Table Of ContentABSTRACT
ALGEBRA
Academic Press
Textbooks in Mathematics
EDITORIAL BOARD:
Ralph P. Boas, Jr., Northwestern University
Herman Gluck, Harvard University
GEORGEBACHMANand LAWRENCENARICI. FunctionalAnalysis
P.R.MASANI,R.C.PATEL,and D.J.PATIL.ElementaryCalculus
WILLIAM PERVIN. Foundations of General Topology
ALBERTl. RABENSTEIN. Introductionto OrdinaryDifferential Equations
JAMESSINGER. Elementsof Numerical Analysis
EDUARD l. STIEFEL. An Introduction to Numerical Mathematics
HOWARDG.TUCKER.An IntroductiontoProbabilityand Mathematical Statistics
CHlH-HAN SAH. Abstract Algebra
Inpreparation
LEOPOLDOV. TORALBALLA. Calculus with Analytic Geometry and Linear Algebra
DONALDW. BLACKETT. ElementaryTopology:ACombinatorial and AlgebraicApproach
I -HAN SAH UH PUDDY—CHHT
ABSTRACT
ALGEBRA
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New York and London
COPYRIGHT © 1967, BYACADEMICPRESSINC.
ALL RIGHTS RESERVED.
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to
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and
Dorothy W. Everitt
PREFACE
This 'book is intended for a one—year algebra course on the
level of advanced undergraduate or beginning graduate work in mathe-
matics.
Our viewpoint is to consider basic techniques and results in
algebra from a position that can be easily generalized. In particular,
concepts are introduced in a general setting; special cases that do not
involve too many technical complications are considered in some detail.
Exercises and excursions are inserted to give some indication of the
possible generalizationsthat have been neglected in the main body of
the text. '
The exercises form an integral part of the book. Aside from
those that have been mentioned in the preceding paragraph, they are
designed to test the reader’s understanding ofthe material presented, to
show the strength as well as the limitations of some of the results, and
to give the reader a chance to practice his skill in writing out complete
proofs. Most of the harder exercises have hints supplied, and in a few
cases the exercises have been worked out in detail. The reader is
encouraged to ignore these hints and solutions. On first reading the
various excursions may be bypassed. However, near the end of the
book a few of the earlier excursions are used in the proofs.
Resultsarereferredtoas ...I.2.3; stands forTheorem,Lemma,
Corollary, Exercise, and so on, I stands for chapter number, 2 stands
for section number, and 3 stands for the number of the particular
result. The chapter number is omitted when the reference is made in
thesamechapter; thesection numberis also omittedwhenthereference
ismadeinthesamesection. Thus Theorem1.2.3 islabeled Theorem2.3
in Chapter I and Theorem 3 in Chapter I, Section 2.
» vii
viii » PREFACE
We now describe the organization of and the prerequisites for
the text.
The reader should have some familiarity with set theory.
Aside from this, he is assumed to have had only the standard beginning
college-level mathematics courses.
The relevant concepts and results from set theory have been
summarized in Chapter 0. A reader who has not studied set theory
beforeshouldgothrough Chapter0andverifythe elementary assertions
stated there. The reader who has already studied set theory should
only glance through Chapter 0 to make certain that he is familiar with
our version of the required results. In any event, our summary is not
intended to be a systematic treatment ofset theory. This can be found
in the references listed at the end of the chapter.
In Chapter I we review the basic properties of the natural
numbers, the integers, and the rational numbers. The reader who has
studied these concepts from only an “informal” or an “intuitive” view—
point should go through Chapter I with some care. Otherwise, he need
only glance through the chapter to make sure that he is familiar with
the present development.
ChapterIIsummarizesratherquicklysomeofthebasicalgebraic
concepts aswell as afew oftheexamples ofthe objects to be considered
later. The reader should come back and review Chapter II every so
often.ThebasicthingtokeepinmindisthatChapterIIisa“dictionary.”
It formalizes the concepts of Chapter 0 and I; it also “classifies” some
of the objects considered in Chapter 0 and I.
In anyevent, Chapters0,I, andII aretheintroductorychapters.
The reader who has already studied some abstract algebra should look
only for things that do not appear to be familiar on the surface.
Chapter III is devoted tothe study of groups. We deliberately
go backward in time and look at groupsfrom the viewpoint of trans-
formation groups. Some indications are made regarding the connection
ofthis viewpoint with geometry. The studyofabstract groups is barely
hinted at. A vast amount of literature exists on the subject of group
theory. The interested reader should consult the references listed at
the end ofthe chapter for more detailed study.
Chapter IV is devoted to a preliminary study of rings. In
particular, it contains a generalization of the divisibility theory of the
rational integers; it points out the connection between rings and com-
mutative groups—namely, rings are viewed as rings of endomorphisms
of commutative groups. The latter is analogous to our viewing groups
as transformation groups.
Chapter V is fairly technical and quite important. The concepts
introduced here are used in many parts of mathematics. This chapter
PREFACE » ix
is deliberately made quite abstract; its results are used and illustrated
later.
Chapter VI presents an abbreviated treatment ofvector spaces.
It is not our intention to study vector spaces exhaustively. The basic
concern is to make certain that the reader becomes familiar with the
results of Chapter V in the special case of a vector space. Included in
this chapter are standard results on the solutions of systems of linear
equations and of systems of linear homogeneous differential equations
with constant coefficients. The latter serve to illustrate the uses of
algebra in another part of mathematics.
In Chapter VII we consider fields from the point of View of
solving polynomial equations. Rather quickly, we convert them into the
problem offorming field extensions and apply our earlier results to the
study of this problem. .
Chapter VIII deals with Galois theory. This is a continuation
of Chapter VII, and most of its results involve putting together 'the
relevant results of group theory, factorization theory, and field theory.
As “applications,” we indicate how some famous problems are solved
by the methods of field theory and Galois theory.
The purpose of the rather short Chapter IX is to provide an
almost completely algebraic proof of the fundamental theorem of
algebra. Again, it illustrates the use of some of our earlier results.
Many interesting topics of algebra have been neglected in our
treatment. Also unmentioned are many different and instructive
approaches to the topics selected. In addition to the usual excuses of
lack of time and space, I must plead my incompetence and my bias.
It is my hope that the reader will find the references at the end ofeach
chapterofsufficientusethathemaydiscovertheotherpossibleapproaches
and explanations.
It is impossible for me to thank everyone who deserves credit
for the preparation of this book. To Alfred Hales I owe a debt for his
critical reading ofthe Various drafts. To Herman Gluck and Academic
PressI owethedebtofmakingthebookpossible. Tomyformerstudents
at Harvard University and the University of Pennsylvania, I owe the
debt of their patience in sitting through incoherent lectures and in
fighting through disorganized lecture notes. To my many friends and
colleagues, especially Oscar Goldman, goes the credit forthe encourage-
ment and numerous useful critical comments I received. Last, but
certainly not least, I must thank Mrs. Madge Goldman for her
infinite patience in rectifying some of the atrocities I committed on
the English language.
C.H.S.
Philadelphia, Penn.
