Undergraduate Thxtsin Mathematics Springer Science+Business Media, LLC Editors s.Axler F. W.Gehring P.R.Halmos Undergraduate Texts in Mathematics Anglin: Mathematics: A Concise History and Philosophy. Readings in Mathematics. Anglin/Lambek: The Heritage of Thales. Readings in Mathematics. Apostol:Introduction toAnalyticNumber Theory. Second edition. Armstrong: Basic'Ibpology. Armstrong: Groups and Symmetry. Axler: Linear Algebra Done Right. Bak/Newman:Complex Analysis.Second edition. Banchoff/Wermer: Linear Algebra Through Geometry. Second edition. Berberian: A First Course in Real Analysis. Bremaud: AnIntroduction toProbabilistic Modeling. Bressoud: Factorization and Primality Testing. Bressoud: Second Year Calculus. Readings in Mathematics. Brickman:MathematicalIntroductionto Linear Programming and Game Theory. Browder:MathematicalAnalysis:An Introduction. Cederberg: A Course in Modern Geometries. Childs:AConcreteIntroductiontoHigher Algebra.Second edition. Chung: Elementary Probability Theory with Stochastic Processes. Third edition. Cox/Little/O'Shea:Ideals,Varieties,and Algorithms. Second edition. Croom: Basic Concepts of Algebraic 'Ibpology. Curtis:LinearAlgebra:AnIntroductory Approach.Fourth edition. Devlin:TheJoyof Sets:Fundamentals of Contemporary Set Theory. Second edition. Dixmier: General 'Ibpology. Driver: Why Math? Ebbinghaus/Flum/Thomas: Mathematical Logic.Second edition. Edgar:Measure,'Ibpology,andFractal Geometry.Elaydi:Introductionto Difference Equations. Exner:AnAccompanimenttoHigher Mathematics. Fischer: Intermediate RealAnalysis. Flanigan/Kazdan: Calculus '!Wo:Linear and Nonlinear Functions. Second edition. Fleming:Functionsof SeveralVariables. Second edition. Foulds:CombinatorialOptimizationfor Undergraduates. Foulds:OptimizationTechniques:An Introduction. Franklin:Methodsof Mathematical Economics. Hairer/Wanner: Analysis by Its History. Readings in Mathematics. Halmos:Finite-Dimensional Vector Spaces. Second edition. Halmos: Naive Set Theory. Hiimmerlin/Hoffmann:Numerical Mathematics. Readings in Mathematics. Hilton/Holton/Pedersen:Mathematical Reflections:In a Room with Many Mirrors. looss/Joseph:Elementary Stabilityand Bifurcation Theory. Second edition. Isaac: The pleasures of Probability. Readings in Mathematics. James: Topological and Uniform Spaces. Jiinich: Linear Algebra. Jiinich: 'Ibpology. Kemeny/Snell:Finite Markov Chains. Kinsey: 'Ibpology of Surfaces. Klambauer: Aspects of Calculus. Lang: A First Course in Calculus.Fifth edition. Lang:Calculusof SeveralVariables.Third edition. Lang:Introduction toLinear Algebra.Second edition. Lang:Linear Algebra.Third edition. Lang: Undergraduate Algebra. Second edition. Lang:Undergraduate Analysis. Lax/Burstein/Lax:Calculuswith Applications and Computing. Volume1. LeCuyer:College Mathematics with APL. Lidl/Pilz:Applied Abstract Algebra. Macki-Strauss:IntroductiontoOptimal Control Theory. Malitz:Introduction toMathematical Logic. (continued after index) B.A.Sethuraman Rings,Fields,and Vector Spaces An Introduction to Abstract Algebra via Geometric Constructibility Springer B.A.Sethuraman Department of Mathematics California State University Northridge Northridge,CA91330 USA Editorial Board S.Axler Department of Mathematics Michigan State University East Lansing,MI48824 USA F.w.Gehring Department of Mathematics University of Michigan Ann Arbor,MI48109 USA Mathematics Subject Classification (1991):12-01,l3-01 Library of Congress Cataloging-in-PublicationData Sethuraman, B.A., P.R.Halmos Department of Mathematics Santa Clara University Santa Clara,CA95053 USA Rings,fields,and vector spaces : an introduction to abstract algebra via geometric constructibility /B.A.Sethuraman. p.cm.- (Undergraduate texts in mathematics) Includes bibliographical references (p.185-186) and index. ISBN 978-1-4757-2702-9 ISBN 978-1-4757-2700-5 (eBook)DOI 10-1007/978 -14757-2700-51.Algebra,Abstract.1.Title.II.Series. QA162.S441996 512'.02-DC2096-32220 Printed on acid-free paper. 1997 Springer Science+Business Media New Yark Originally published by Springer-Verlag New York,Inc.in 1997 Softcover reprint ofthe hardcoverIst edition 1997 Al!rightsreserved.This workmaynot betranslatedor copiedinwholeor inpart without the written permission ofthe publisher Springer Science+Business Media,LLC, except for brief excerpts in connection with reviews or scholarly analysis.Use in connection with any formof information storage and retrieval,electronicadaptation,computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptivenames, trade names,trademarks,etc.,inthispublication,even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the ltade Marks and Merchandise Marks Act,may accordingly be used freely by anyone. Producion managed by Lesley poliner;manufacturing supervised by Jacqui Ashri. Camera-ready copy prepared by the author. 