Basic Algebra - Mathematics and Statistics

1 Basic ,withanappendix ElementaryComplexAnalysis AdvancedRealAnalysisAnthony W. KnappBasic AlgebraAlong with a Companion VolumeAdvanced AlgebraDigital Second Edition, 2016 Published by the AuthorEast Setauket, New , aknappTitle:BasicAlgebraCover:Constructi onofaregularheptadecagon,thestepsshownin colorsequence; (2010):15 01,20 01,13 01,12 01,16 01,08 01,18A05, ,ISBN-13978-0-8176-3248-9c auserBostonDigitalSecondEdition,nottobes old,noISBNc ,images,andotherdatacontainedinthisfile, whichisinportabledocumentformat(PDF),are proprietarytotheauthor,andtheauthorretai nsallrights,includingcopyright, ,trademarks,servicemarks,andsimilaritems ,eveniftheyarenotidentifiedassuch, userBoston,c/oSpringerScience+BusinessMe diaInc.,233 SpringStreet,NewYork,NY10013,USA, ,scholarship,andresearch,andforthesepurp osesonly, ,postitonline,andtransmititdigitallyforp urposesofeducation,scholarship, ( ,EPUB),theymaynoteditit, ,usersmustchargenofee, ,noextractsorquotationsfromthisfilemaybe usedthatdonotconsistofwholepagesunlesspe rmissionhasbeengrantedbytheauthor(andbyB irkh userBostonifappropriate).

2 Thepermissiongrantedforuseofthewholefile andtheprohibitionagainstchargingfeesexte ndtoanypartialfilethatcontainsonlywholep agesfromthisfile, (andbyBirkh userBostonifappropriate).Inquiriesconcer ningprintcopiesofeithereditionshouldbedi rectedtoSpringerScience+ ,SarahandWilliam,andToMyAlgebraTeachers: RalphFox,JohnFraleigh,RobertGunning,John Kemeny,BertramKostant,RobertLanglands,Go roShimura,HaleTrotter, ,POLYNOMIALS, ,R, ,LinearIndependence, (Continued) ,manyreadershavereactedtothebookbysendin gcomments,suggestions, , , ,andtherewereenoughcorrections,perhapsah undredinall, , , ,theauthorgrantedapublishinglicensetoBir kh auserBostonthatwaslimitedtoprintmedia, ,andaPDFfile,calledthe digitalsecondedition, : sLemma,theearlierproofhavinghadagap. Anumberofproblemshavebeenaddedattheendso fthechapters, :(a)(ChapterII)therelationshipintwoandth reedimensionsbetweendeter-minantsandarea sorvolumes,(b)(ChaptersVandIX)furtherasp ectsofcanonicalformsformatricesandlinear mappings,(c)(ChapterVIII)amplificationof usesoftheFundamentalTheoremofFinitelyGen eratedModulesoverprincipalidealdomains,x ixiiPrefacetotheSecondEdition(d)(Chapter IX)theinterplayofextensionofscalarsandGa loistheory,(e)(ChapterIX)propertiesandex amplesoforderedfieldsandrealclosedfields .

3 Somerevisionshavebeenmadetothechapteronf ieldtheory(ChapterIX).Itwasoriginallyexp ected,anditcontinuestobeexpected, ,theoriginalplacementofthebreakbetweenvo lumesleftsomepossibleconfusionaboutthero leof normalextensions infieldtheory,andthatmatterhasnowbeenres olved. Characteristicpolynomialsinitiallyhaveav ariable , throughoutmostofthebookintheoriginaledit ion, fromChapterVon,andcharacteristicpolynomi alshavebeentreatedunambiguouslythereafte rasabstractpolynomials. :theanalogybetweenintegersandpolynomials inonevariableoverafield,theinterplaybetw eenlinearalgebraandgrouptheory, ;anexampleofsuchanotionis universalmappingproperty. Readerswillbenefitfromlookingfortheseand othersuchthemes, ,Birkh ausermathematicseditorinNewYork,whoencou ragedthewritingofasecondedition,whomadea numberofsuggestionsaboutpursuingit, , , , ,Iplantopointtoalistofknowncorrectionsfr ommyownWebpage, ,whetherpureorapplied, ,itsuse, , ,particularlyinBasicAlgebrabutalsoinsome ofthechaptersofAdvancedAlgebra, : ,asthetopiclikelytobebetterknownbytherea deraheadoftime,andthenalittlegrouptheory isintroduced,withlinearalgebraprovidingi mportantexamples.

