Table Of ContentSc
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
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© C. W. Norman, 1986
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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.