987654321 ISBN 978-1-4757-2702-9SPIN10545117 This book is dedicated toPrabha, who gave me so much, and taught me so much more. Preface This bookisan attempt tocommunicatetoundergraduatemath-ematicsmajorsmyenjoymentof abstractalgebra.It grewoutof acourseofferedatCaliforniaStateUniversity,Northridge,inour teacher preparation program, titled Foundations of Algebra, that was intended to provide an advanced perspective on high-school mathe-matics. When I first prepared to teach this course,I needed to select aset of topicstocover.Thematerial that Iselected wouldclearly have to have some bearing on school-level mathematics, but at the same time would have to be substantial enough for a university-level course.It would have to be something that would give the students aperspectiveintoabstractmathematics,afeelfortheconceptual eleganceand grandsimplifications broughtabout by thestudy of structure.It would haveto beof akind that wouldenable thestu-dents todevelop their creative powers and their reasoning abilities. And of course,it would all have to fitinto a sixteen-week semester. Thechoicetomewasclear:weshouldstudyconstructibility. Themathematics that leads totheproof of thenontrisectibilityof an arbitrary angle is beautiful,it is accessible,and it is worthwhile. Every teacher of mathematics would profit from knowing it. Nowthat I had decided on the topic,I had todecideon how to develop it. All the students in my course had taken an earlier course .. Vll viiiPreface on sets and functions, but many had not progressed any further into abstract mathematics. What I needed to do, therefore, was to develop enough algebra to lead to the proofs of the nonconstructibility results without getting bogged down in technicalities. But since this course was going to be the only algebra course that several of my students would take,the material I developed needed be rich enough so that everybody would get a good sense of what the subject was all about. Given this goal for the course,I set out tofind a textbook. There certainly isawealthof rather excellent textbooksonintroductory abstract algebra, but they seem to be designed with adifferent pur-pose in mind: todevelop technical mastery of the subject. As such, they delve into the details of the subject, rather than focusing on an overview.For me to have culled from existing textbooks the mathe-matics that I wanted to cover in my course at the level that I wanted tocover it would have been a horrendous task.Idecided instead to write my own book. Thisbookhasbeenwritteninaconversationalstyle,astyle thatmirrorsmyownapproachtoteaching.Thefocusison expo-sition, on conveying mathematical intuition to an audience that will havecareersinmathematics,but forthemostpart willnot goon togetaph.D.in mathematics.Familiarity withthematerialisde-veloped by exposing the students to lots of examples;sacrificing,if necessary, the desire to prove lots of theorems. The text is peppered liberally with questions, designed to encourage the students to learn the subject by thinking through the material themselves. This is par-ticularly trueof thesections that deal withexamples:many of the questionsasked withintheseexamplescould serve just aswellas formalexercises. The book begins with an essay on how to learn mathematics,a topicthatIfeeliswellworthspending sometimeoninintroduc-tory courses in abstract mathematics.This is followedin Chapter 1 by astudy of divisibility in the integers,and in Chapter 2 by a gen-eral introduction to rings and fields.Vector spaces are introduced in Chapter 3 so as to make it possible to measure degrees of field exten-sions.Chapter 4 discusses how degrees offield extensions behave in towers,and studies theconcept of an element in afieldextension beingalgebraicoverthe basefield.Thenotionof irreducibilityof polynomialsandthephenomenon of uniquefactorizationinpoly-PrefaceIX -------------------------------------------------nomialringsarestudied inChapter 5,and immediately after,the relation between thedegreeof thefieldgenerated by anelement and the degreeof itsminimal polynomial isderived in Chapter 6. Finally,theseresults areput together in Chapter 7 toarriveat the algebraic criterion for the constructibility of a real number. Although the treatment of divisibility in the integers in Chapter 1 is somewhat standard, the remaining chapters are a little less tra-ditional. As described above, the goal is to get to constructibility with minimum fuss, but without sacrificing richness. For instance, in the chapter on rings and fields(Chapter 2),Idiscussnumerous exam-ples of such objects and Idiscuss subrings generated by elements, but I avoid talking of ideals since I do not have a formal need for this concept.