4 Chaptersonlinearalgebradevelopnotionsrel atedtovectorspaces,thetheoryoflineartran sformations,bilinearforms,classicallinea rgroups,andmultilinearalgebra. Chaptersonmodernalgebratreatgroups,rings ,fields,modules,andGaloisgroups,includin gmanyusesofGaloisgroupsandmethodsofcompu tation. Threeprominentthemesrecurthroughoutandbl endtogetherattimes:theanalogybetweeninte gersandpolynomialsinonevariableoverafiel d,thein-terplaybetweenlinearalgebraandgr ouptheory,andtherelationshipbetweennumbe rtheoryandgeometry. Thedevelopmentproceedsfromtheparticulart othegeneral,oftenintroducingexampleswell beforeatheorythatincorporatesthem. Morethan400problemsattheendsofchaptersil luminateaspectsofthetext,developrelatedt opics, HintsforSolutionsofProblems attheendofthebookgivesdetailedhintsformo stoftheproblems,completesolutionsformany . ApplicationssuchasthefastFouriertransfor m,thetheoryoflinearerror-correctingcodes ,theuseofJordancanonicalforminsolvinglin earsystemsofordinarydifferentialequation s, , , ,inthe1920sEmmyNoetherintroducedvectorsp acesandlinearmappingstoreinterpretcoordi natespacesandmatrices,andshedefinedthein gredientsofwhatwasthencalled modernalgebra theaxiomaticallydefinedrings,fields,andm odules, , ,withmanyapplications, ,cryptography,andadvancesinphysicsandche mistryhavechangedallthat, ,andtheytoohaveextensiveapplicationsinsc ienceandengineering, , ,rings,fields, ;ofthese,theWedderburntheoryofsemisimple algebras,homologicalalgebra, VIItreatlinearalgebraandgrouptheoryatvar iouslevels,exceptthatthreesectionsofChap terIVandoneofChapterVintroduceringsandfi elds,polynomials,categoriesandfunctors, ,withemphasisonuniquefactorization;Chapt erIXconcernsfieldextensionsandGaloistheo ry,withemphasisonapplicationsofGaloisthe ory.

5 VandpartsofChaptersVIIIandIX, ,itmaybepossibletoskimmuchofthefirstthre echaptersandsomeofthebeginningofthefourt h;thentimemayallowforsomeofChaptersVIand VII,oradditionalmaterialfromChaptersVIII andIX, , ,plusthefirstsixsectionsofChapterIVandas muchasreasonablefromChapterV; ;thecoursewillperhapstreattheremainderof ChapterIV,thefirstfiveorsixsectionsofCha pterVIII,andatleastSections1 GuidefortheReader onpagesxxi ,IhavebuiltonecoursearoundChaptersI III,Sections1 6ofChapterIV,allofChapterV, ,alittleofChapterVII,Sections1 6ofChapterVIII,andSections1 ,someblocksofproblemsformadditionaltopic sthatcouldhavebeenincludedinthetextbutwe renot;theseblocksmayeitherberegardedasop tionaltopics, ,VII, , ,someareexamplesshowingthedegreetowhichh ypothesescanbestretched, , , ,themostimportantprereq-uisiteforusingBa sicAlgebraisthatthereaderalreadyknowwhat aproofis,howtoreadaproof, ,orfromacourseinlinearalgebra,orfromafir stjunior ,itisassumedthatthereaderiscomfortablewi thasmallamountoflinearalgebra,includingm atrixcomputations,rowreductionofmatrices ,solutionsofsystemsoflinearequations, ,propositions,lemmas, ,andonecanfindthepagereferenceforeachfig urefromthetableonpagesxvii.

