Invitation to Algebra: A Resource Compendium for Teachers, Advanced Undergraduate Students and Graduate Students in Mathematics

452 Pages·2020·6.581 MB·English

by  Vlastimil Dlab, Kenneth S Williams

Invitation
to
Algebra
A Resource Compendium for Teachers,
Advanced Undergraduate Students and Graduate Students in
Mathematics
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Invitation
to
Algebra
A Resource Compendium for Teachers,
Advanced Undergraduate Students and Graduate Students in
Mathematics
Vlastimil Dlab • Kenneth S Williams
Carleton University, Canada
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
Published by
World Scientific Publishing Co. Pte. Ltd.
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UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Control Number: 2020937550
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
INVITATION TO ALGEBRA
A Resource Compendium for Graduate Students and Advanced Undergraduate Students in Mathematics
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
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For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
ISBN 978-981-121-997-9 (hardcover)
ISBN 978-981-121-998-6 (ebook for institutions)
ISBN 978-981-121-999-3 (ebook for individuals)
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/11818#t=suppl
Desk Editor: Soh Jing Wen
Printed in Singapore
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To our wives Helena and Carole
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Preface
Up until towards the end of the nineteenth century, algebra loosely comprised the study
of a variety of concrete systems such as vectors, polynomials, quaternions, and matrices.
However,inthedecadeorsobeforetheendofthatcentury,mathematiciansrecognizedthat
byabstractingthecommoncontentofsuchsystemsthesedifferentrealmsofalgebracouldbe
brought efficiently together. The foremost abstraction was to treat them as sets of elements
subject to an operation specified by certain abstract properties. Thus, for example, the set
of equivalence classes of binary quadratic forms ax2+bxy+cz2 treated by Gauss and the
class of transformations
az+b
z
→ cz+d
in the complex plane studied by M¨obius were brought efficiently together under that part of
abstract algebra known as the theory of groups.
Successful students know that learning an abstract subject like mathematics requires an
appreciation of concrete examples which illustrate and motivate the underlying concepts.
The learning process should imitate the way in which children learn new notions. A child
does not learn what a table is through an abstract description of a certain construction but
rather, after being shown many different tables, learns to single out tables from collections
of furniture. In the same way an abstract notion should be built on well-chosen concrete
examples. If a student is provided with a variety of concrete groups, then the definition of a
group will arise naturally. The pedagogical principle which we shall follow is well expressed
in a Chinese proverb attributed to Confucius (551 - 479 BCE)
I hear and I forget. I see and I remember. I do and I understand.
Theusualdesireofmathematicianstopresenttheirsubjectstrictlylogicallyfromthebegin-
ning often obscures the learning process, kills the joy of discovery, and does not contribute
to a lasting and deeper understanding of the substance of new knowledge.
Inwritingabooksuchasthisitisvitaltokeepinmindwhoitisintendedfor. Ourinten-
tionisthatthebookbeausefulresourceforgraduatestudentsandadvancedundergraduates
in mathematics. We hope too that it will be helpful to mathematics teachers at all levels
as a source book and this is why so much emphasis has been placed on the development of
the integers and rational numbers in the beginning chapters. Hopefully too a general reader
of mathematics will find topics of interest. The material is presented in a challenging way
so that the reader will be led to a deeper understanding of the subject. A familiarity with
elementary linear algebra is assumed.
The origins of the book lie in a set of lecture notes prepared by the first author for
vii
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viii PREFACE
students in an introductory algebra course given at Carleton University in Ottawa in the
academic year 2005-2006. These notes were subsequently revised for courses given by the
first author at Charles University in Prague during the years 2008 to 2012. More recently
they have been revised again in conjunction with the second author taking into account the
many valuable suggestions received from colleagues. They have also benefited from notes
given in a variety of algebra courses by both authors. Tested in this way our book should
provide useful guidance and reference to graduate students and advanced undergraduates in
mathematics. No book lives in isolation from earlier publications and we have learned and
benefited from textbooks by other authors. What we have learned from these books has
helped improve our formulation of the theory and to hopefully avoid certain pitfalls in the
exposition.
This book emphasizes the relationship between algebra and other parts of mathematics,
especiallygeometrybutalsocombinatoricsandnumbertheory,whichareeithermissingfrom
present algebra textbooks or are mentioned only marginally. It contains varied and diverse
material that every mathematics graduate student and advanced undergraduate should be
acquainted with. Its aim is to present beautiful, inspiring, and interesting topics, and to
present them in such a way as to arouse interest in the reader to study and expand her/his
horizons. This book roughly follows a prescribed scheme. Each chapter begins with some
motivationandexamples,followedbythecentralcorematerialofthechapterwithassociated
exercises, possibly followed in turn by more advanced topics and problems. The reader
shouldunderstandthattheexercisesformanimportantpartofthebook. Thereadershould
work through them to develop a proper understanding of the basic theoretical material. A
summary of definitions of abstract concepts that have been introduced as examples earlier,
is given in Chapter VI. There are also sections which deal with special topics to give a
taste of contemporary algebra, some of which are left to Chapter X (Appendix). One of
our objectives is to remove rigid boundaries between parts of mathematics and to stress
the unity of mathematics. In particular, we have tried to show the very close relationship
between algebra and geometry.
Finally, we make a few comments on the presentation. Exercises are integrated with the
development of the theory in order to clarify the new concepts. Generally routine exercises
are avoided. Solutions to some of the exercises appear at the end of the book. Sections
markedby?serveprimarilytoextendourunderstandingoftherelationshipwithotherparts
ofmathematics. Materialmarkedinthiswaywillnotbereferredtolaterandcanbeskipped
overatfirstreading. Anumberofnotionsareintroducedinformallyinthefirstfewchapters;
the reader will find a brief more formal summary of these concepts in Chapter VI.
The presentation of the book received substantial assistance from Piroska Lakatos and
Nadiya Gubareni. The book would not appear in the present form without their support
and encouragement. The authors would like to thank them for many valuable remarks and
corrections, and for their patience with frequent changes and amendments. We also wish to
thankJohnDixonforproofreadingpartsofthemanuscript;hiscommentsandadviceleadto
anumberofimprovements. WealsothankProfessorMasahisaSatoofYamanashiUniversity,
Japanforhisverycarefulreadingofourmanuscriptwhichenabledustocorrectmanytypos
and errors. The first author is happy to acknowledge assistance of his son Daniel in the
final form of presentation. Both authors thank the staff of World Scientific, particularly
RochelleKronzek(USA)andSohJingWen(Singapore),fortheirgreatcareandcooperation
in bringing our book into print.
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PREFACE ix
Learningnewconceptsisnotasimpleprocess. Adeeperunderstandingofnewconceptsis
aprocessconsistinginattainingahigherlevelofmaturity. Thecoursetofollowiseloquently
expressed by G. Chrystal in his classic textbook Algebra. An Elementary Text-Book for the
Higher Classes of Secondary schools and for Colleges (Adam and Charles Black, Edinburgh,
1889):
Every mathematical book that is worth reading
must be read ‘‘backwards and forwards’’ if I may
use the expression. I would modify Lagrange’s
advice a little and say, ‘‘Go on, but often return
to strengthen your faith.’’ When you come on a hard
and dreary passage, pass it over; and come back to
it after you have seen its importance or found the
need for it further on.