(On theother hand,in Chapter6,theset denoted IF,ais after all just an ideal of F[x],soI take advantage of this opportunity to givethem asequence of exercises concerning ideals in general.) Similarly,by workingwithinafixedfieldextensionKIF,Inever deal with the abstract construction of field extensions generated by rootsof polynomials,and instead focuson fieldextensionsgener-ated by specific elements of the overfield K. (In fact,the issue of an element aof Kbeing algebraic or transcendental over the subfield Fismotivated by the question of when thefieldgenerated over F by a equals the ring generated over Fby a.) Along the way,I have tried todevelop topics that a high-school mathematicsteachermightfindinteresting.Forinstance,inthe chapter on divisibility in the integers (Chapter 1), I include problems that show the validity of various divisibility tests (such as tests of di-visibility by 3 and divisibility by 11).In the same chapter, I include a discussion on the Euclidean algorithm for finding the greatest com-mon divisor of two integers, following an exercise where one has to show that if a=bq+r,then gcd(a, b)=gcd(b, r).In Chapter 2, I include aproblem that shows in aseries of steps that unique fac-torization failsin a very natural "number system," an exercise that I hope will help the reader appreciate the significance ofunique prime factorizationin theintegers.In Chapter 4,I introducetheconcept of algebraic and transcendental numbers, and then discuss the tran-scendentality of certainspecificnumbers.Iincludeaproblemon showing that e isirrational,and in the notes tothis same chapter,I explain why there are "so many more" transcendental numbers than X Preface ----------------------------------------------------there are algebraic numbers. In the chapter on polynomials (Chapter 5),Iincludediscussionson theFundamental Theoremof Algebra and on rootsof polynomials.Intheexercisestothischapter,Iin-clude problems that show why complex nonreal roots of polynomial equations with real coefficients come in pairs,why polynomials of odddegreewithcoefficientsin therealshavearootinthereals, how the coefficients of apolynomial are related to its roots,how to obtain all n nth roots of acomplex number given anyone root,why the Lagrange interpolation polynomial is unique, and why synthetic division works the way it does.As well,in the notes to this chapter, I include discussions on the general problem of solving polynomial equations by radicals,and I outline Cardano's solution of the cubic. A fewwords about the notes and the exercises.First thenotes-they are meant to be informal.They started off as a vehicle by which Icould try toprovide glimpses intomore advanced areasof math-ematicsaswellasavehicle by whichIcouldcommunicatesome of my own excitement about these areas.Very soon,however,they developed into aconvenient receptacle for all sorts of remarks that I wanted to make,remarks that I felt should not be made in the text either forfear of derailing thecourse or forfearof giving away too much too soon. Assuch, one will find in the notes, besides pointers to theories beyond the scope of this book, comments on certain def-initions,noteson certainproofs,remarkson specificexamples,as well as occassional hints tosome of the questions I ask in the text. Nowfortheexercises.Ihavealreadynotedabovethatmany of thequestionsIask withinsomeof theexamplesIdevelopcan serve as formalassignments.(Tbtake an instance at random,in Ex-amples3.11inChapter 3,thequestionsaskedinExamples3.11.1, 3.11.2,3.11.7,and 3.11.8can all be assignedformallyasproblems.) For the most part, such problems assigned from the various examples Idevelopwillbeof aroutinenature,designed tobuildfamiliarity with thematerial.Asfortheexercises at theend of each chapter,I have attempted tomake many of them of some substance, exercises fromwhichstudents willhopefully learnsomesignificantmathe-matics.Of course,thereis always adanger with such aphilosophy in a beginning class,the danger that this approach may be too diffi-cult forthestudents.Tbmitigatethissomewhat,Ihave brokenup many exercises into several digestible chunks,and I have provided PrefaceXl copious hints.