Introduction to Abstract Algebra

298 Pages·1975·13.58 MB·English

by  J. Strother Moore (Auth.)

INTRODUCTION TO
ABSTRACT ALGEBRA
J. T. MOORE
THE UNIVERSITY OF WESTERN ONTARIO
ACADEMIC PRESS New York San Francisco London
A Subsidiary of Harcourt Brace Jovanovich, Publishers
To G.R.M.
Teacher, Colleague, Friend
COPYRIGHT © 1975, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Moore, J T
Introduction to abstract algebra.
Bibliography: p.
Includes index.
1. Algebra, Abstract. I. Title.
QA162.M66 512'.02 74-17985
ISBN 0-12-505750-4
PRINTED IN THE UNITED STATES OF AMERICA
To G.R.M.
Teacher, Colleague, Friend
COPYRIGHT © 1975, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR
TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC
OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY
INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT
PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Moore, J T
Introduction to abstract algebra.
Bibliography: p.
Includes index.
1. Algebra, Abstract. I. Title.
QA162.M66 512'.02 74-17985
ISBN 0-12-505750-4
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
This book is designed as a text for a first course in abstract algebra. In
view of the difficulty usually associated with a student's initial exposure to
abstract mathematics, I have introduced abstraction very slowly and have
taken almost nothing for granted in connection with what is known as
"mathematical maturity". Many details which would be quickly glossed
over in an advanced course are examined here in great detail, and so the
text could be used quite independent of an instructor if this is desired.
Each section contains enough examples to enable the reader to gain an
insight into the methods of abstract algebra as they are applicable to the
topic at hand.
In writing this text, I have had two kinds of courses in mind :
(1) A minimal core course, suitable for an average class of one term
duration, built around the following sections :
Chap 0: Sees 0.1-0.2, and the rest of the chapter if complex numbers
are not familiar.
Chap 1 : All sections
Chap 2 : All sections
Chap 3: Sees 3.1-3.4
Chap 4: Sees 4.1-4.3 (optional, as time allows)
Chap 5: Sees 5.1-5.6 (lightly on Sees 5.2-5.4)
vii
vîii Preface
(2) A typical course for one academic year, with coverage of essentially
the whole book.
The sections listed above for the core course are interdependent, but they
do not depend on the other sections of the book. It is my best judgment that
as close as possible to three class sessions should be devoted to most of the
individual sections, with a generous amount of time spent on the problem
sets.
I follow the familiar custom of using a star (*) to indicate a more diffi­
cult problem, and most of these problems should not be attempted by
students of the core course. In Chap 5, several of the problem sets include
groupings of problems (usually near the end of the set) which, while not
all starred, do depend on material not in the core sections and should be
avoided by students in a minimal course.
The material in the text proceeds from sets, semigroups, and groups to
rings, in what I regard as the logical approach to abstract algebra. While
theorems and proofs are the essence of this subject, I have tried to avoid
the definition-theorem-proof format insofar as this seemed to be feasible.
The theorems are numbered sequentially in each chapter, the designation
denoting the section and number of each theorem. For reference purposes,
the chapter is indicated only if the theorem occurs in a different chapter.
For example, in the context of Chap 2, any reference to Theorem 2.2 of
Chap 2 would be simply to Theorem 2.2; but, if the same theorem is re­
ferred to in some other chapter, the reference would be to Theorem 2.2
(Chap 2). Diagrams are identified by numbers in sequence for each chap­
ter, but with no reference to the section in which one is located. For ex­
ample, the fifth diagram in Chap 0 is designated as Fig 0-5, but this
particular diagram happens to be in Sec 0.4. It would have been easy to
use a more elaborate and definitive system of numeration for theorems and
diagrams, but I felt that the disadvantages of this would outweigh the
advantages.
It is customary to begin a book on abstract algebra with a chapter on
"basic concepts", but it has been my experience that this is usually the
most difficult chapter in the whole book for the students. Accordingly, I
have included no such introductory cluster of abstraction here, but I in­
troduce each basic concept just as it is about to be used. For example, the
concept of a "relation' ' is not mentioned until Chap 4, where it is used for
a study of congruences. However, I have included a Chap 0, which is
designed to serve a variety of purposes: (a) It is seldom that a course gets
under way the first day of scheduled classes (and often not even the first
week!), and so it may be desirable for the instructor to have a "breather"
before beginning the course proper; (b) the first two sections contain an
Preface ix
intuitive survey of the various kinds of real numbers, with the inclusion
of several very important (but for the most part familiar) results on in­
tegers to which frequent reference is made in later sections of the book ; (c) a
simple development of complex numbers is given in the final three sections
(which may be omitted if these numbers are familiar), and we make fre­
quent use of these numbers in the sequel.
In the Answer section at the rear of the book, I have given answers or
hints to most of the odd-numbered and unstarred problems. Many of the
sections contain a True-False problem but, since I regard these problems
as best suited for class discussion, I have included no answers for them.
ACKNOWLEDGMENTS
There are many people to whom I owe a debt of gratitude at this time for
their assistance in bringing this book to publication. Most of all, perhaps,
I am indebted to the many students who have studied abstract algebra in
my classes over the years and who have contributed unwittingly to the
evolution of the manuscript material. Several anonymous reviewers pro­
vided me with very encouraging comments on the manuscript, but I would
like to identify Professor Gordon Brown of the University of Colorado as
one whose comments and suggestions were particularly appreciated and
very useful to me. However, it is to Professor Ruth Afflack of California
State College at Long Beach to whom I owe a special allotment of thanks.