6 Tofacilitatethisprocedure,eachoccurrence oftheword PROOF or PROOF ismatchedbyanoccurrenceattherightmargino fthesymbol ,whomadedetailedcommentsaboutmanyaspects ofapreliminaryversionofthebook,andtoDavi dKramer, , + {TX} (x 1)x(x+1) , , , 719, #Sor|S|numberofelementsinS emptyset{x E|P}thesetofxinEsuchthatPholdsEccompleme ntofthesetEE F,E F,E Funion,intersection,differenceofsets E , E union,intersectionofthesetsE E F,E FEiscontainedinF,EcontainsFE F,E FEproperlycontainedinF,properlycontainsF E F, s SXsproductsofsets(a1,..,an),{a1,..,an}or deredn-tuple,unorderedn-tuplef:E F,x f(x)function,effectoffunctionf gorfg,f Ecompositionofgfollowedbyf,restrictionto Ef( ,y)thefunctionx f(x,y)f(E),f 1(E)directandinverseimageofaset ijKroneckerdelta:1ifi=j,0ifi =j nk binomialcoefficientnpositive,nnegativen> 0,n<0Z,Q,R,Cintegers,rationals,reals,com plexnumbersmax(andsimilarlymin)maximumof afinitesubsetofatotallyorderedset or sumorproduct,possiblywithalimitoperation countablefiniteorinone-onecorrespondence withZ[x]greatestinteger xifxisrealRez,Imzrealandimaginarypartsof complexz zcomplexconjugateofz|z|absolutevalueofz1 multiplicativeidentity1orIidentitymatrix oroperator1 XidentityfunctiononXQn,Rn,Cnspacesofcolu mnvectorsdiag(a1.)

7 ,an)diagonalmatrix =isisomorphicto, , ,theinterplaybetweenlinearalgebraandgrou ptheory, ,reals, , , , ;theemphasisisonnotationandtheIntermedia teValueTheorem, ,whicharenormallyomittedinanundergraduat ecourse, , ,asmentionedabove,establishesuniquefacto rizationfortheintegersandforpolynomialsi noneindeterminateovertherationals,reals, ; , ,ChaptersII ,subspaces,andlinearmappingsisnotinclude dinthechapter, universalmappingproperty appearsforthefirsttimeinChapterII, , ,ifnotmoreso;attheleast, 6formthefirstpart, 3introducegroupsandsomeassociatedconstru ctions, , ,manyexamplesofgroupsariseinthecontextof groupactions, , 5areadigressiontodefinerings,fields,andr inghomomorphisms,andtoextendthetheoriesc oncerningpolynomialsandvectorspacesaspre sentedinChaptersI ,theirassociatedmultiplicativegroups, ,ratherthanjusttheGuidefortheReaderxxiii rationalsorrealsorcomplexnumbers, , , , CategorytheoryasksofeverytypeofMathemati calobject: Whatarethemorphisms?

8 ; (homo)morphismsislargelyduetoEmmyNoether ,whoemphasizedtheuseofhomomorphismsofgro upsandrings. , , , ; permanenceofidentities. ChapterVIlargelyconcernsbilinearformsand tensorproducts, ,butitisnotneededinChaptersVII 22attheendofthechapterdiscussuniversalma ppingpropertiesinthegeneralcontextofcate gorytheory, , 3concernfreegroupsandthetopicofgenerator sandrelations; , ,givingmanyexamplesinSection1, ,onceagainillustratingthefirsttheme ,therelationshipbetweennumbertheoryandge ometry, , 5ofChapterIXgivethefoundationaltheory, 8introduceGaloistheory, 11giveafirstroundofapplicationsofGaloist heory:Gauss stheoremaboutwhichregularn-gonsareinprin cipleconstructiblewithstraightedgeandcom pass,theFundamentalTheoremofAlgebra,andt heAbel 13giveasecondroundofapplications:Gauss smethodinprincipleforactuallyconstructin gtheconstructibleregularn-gonsandaconver setotheAbel 17makeuseofSections7 11ofChapterVIII,provingthat , , AlgebraCHAPTERIP reliminariesabouttheIntegers,Polynomials , ,discussinguniquefactorizationofpositive integers,uniquefactorizationofpolynomial swhosecoefficientsarerationalorrealorcom plex,signsofpermutations, , ,includingtheChineseRemainderTheoremandt heevaluationoftheEuler , , , ,scalarmultiplication, , , ,subtraction,andmultiplicationwithinZase stablished, ,thatkisadivisorofn, | ,anyproductformulan=kl1 ,Polynomials,andMatricesof1,henceiseithe r+1or = >1issaidtobeprimeifithasnonontrivialfact orizationp= ,whichwillbegivenpreciselyinSection2, Euclideanalgorithm, , (divisionalgorithm).