(Isuspect that "guideddiscovery"isthesurest way tolearn mathematics,even when thestudents areseeing abstract mathematics for the first time.) I believethat this book would servevery wellasagentleone-semester introduction to abstract algebra, after the students have had a basic introduction tosets and functionssuch as the introduction one gets in atypical undergraduate "discrete mathematics" course. Bydwelling just a bit on thechapter on polynomials,working out all the exercises therein,acourse taught out of this book would ad-ditionallyprovidesomeinsight into what used togoby thename of "Theory of Equations!' Such insight would be particularly useful to anybody teaching high-school mathematics. This book could also beused byanybodylearningalgebraontheirown;thefocuson exposition is designed tofacilitateself-study. Several colleagues have been of enormous help to me during the writingof this book.PatMorandi wasvery encouraging about the worth of the project, and being a fellow-author,listened sympathet-icallytomytravails.Besides,heputupwithendlessdiscussions on things like thedefinition of the greatest common divisor,when he would much rather be having endless discussions on things like the definition of etale cohomology. Also,he bravely volunteered to teach out of this book while on sabbatical at Indiana University. Jerry Gold,another brave soul,agreed to teach out of this book at Califor-nia State University, Northridge, and provided several very valuable suggestions for improvement based on his experience. Ann Watkins read through portions of the book and made numerous comments thatwereextremelyperceptive.SheandReinhardLaubenbacher both introduced me to themechanics and the culture behind book publishing. Manyfriendsand familymembers helped aswell.My brother Ananthandsister-in-lawVidyareadthroughthepreliminary ver-sion of the first few chapters and provided several suggestions. So did my college buddies KPand Shanks, as well as their spouses Malathi andBrinda.MygoodfriendsHenriand Stan readthroughtheIn-troduction,and insisted that I retain thereference tothe equitable distribution of pastry. Andof course,someof themosthelpfulindividualswerethe studentsintheFoundationsof AlgebracourseatCaliforniaState XlIPreface University,Northridge.Theyareunfortunatelytoonumerousto mention by name, but all these students should know that they were a joy to teach, and that it was they who were the fundamental reason why I wrotethis book.It givesmeparticular pleasure tonotethat most of them are now established teachers themselves. TheNationalScienceFoundation,aswellastheOfficeof Re-searchandSponsoredProjectsandtheCollegeofScienceand MathematicsatCaliforniaStateUniversity,Northridge,provided generous support while this book was being written. Thall these people and organizations,I am very grateful. B.A. Sethuraman August 1996 Contents Preface Introduction 1Divisibility in the Integers 2Rings and Fields 3Vector Spaces 4Field Extensions 5Polynomials 6The Field Generated by an Element 7Straightedge and Compass Constructions References Index vii 1 9 29 63 97 119 155 169 185 187 Xlll Introduction Most of us are introduced to number systems very early in our lives, when wefirst learn how to count. Webegin by learning to add,us-ing thenumbersI,2,3,.... Then,welearn 'about theprocessof "taking away:'that is,theprocessof subtracting onenumber from another, and as a consequence, we learn about the number 0 as well as the numbers -I, -2, -3, .... Thus acquainted with the integers, welearnmultiplicationasashortcuttoaddition-addingfour3s togetheristhesameasmultiplying3 by 4.Afterseveralyearsof multiplication tables, we are taught fractions (usually in the context of dividingtwopiesamong fivepeople),and asaresult,welearn about the rational numbers. Ourintroductiontotherealnumberscomestousfromtwo sources.Ontheonehand,welearnaboutsquarerootsandcube roots,and arethus introduced tonumbers like,J2 and 4'3.On the other hand,welearnfromgeometry theconceptof thelengthof aline.Wearetoldthatrealnumbersarethenumbersthat corre-spond tolengths of line segments,that is,topoints on the number line. Finally,itispointed out tous that althoughit seems as if only positive numbers can have square