Abstract Algebra [corrected scan] PDF
Preview Abstract Algebra [corrected scan]
ABSTRACT 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 AAAAAAAAAAAAA New York and London COPYRIGHT © 1967, BYACADEMICPRESSINC. ALL RIGHTS RESERVED. NO PART OFTHIS BOOK MAY BE REPRODUCE‘D IN ANYFORM, 3? PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRI’ITEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS INC. 111 FifthAvenue, NewYork, NewYork10003 UnitedKingdom Editionpublishedby ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.1 LIBRARY OF Concms CATALOG CARD NUMBER: 66-29641 PRINTED IN THE UNITED STATES OF AMERICA to William L. Everitt 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.
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