Basic Algebra - McGill University

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 .

5 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; VandpartsofChaptersVIIIandIX, ,itmaybepossibletoskimmuchofthefirstthre echaptersandsomeofthebeginningofthefourt h;thentimemayallowforsomeofChaptersVIand VII,oradditionalmaterialfromChaptersVIII andIX, , ,plusthefirstsixsectionsofChapterIVandas muchasreasonablefromChapterV.

6 ThecoursewillperhapstreattheremainderofC hapterIV,thefirstfiveorsixsectionsofChap terVIII,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.

7 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.)

8 ,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?

9 ; (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.

10 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).Ifaandbareintegerswi thb =0,thenthereexistuniqueintegersqandrsuch thata=bq+rand0 r<|b|.