roots,this is in fact not true. The number i is introduced to us as the square root of -I, and we are told 1 B. A. Sethuraman, Rings, Fields, and Vector Spaces Springer Science+Business Media New York 19972Introduction that from this,we get a new system of numbers (the complex num-bers) by considering all expressions of the form a+ ib,where a and b are realnumbers.Welearn toadd,subtract,multiply,and divide with these numbers. Welearn about the geometric interpretation of complex numbers and about deMoivre's theorem,and weare told that at least in theory,wecan solveany polynomial equation over the complex numbers. Needless to say,in spiteof our developing great mechanical fa-cilitywiththecomplexnumbers,they remainamystery tomost of us.Somehow,it stilldoesnot seem correctthat anegativenum-ber could have a square root!Merely defining ito be the square root of -1 seems rather contrived,yet theseabstract expressions of the form a+ ib indeed seem to give us a set of numbers with wonderful properties. The complex numbers are not the only numbers that we wonder about.Atsomepoint,weallwonder abouteven themost basic of numbers,thepositiveintegers.They haveendlessfascinationfor us,and there isawealth of questions that weask ourselvesabout these numbers.(Someof these,such as Goldbach's conjecture,that every positive even integer greater than 4 can be written as a sum of two odd primes,or the question of the existence of infinitely many "twin" primes, that is,primes that differ by two,remain unsettled to this day.) At other times, we wonder about the rationals, this process of forming fractions,that seems to take care of dividing objects into equal parts.Wewonder abouttheother numbers on thereal line, how some of them have decimal expansions that go on forever with-out any repetition, and how they can all be approximated arbitrarily closely by rational numbers. And of course,we continue to wonder about this mysterious square root of -1. It is precisely this wonder about numbers that has been respon-sibleforthedevelopmentof muchof themathematicsof thelast two centuries, and in particular, of what is often referred to today as "abstract" algebra.The attempt tounderstand the structure of these numbers and tosolvesomeof theoutstanding problemsconcern-ing them has led totheintroductionof somevery deepconcepts. These concepts have in turn shed light on other areas ofmathemat-ics,as wellason areasof scienceand engineering,and havethus considerably enriched human knowledge. Innoduction3 Oneof theproblems that theseconcepts have solved isthat of constructibility.This isaproblem that had baffled the Greeks,and had remained unsolved for about two thousand years: Can arbitrary geometric figures be constructed using just a straightedge and a com-pass? Themost famousversionof this question askswhether it is possibletotrisect an arbitrary angle,that is,whether it ispossible toconstruct an angle whose measure is one-third that of any given angle, using just a straightedge and a compass. As it turns out, a com-plete answer can be given tothis question using only introductory algebraic concepts. Our goal in this book will be tolearn these introductory concepts and then apply them tothe solution of the constructibility problem. Our path will take us through rings,fields,and vector spaces. We will learn about field extensions and learn to differentiate between algebraicand transcendentalnumbers.Wewillstudy thedivision algorithm forpolynomials and the notion of an irreducible polyno-mial,and wewillrealizethat theseconceptsareexactanalogsof the corresponding division algorithm forintegers and the notion of aprime integer.(In fact,we willstart our studies by examining di-visibility and primes in the integers.) Wewill learn about the degree of afieldextension,and we will relate this degree todimensions of certain vectorspaces.Finally,wewillseehow thisdegreeaffects constructibility. A thorough understanding of these introductory concepts will en-able you to proceed further into mathematics and understand some of the questions we have described above.For instance,amore ad-vanced course willdetailthealgebra behind theprocess by which the complex numbers are formed from the reals and will discuss the concept of an ordered field.This will hopefully settle your confusion about negative numbers having square roots,and you will hopefully seethattheformationof thecomplexnumbersfromtherealsis really not the contrived process it firstseems, but isinstead