9 Ifaandbareintegerswithb =0,thenthereexistuniqueintegersqandrsuch thata=bq+rand0 r<|b|. q,wemayassumethatb> aareboundedaboveby|a|,andthereexistssuch ann,namelyn= |a|.Thereforethereisalargestsuchinteger, sayn= randa=bq+ b,thenr b 0saysthata=b(q+1)+(r b) b(q+1).Theinequalityq+1>qcontradictsthem aximalityofq,andweconcludethatr< >0,supposea=bq1+r1=bq2+ ,weobtainb(q1 q2)=r2 r1with|r2 r1|<b,andthisisacontradictionunlessr2 r1=0. >0suchthatd|aandd| ,forexample,isnonzero,thenanysuchdhas|d| |b|, (a,b).Letussupposethatb = ( )toaandbuntiltheremaindertermrdisappears :a=bq1+r1,0 r1<b,b=r1q2+r2,0 r2<r1,r1=r2q3+r3,0 r3<r2,..rn 2=rn 1qn+rn,0 rn<rn 1(withrn =0,say),rn 1=rnqn+ +1equalto0inthiswaysinceb>r1>r2> ,namelyrnabove, ,thestepsread13=5 2+3,5=3 1+2,3=2 1+1,2=1 =0,andletd=GCD(a,b).Then(a)thenumberrnin theEuclideanalgorithmisexactlyd,(b)anydi visord ofbothaandbnecessarilydividesd,(c)theree xistintegersxandysuchthatax+by= , ,asappliedintheaboveexamplewitha=13andb= 5,soastoyieldsuccessivesubstitutions:13= 5 2+3,3=13 5 2,5=3 1+2,2=5 3 1=5 (13 5 2) 1=5 3 13 1,3=2 1+1,1=3 2 1=(13 5 2) (5 3 13 1) 1=13 2 5 +5ywithx=2andy= , 1=a,sothatrk 2=rk 1qk+rkfor1 k n.

10 ( ) ,Polynomials, ,fromrn 1=rnqn+1,wehavern|rn n,andassumeinductivelythatrndividesrk 1,..,rn 1, ( )showsthatrndividesrk 1,r0,..,rn , +by= ,weshowbyinductiononkfork nthatthereexistintegersxandywithax+by= 1andk=0, 1isgivenandiftheresultisknownfork 2andk 1,thenwehaveax2+by2=rk 2,ax1+by1=rk 1( )forsuitableintegersx2,y2,x1, ( )byqk,subtract,andsubstituteinto( ).Theresultisrk=rk 2 rk 1qk=a(x2 qkx1)+b(y2 qky1), +by= (a),(b),and(c). >0dividesbothaandb,theresultofStep2shows thatd | rn, (a);(b)followsfrom(a)sinced |rn,and(c)followsfrom(a)andStep2. ,ifcisanonzerointegerthatdividesaproduct mnandifGCD(c,m)=1, +my= ,weobtaincnx+mny= , ,whichisn. ,ifaandbarenonzerointegerswithGCD(a,b)=1 andifbothofthemdividetheintegerm, +by= ,weobtainamx+bmy=m,whichwerewriteininteg ersasab(m/b)x+ab(m/a)y= ,itdividestherightside,whichism. (FundamentalTheoremofArithmetic).Eachpos itiveintegerncanbewrittenasaproductofpri mes,n=p1p2 pr, :ifn=q1q2 qsisanothersuchfactorization,thenr=sand, aftersomereorderingofthefactors,qj=pjfor 1 j , ,ifpisaprimeandpdividesaproductab, ,GCD(a,p)= ,n=b,andc= ,weseethatpdividesb.