{ "query": "what is algebra based on your context?", "namespace": "6780d7f7-c643-42bc-b383-40a6b7fd063b", "messages": [ { "role": "assistant", "content": "Hallo Herr Nhyll Test,\n\nich bin Simon, dein KI-Chatbot, der mit dem Large Language Model claude-3-haiku-20240307 ausgestattet ist, um all deine Fragen pr\u00e4zise und umfassend zu beantworten. \n\nIn unserer Wissensdatenbank \"Test 10-21\" findest du eine Vielzahl von Dokumenten, die dir bei deinen Recherchen helfen k\u00f6nnen. \n\nHier sind einige Themen, die dich vielleicht interessieren k\u00f6nnten:\n\n* **Datenanalyse und -mining:** \n * Wie kann man komplexe Muster in Daten erkennen und interpretieren? \n * Welche Herausforderungen gibt es bei der Analyse gro\u00dfer Datens\u00e4tze? \n * Wie kann man Ausrei\u00dfer in Daten identifizieren und interpretieren? \n* **Algebra und Mathematik:** \n * Wie h\u00e4ngen die verschiedenen Bereiche der Algebra zusammen? \n * Welche grundlegenden Konzepte der Algebra sind wichtig zu verstehen? \n * Wie kann man die Axiom of Choice und Zorn's Lemma in der Praxis anwenden? \n* **Sauna-Steuerungssysteme:** \n * Wie funktioniert ein Sauna-Steuerungssystem? \n * Welche Sicherheitsvorkehrungen sind bei der Installation und Wartung eines Sauna-Steuerungssystems zu beachten? \n * Welche Funktionen bietet ein modernes Sauna-Steuerungssystem? \n\nWas m\u00f6chtest du dir aus dieser Liste genauer ansehen? \ud83d\ude0a \n\n\nwhat is algebra based on your context?\n\n FOR ADDITIONAL CONTEXT:\n\nKnowledgebase name: Test 10-21\nKnowledgebase files:\nA First Encounter with Machine Learning - Max Welling (PDF).pdf\nData Mining Concepts and Techniques - Jiawei Han, Micheline Kamber, Jian Pei (PDF).pdf\nAdvanced Algebra - Anthony W. Knapp (PDF).pdf\nECON_D1-R4.35_-_MA_de.pdf\n15-MB-docx-file-download.docx\nKnowledgebases crawlers:\n" }, { "role": "user", "content": "what is algebra based on your context?" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":4939:1729570515779", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "claude-3-haiku-20240307", "knowledgebase": "ki-dev-large", "seed": 1118, "client_id": 1729570515779, "all_context": false, "follow_up_for": null, "knowledgebase_files_count": 5, "override_command": null, "disable_clarity_check": false, "custom_primer": "", "logging": true, "query_route": "" } INITIALIZATION Knowledgebase: ki-dev-large Base Query: what is algebra based on your context? 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Focus on providing precise and accurate information in your answers. **Elapsed Time: 0.17 seconds** GEMINI ERROR -- FALLBACK TO GPT ================================================== FINAL QUERY Final Query: CONTEXT: ########## File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 1 Context: AdvancedAlgebra #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 27 Context: AdvancedAlgebra #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 13 Context: PREFACETOTHEFIRSTEDITIONAdvancedAlgebraanditscompanionvolumeBasicAlgebrasystematicallyde-velopconceptsandtoolsinalgebrathatarevitaltoeverymathematician,whetherpureorapplied,aspiringorestablished.Thetwobookstogetheraimtogivethereaderaglobalviewofalgebra,itsuse,anditsroleinmathematicsasawhole.Theideaistoexplainwhattheyoungmathematicianneedstoknowaboutalgebrainordertocommunicatewellwithcolleaguesinallbranchesofmathematics.Thebooksarewrittenastextbooks,andtheirprimaryaudienceisstudentswhoarelearningthematerialforthefirsttimeandwhoareplanningacareerinwhichtheywilluseadvancedmathematicsprofessionally.Muchofthematerialinthetwobooks,includingnearlyallofBasicAlgebraandsomeofAdvancedAlgebra,correspondstonormalcoursework,withtheproportionsdependingontheuniversity.Thebooksincludefurthertopicsthatmaybeskippedinrequiredcoursesbutthattheprofessionalmathematicianwillultimatelywanttolearnbyself-study.Thetestofeachtopicforinclusioniswhetheritissomethingthataplenarylectureratabroadinternationalornationalmeetingislikelytotakeasknownbytheaudience.KeytopicsandfeaturesofAdvancedAlgebraareasfollows:•Topicsbuildonthelinearalgebra,grouptheory,factorizationofideals,struc-tureoffields,Galoistheory,andelementarytheoryofmodulesdevelopedinBasicAlgebra.