some-thing very natural. Similarly,a more advanced course that includes real analysis (or "advanced calculus") willillumine the relationship of therationalstothereals.Asaresultof such acourse,you will hopefully realizethat therealnumbers arepreciselythenumbers that arise when one tries tocome togrips with the concept of deci-mal expansions that goon forever without repetition,and you will 4Inttoduction hopefully understand that it is perfectly natural that the rationals be densein the reals,that is,that every real number be approximated arbitrarily closely by rational numbers. Howshould youread this book? Theanswer,which appliesto every book on mathematics, can be given in one word-actively. You may have heard this before,but it can never be overstressed-you can only learnmathematics by doing mathematics.This means much morethanattempting alltheproblemsassignedtoyou(although attempting every problem assigned to you is a must). What it means is that you should take time out to think through every sentence and confirm every assertion made. Youshould accept nothing on trust; instead, not only should you check every statement, you should also attempt to go beyond what is stated,searching for patterns, looking for connections with other material that you may have studied, and probing for possible generalizations. Let us consider an example.On page34in Chapter 2,you will find the following sentence: Yet,eveninthisextremelyfamiliarnumbersystem, multiplication is not commutative; for instance, ~ ~ ) . ~ ~ ) f ~ ~ ) . ~ ~ ) .(The "number system" referred to is the set of 2 x2 matrices whose entries are realnumbers.) When you read asentence such asthis, the first thing that you should do is verify the computation yourselves. Mathematical insight comes from mathematical experience, and you cannot expect to gain mathematical experience if you merely accept somebody else's word that the product on the left side of the equation does not equal the product on the right side. The very process of multiplying out these matrices will make the set of 2 x2 matrices amore familiar system of objects,but as you dothe calculations,more things can happen if you keep your eyes and ears open. Some or all of the following may occur: 1.Youmay notice that not only are the two products not the same, but that the product on theright side gives you the zeromatrix. This should make you realize that although it may seem impos-siblethat twononzero"numbers" can multiply out tozero,this isonly becauseyouareconfiningyourthinkingtotherealor Inttoduction5 complex numbers. Already, the set of 2 x 2 matrices (with which you haveat least somefamiliarity)containsnonzeroelements whose product is zero. 2.Intrigued by this, you may want to discover other pairs of nonzero matricesthatmultiplyouttozero.Youwilldothisbytaking arbitrary pairsof matrices and determining their product.It is quite probable that you willnot find an appropriate pair.At this point you may be tempted to give up.However, you should not. Youshould try to be creative,and study how the entries in the various pairs of matrices you have selected affect the product. It may be possible foryou tochange one or twoentries in such a way that the product comes out to be zero.For instance, suppose you consider the product Youshould observe that nomatter what theentries of thefirst matrix are,the product willalways havezeros in the(I, 2)and the (2,2) slots. This gives you some freedom to try toadjust the entries of the first matrix so that the (I, 1) and the (2, 1) slots also come out to be zero. After some experimentation, you should be able to do this. 3.Youmay notice a pattern in the two matrices that appear in our inequality on page 4.Both matrices have only one nonzero entry, and that entry isa1.Of course,the1occursin differentslots inthetwomatrices.Youmaywonderwhatsortsof products occur if you take similar pairs of matrices, but with the nonzero 1occuring at other locations.Thsettleyour curiosity,you will multiply out pairs of such matrices,such as ~ ~ ) . ~ ~ ) ,or You will try to discern a pattern behind how such matrices mul-tiply. Th help you describe this pattern,you will let eijstand for 6Introduction the matrix with 1 in the(i,i)-th slot and zeros everywhereelse, and you will try todiscover aformula forthe product of ei,jand ek,l,where i,i, k,and 1 can each be any element of the set {l,2}. 