex-student, Geoffrey G. Silver, for encouraging me to `go into print'. Many improvements to the text were made by Professor B. C. Mortimer, Dr M. Walker, Colleen Farrow, John Bentin, and others who read preliminary drafts; I am grateful to them all and also to Mrs G. A. Place for her first-class typing of the manuscript. Royal Holloway and Bedford New College C. W. N. August 1985 Contents I Preliminary concepts I Sets 1 Subsets. Intersection. Union. Complement. Venn diagrams. De Morgan's laws. Cartesian product. Mappings 12 Image element. Domain, codomain. Composition. Injections, surjec- tions, bijections. Inverse mapping. Identity mapping. Equivalence relations 21 Reflexive, symmetric, and transitive laws. Equivalence classes. Parti- tions. Natural mapping. The field 712. PART I: RINGS AND FIELDS 2 Rings, fields, and complex numbers 31 Binary operations. Ring laws. Boolean ring of subsets. Elementary properties of rings. Cancellation. Integral domains. Fields. The complex field 46 Basic properties of C 46 Addition and multiplication of complex numbers. Conjugation. Argand diagram. Modulus. Argument. Geometric properties of C 55 Translations. Parallelogram construction. Triangle inequality. Radial expansion. Rotations. Argument formula. Trigonometric formulae. De Moivre's theorem. Roots of a complex number. 3 Integers 67 Order properties 67 Ordered integral domains. Well-ordering principle. Induction. Division properties 74 The division law. Divisors. Greatest common divisors. Euclidean algorithm. Primes. The fundamental theorem of arithmetic.