She made a very careful reading of the entire manuscript, and pointed out
many ways in which it could be improved. In proofreading, I had most
welcome assistance from my good friends Professors Paul Campbell and
Bill Cannon of Presbyterian College, South Carolina, and my colleague
Professor Jay Delkin. Each of these persons made a significant contribution
in the elimination of errors of various kinds, and I express my deep appre­
ciation to them. Finally, I wish to thank all persons who have worked so
expeditiously on the various stages of the publication process and, in
particular, I take this opportunity to thank the staff of Academic Press for
their very capable assistance.
xi
0
Chapter
NUMBERS
0.1 A Naïve Survey of Real Numbers
It is only natural to begin a study of numbers with the natural numbers,
used historically for the purpose of counting the objects in various as­
semblages:
1,2,3, ·..
These numbers are ordered in a natural way, and they may be combined
by the operations of addition and multiplication under rules which are
familiar to everyone from the early grades of elementary school. While
the ancient Greek mathematicians considered the concepts of "point"
and "line segment" to be central in the mathematics of their era, it became
a guiding principle in the nineteenth century that all mathematical state­
ments are ultimately reducible to statements about the natural numbers.
In the (translated) words of Leopold Kronecker (1823-1891) :
God created the integers,
the rest is the work of man.
By themselves, however, the natural numbers were not adequate for the
needs of scientific theory and practice, and so successive extensions were
1
2 0 Numbers
made of the number concept. In each of these extensions, the idea was to
introduce numbers that would be more useful but which would obey as
many as possible of the rules of operation of natural numbers. The more of
these rules that were preserved in any extension, the more justification
there would be in referring to the new entities as "numbers".
With no little hesitation, the number zero and negative numbers were
accepted to extend the collection of numbers to the integers:
...,-3,-2,-1,0,1,2,3,·.·
The natural numbers are included here as the positive integers, while the
"negative sign" is associated with the negative integers; the number zero
(denoted 0) is regarded as neither positive nor negative.
The operations of addition, multiplication, and subtraction can be carried
out quite freely with integers, and every equation of the form
x + a = 6
with a and 6 integers has an integral solution for x.
The demands of arithmetic and greater precision in measurement led
mathematicians from integers to rational numbers or fractions of the form
m
n
where m and n (^0) are integers. With rational numbers, we are able to
add, subtract, multiply, and divide without restriction except by 0. More­
over, in addition to being able to solve equations of the form x + a = ò,
with a and 6 rational numbers we can solve any equation of the form
ax = b
1
for x rational, provided only that a -^ 0. It is assumed that the reader is
familiar with the rules of operation for both the integers and the rational
numbers.
With the emergence of rational numbers, all practical requirements for
numbers in the science of quantitative measurement were met, but there
remained strong theoretical reasons for a further extension of the number
concept. Even such simple equations as x2 = 2 have no rational number
solutions, and the parts of certain common geometric figures can not be
measured with rational numbers. The School of Pythagoras had made the
surprising—and very disturbing—discovery that it was not possible to use
a rational number (see Prob 5) to measure a diagonal of a unit square!
In order to take care of these and other deficiencies that are inherent in
the system of rational numbers, it was necessary to introduce nonrational
0.1 A Naïve Survey of Real Numbers 3
or irrational numbers. The irrational numbers include all so-called "rad­
icals," such as V2 , \/3 > yfi > as well as "transcendental·' (see Prob 14)
numbers like π and e. The totality of rational and irrational numbers
constitutes what we call the real numbers. In the context of real numbers,
we may perform quite freely the operations of addition, subtraction,
multiplication, and division except by 0; we may solve any equation of
the form x + a = ò or (if a ^ 0) ax = b for a real number x; and we may
also extract roots of nonnegative numbers, which implies that any equation
xn = a
where n is a positive integer and a is a nonnegative real number, has the
real number \/a as a solution. Throughout this book, we shall use the symbols
N, Z, Q, R to denote the natural numbers, integers, rational numbers, and real
numbers, respectively.
While an intuitive understanding of natural numbers, integers, and
rational numbers is easy, the irrational real numbers have about them
an esoteric nature that requires a penetrating study of analysis for their
understanding. We shall not attempt any such study here, but it is in­
structive to review the geometric approach to real numbers. The reader
will be familiar with the custom in analytic geometry of associating
numbers with points on a number axis, and so it should not seem un­
reasonable to suppose that there is a number of some sort associated with
each and every point of this axis. This is in fact the fundamental assump­
tion of analytic geometry: Each point of a number axis represents a unique
real number, and each real number may be represented by a unique point of a
number axis. A typical number axis is often referred to as the real linei
a partial sketch being shown in Fig 0-1.
It is likely that the reader is familiar with the representation of a real
number as an "infinite decimal", and in fact real numbers are sometimes
defined this way. Some of these decimal representations have "repeating"
digits, while others do not. For example, 7/33 = 0.212121 · · · in which 21
repeats indefinitely, and 288/55 = 4.1454545 · · · in which 45 repeats
indefinitely, while numbers such as π = 3.14159 ···, V2 = 1.4142 ···
and 0.010010001 · · · can be shown to have no repeating digits. It is the
rational numbers—and only these—whose decimal representations have
repeating digits, it being understood that a "terminating" decimal may be
3 1 1 1
~2 ~2 2 2
_J i I i I i I i I i I i I i I i I
- 4 - 3 - 2 -1 0 1 2 3 4
FIG 0-1