4.Youmay wonder whether the fact that we considered only 2 x2 matrices is significant when considering noncommutativemul-tiplication or when considering the phenomenon of two nonzero elementsthatmultiplyouttozero.Youwillaskyourselves whether thesamephenomena occur in theset of 3x3matri-ces or 4 x4 matrices. You will next ask yourselves whether they occur in the set of nxnmatrices,where nis arbitrary.But you will caution yourselves about letting n be too arbitrary. Clearly n needs to be a positive integer,since linxnmatrices" is meaning-less otherwise, but you will wonder whether n can be allowed to equal 1 if you want such phenomena tooccur. 5.Youmay combine 3 and 4 above,and try todefine thematrices ei,janalogously in the general context of nxnmatrices. Youwill study theproduct of such matricesin thisgeneralcontextand try todiscover aformula for their product. Noticethat asinglesentencecan leadtoanenormousamountof mathematicalactivity!Everysteprequiresyoutobealertandac-tivelyinvolvedinwhatyouaredoing.Youobservepatternsfor yourselves,youaskyourselvesquestions,andyoutrytoanswer thesequestions on your own.In theprocess,you discover most of themathematicsyourselves.Thisisreallytheonlywaytolearn mathematics(andinparticular,itisthewayeveryprofessional mathematician has learned the subject). Mathematical concepts are developedprecisely becausemathematiciansobservepatternsin variousmathematicalobjects(suchasthe2x2matrices),and to have a good understanding of these concepts you must try tonotice these patterns for yourselves. 1bhelpyoualong,brief notesforeachchapterhavebeenin-cluded.Thesenotescontain hintstosomeof thequestionsasked in thechapter,as wellas generalcomments about some of thedef-initions,examples,andtheoriespresentedinthechapter.Donot rush to read these notes; you need to think independently about the material first! Exercises7 Besidesthewillingnesstoreadthisbookactively,theprereq-uisitesforthisbookaresmall.Youareexpectedtohavesome familiarity with the integers,as well as with the rationals,real and complex numbers,polynomials, .and matrices.It would be helpful tobe abletodoproofs by induction.Arudimentary knowledgeof set theory is assumed. Exercises 1.Carry out the program in steps (1)through (5)above. CHAPTER Divisibility in the Integers Wewill begin our study with avery concrete set of objects,the in-tegers,thatis,theset{O,I, -I, 2,-2, ... }.Thissetistraditionally denoted Zand isveryfamiliartous-in fact,wewereintroduced tothis set so early in our lives that we think of ourselves as having grown up with the integers!Moreover,we view ourselves as having completely absorbed theprocess of integer division;weunhesitat-ingly describe 3 as dividing 99and equally unhesitatingly describe 5 as not dividing lOI. Asit turns out,this very familiar set of objects has an immense amount of structuretoit.It turnsout,forinstance,that thereare certaindistinguishedintegers(theprimes)thatserveasbuilding blocksforallother integers.These primes are rather beguiling ob-jects;their existence has been known for over two thousand years, yet there arestill severalunanswered questions about them.They serve as building blocks in the followingsense:every positive inte-ger greater than 1 can be expressed uniquely as a product of primes. (Negative integers less than -1 also factor into a product of primes, except that they have a minus sign in front of the product.) The fact that nearly every integer breaks up uniquely into build-ing blocksisan amazing one;thisisaproperty that holdsin very fewnumber systems.(In the exercises toChapter 2 we willsee an 9 B. A. Sethuraman, Rings, Fields, and Vector Spaces Springer Science+Business Media New York 1997101.Divisibility in the Integers example of a number system whose elements donot factor uniquely into building blocks. Chapter 2 will also contain a discussion of what a"number system"is-see Remark 2.5.)On theother hand,there are some number systems where such a property does hold, notably polynomials,and wewillfindthat thefactthatpolynomialsalso break up uniquely into building blocks is crucial to our treatment of constructibility. Wewill study polynomials in Chapter 5. Ourgoalinthischapteristoprovethatintegerscan befac-tored uniquely into primes. Wewill begin by examining thenotion of divisibilityanddefiningdivisorsandmultiples.Wewillstudy thedivisionalgorithmandhowitfollowsfromtheWellOrder-ingPrinciple.Wewillexploregreatestcommondivisorsandthe :p.otionof relativeprimeness.Wewillthenintroduceprimesand proveourfactorizationtheorem.Finally,wewilllookatwhatis widely considered as the ultimate illustration of the elegance of pure mathematics- Euclid's proof that there are infinitely many primes. Letusstart withsomethingthatseemsveryinnocuous,butis actually rather profound. WriteN forthe nonnegative integers that is,N={O,1,2,3, ... }.(Nstands for"naturalnumbers,"as thenon-negative integers are sometimes referred to.) Let Sbe any nonempty subset of N.For example,Scould be the set {O,5,10, IS, ... },or the set {I, 4,9,16, ... },or else the set {lOO,lOOO}.The following is rather obvious:thereisan element in Sthat issmaller than every other elementinS,thatis,Shasasmallestorleastelement.Thisfact, namely that every nonempty subset of N has a least element, turns out to be a crucial reason why the integers possess all the other beau-tiful properties (such as a notion of divisibility,and the existence of prime factorizations)that make them so interesting. Contrast the integers with another very familiar number system, the rationals,that is,the set {alb I a and b are integers, with bfa}. (This set is traditionally denoted by Q.) Can you think of a nonempty subset of the positive rationals that failsto have a least element? We will take this property of the integers as a fundamental axiom, that is,we will merely accept it as given and not try to prove it from more fundamental principles. Also,we will giveit a name: WellOrderingPrinciple:Everynonempty subsetof thenon-negative integers has a least element. 1.Divisibility in the Integers11 Now let us look at divisibility. Why do we say that 2 divides 6? It is because there is another integer,namely 3,such that the product 2 times 3 exactly gives us 6.On the other hand, why do we say that 2 does not divide 7? This is because no matter how hard we search, we will not be able to find an integer b such that 2 times b equals 7. This idea will be the basis of our definition: Definition 1.1 A (nonzero)integer dis said to dividean integer a(denoted dla)if there existsan integer bsuch that a=db.If ddivides a,then dis referred toas a divisor of aor afactorof a,and aisreferred toas a multiple of d. Observe that this is a slightly more general definition than most of us are used to-according to this definition,-2 divides 6 as well, sincethereexistsaninteger,namely-3,suchthat-2 times-3 equals 6.Similarly,2 divides -6, since 2 times -3 equals -6. More generally, if d divides a, then all of the following are also true: dl-a, -dla, -dl-a. (Th.kea minute to prove this formally!)It is quite rea-sonable to include negative integers in our concept of divisibility, but forconvenience,we will often focuson the casewhere the divisor is positive. The followingeasy result will be very useful: Lemma 1.2 If d is anonzero integer such that dla and dlb fortwo integers a and b, then for any integers x and y,dl(xa+ yb).(In particular;dl(a+ b)and dl(a - b).) ProofSince dla,a=dm forsome integer m.Similarly,b=dnfor some integer n. Hence xa + yb=xdm + ydn=d(xm + yn). Since we have succeeded in writing xa+yb as dtimes the integer xm+ yn, wefindthat dl(xa+yb).Asforthestatement in theparentheses, taking x=1 and y=I,wefind that dla+b,and taking x=1 and y=-I, wefind that dla - b.0 Thefollowinglemma holdsthekey tothedivisionprocess.Its statementisoftenreferredtoasthedivisionalgorithm.TheWell Ordering Principle plays a central role in its proof. 121.Divisibility in the Integers Lemma1.3(Division Algorithm) Given integers aand bwith b>0,there exist unique integers q and r, with 0::::r-1 . vEW (asWisclosed with respect toscalar multiplication)=>-v EWis correct even when there is no vector v in Wto begin with!) Now let us use the fact that Wis nonempty. Since Wis nonempty, it contains at least one vector,call it v.Then, by what weproved above,-v is alsoinW.SinceWisclosed under vector addition,v+(-v) isin W,and so 0 isin W.Wehave thus shown that (W,+) isan abelian group. It remainstobe shown that thefouraxiomsof scalar multipli-cationalsoholdforW.Butforanyrand sinFandvandwin Subspaces87 W,we may consider v and w to be elements of V,and as elements of V,wecertainlyhavetherelationsr (v+w)=r v+r. w, (r+ s) . v=r v+ s v,(rs) . v=r (s v),and 1 . v=v.Hence,the axioms of scalar multiplication hold forW. This proves that Wis asubspace of V.0 Here are some examples of subspaces.In each case,check that the conditions of Theorem 3.22 apply. Examples 3.23 1.If youthink of lR2 asthevectors lying along thexyplaneof 3-dimensional xyz space,then lR2 becomes asubspace of lR3 . 2.Foranynonnegativeintegersnand mwithn