•Individualchapterstreatvarioustopicsincommutativeandnoncommutativealgebra,togetherprovidingintroductionstothetheoryofassociativealgebras,homologicalalgebra,algebraicnumbertheory,andalgebraicgeometry.•Thetextemphasizesconnectionsbetweenalgebraandotherbranchesofmath-ematics,particularlytopologyandcomplexanalysis.Allthewhile,itcarriesalongtwothemesfromBasicAlgebra:theanalogybetweenintegersandpolynomialsinonevariableoverafield,andtherelationshipbetweennumbertheoryandgeometry.•SeveralsectionsintwochaptersintroducethesubjectofGr¨obnerbases,whichisthemoderngatewaytowardhandlingsimultaneouspolynomialequationsinapplications.•Thedevelopmentproceedsfromtheparticulartothegeneral,oftenintroducingexampleswellbeforeatheorythatincorporatesthem.xiii #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 15 Context: cpointofviewofsimultaneoussystemsofpolynomialequationsinseveralvariables,fromthenumber-theoreticpointofviewsuggestedbytheclassicaltheoryofRiemannsurfaces,andfromthegeometricpointofview.ThetopicsmostlikelytobeincludedinnormalcourseworkincludetheWedderburntheoryofsemisimplealgebrasinChapterII,homologicalalgebrainChapterIV,andsomeoftheadvancedmaterialonfieldsinChapterVII.Achartonpagexvitellsthedependenceofchaptersonearlierchapters,and,asmentionedabove,thesectionGuidefortheReadertellswhatknowledgeofBasicAlgebraisassumedforeachchapter.Theproblemsattheendsofchaptersareintendedtoplayamoreimportant #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 19 Context: GUIDEFORTHEREADERThissectionisintendedtohelpthereaderfindoutwhatpartsofeachchapteraremostimportantandhowthechaptersareinterrelated.Furtherinformationofthiskindiscontainedintheabstractsthatbegineachofthechapters.Thebooktreatsitssubjectmaterialaspointingtowardalgebraicnumbertheoryandalgebraicgeometry,withemphasisonaspectsofthesesubjectsthatimpactfieldsofmathematicsotherthanalgebra.Twochapterstreatthetheoryofassociativealgebras,notnecessarilycommutative,andonechaptertreatshomologicalalgebra;boththesetopicsplayaroleinalgebraicnumbertheoryandalgebraicgeometry,andhomologicalalgebraplaysanimportantroleintopologyandcomplexanalysis.Theconstantthemeisarelationshipbetweennumbertheoryandgeometry,andthisthemerecursthroughoutthebookondifferentlevels.ThebookassumesknowledgeofmostofthecontentofBasicAlgebra,eitherfromthatbookitselforfromsomecomparablesource.SomeofthelessstandardresultsthatareneededfromBasicAlgebraaresummarizedinthesectionNotationandTerminologybeginningonpagexxi.TheassumedknowledgeofalgebraincludesfacilitywithusingtheAxiomofChoice,Zorn’sLemma,andelementarypropertiesofcardinality.AllchaptersofthepresentbookbutthefirstassumeknowledgeofChaptersI–IVofBasicAlgebraotherthantheSylowTheorems,factsfromChapterVaboutdeterminantsandcharacteristicpolynomialsandminimalpolynomials,simplepropertiesofmultilinearformsfromChapterVI,thedefinitionsandelementarypropertiesofidealsandmodulesfromChapterVIII,theChineseRemainderTheoremandthetheoryofuniquefactorizationdomainsfromChapterVIII,andthetheoryofalgebraicfieldextensionsandseparabilityandGaloisgroupsfromChapterIX.AdditionalknowledgeofpartsofBasicAlgebrathatisneededforparticularchaptersisdiscussedbelow.Inaddition,somesectionsofthebook,asindicatedbelow,makeuseofsomerealorcomplexanalysis.Therealanalysisinquestiongenerallyconsistsintheuseofinfiniteseries,uniformconvergence,differentialcalculusinseveralvariables,andsomepoint-settopology.Thecomplexanalysisgenerallyconsistsinthefundamentalsoftheone-variabletheoryofanalyticfunctions,includingth #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 475 Context: CHAPTERVIIIBackgroundforAlgebraicGeometryAbstract.Thischapterintroducesaspectsofthealgebraictheoryofsystemsofpolynomialequationsinseveralvariables.Section1givesabriefhistoryofthesubject,treatingitasoneoftwoearlysourcesofquestionstobeaddressedinalgebraicgeometry.Section2introducestheresultantasatoolforeliminatingoneofthevariablesinasystemoftwosuchequations.AfirstformofBezout’sTheoremisanapplication,sayingthatiff(X,Y)andg(X,Y)arepolynomialsofrespectivedegreesmandnwhoselocusofcommonzeroshasmorethanmnpoints,thenfandghaveanontrivialcommonfactor.Thisversionofthetheoremmayberegardedaspertainingtoapairofaffineplanecurves.Section3passestoprojectiveplanecurves,whicharenonconstanthomogeneouspolynomialsinthreevariables,twosuchbeingregardedasthesameiftheyaremultiplesofoneanother.VersionsoftheresultantandBezout’sTheoremarevalidinthiscontext,andtwoprojectiveplanecurvesdefinedoveranalgebraicallyclosedfieldalwayshaveacommonzero.Sections4–5introduceintersectionmultiplicityforprojectiveplanecurves.Section4treatsalineandacurve,andSection5treatsthegeneralcaseoftwocurves.ThetheoryinSection4iscompletelyelementary,andaversionofBezout’sTheoremisprovedthatsaysthatalineandacurveofdegreedhaveexactlydcommonzeros,providedtheunderlyingfieldisalgebraicallyclosed,thezerosarecountedasoftenastheirintersectionmultiplicities,andthelinedoesnotdividethecurve.Section5makesmoreserioususeofalgebraicbackground,particularlylocalizationsandtheNullstellensatz.Itgivesanindicationthatostensiblysimplephenomenainthesubjectcanrequiresophisticatedtoolstoanalyze.Section6provesaversionofBezout’sTheoremappropriateforthecontextofSection5:ifFandGaretwoprojectiveplanecurvesofrespectivedegreesmandnoveranalgebraicallyclosedfield,theneithertheyhaveanontrivialcommonfactorortheyhaveexactlymncommonzeroswhentheintersectionmultiplicitiesofthezerosaretakenintoaccount.Sections7–10concernGr¨obnerbases,whicharefinitegeneratingsetsofaspecialkindforidealsinapolynomialalgebraoverafield.Section7setsthestage,introducingmonomialordersandde #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 14 Context: hilefindingrationalsolutionsisalgebraicgeometry.Experienceshowsthatthisisanartificialdistinction.Althoughalgebraicgeometrywasinitiallydevelopedasasubjectthatstudiessolutionsforwhichthevariablestakevaluesinafield,particularlyinanalgebraicallyclosedfield,theinsistenceonworkingonlywithfieldsimposedartificiallimitationsonhowproblemscouldbeapproached.Inthelate1950sandearly1960sthefoundationsofthesubjectweretransformedbyallowingvariablestotakevaluesinanarbitrarycommutativeringwithidentity.Theveryendofthisbookaimstogivesomeideaofwhatthosenewfoundationsare.Alongthewayweshallobserveparallelsbetweennumbertheoryandalgebraicgeometry,evenaswenominallystudyonesubjectatatime.Thebookbeginswith #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 488 Context: 460VIII.BackgroundforAlgebraicGeometryd>0toapolynomialf(X,Y)isgivenbyf(X,Y)=F(X,Y,1).ThemappingF7→fisasubstitutionhomomorphism,anditthereforerespectsproducts.However,thedegreemaydropintheprocess,andinparticularf(X,Y)isaconstantifandonlyifF(X,Y,W)isamultipleofWd.Inthereversedirectioniff(X,Y)isapolynomialofdegreee,thenf(X,Y)arisesfromapolynomialF(X,Y,W),butwehavetospecifythedegreedofFandwemusthaved∏e.OperationallyweobtainFbyinsertingapowerofWintoeachtermofftomakethetotaldegreeofthetermbecomed.Forexample,withf(X,Y)=Y2+XY+X3ifthedesireddegreeis3,thenF(X,Y,W)=Y2W+XYW+X3.Ontheotherhand,ifthedesireddegreeis4,thenF(X,Y,W)=Y2W2+XYW2+X3W.TheformulaforthisreverseprocessisF(X,Y,W)=Wdf(XW−1,YW−1).Thatis,Fisgivenbyasubstitutionhomomorphism,followedbymultiplicationbyapowerofW.Fromthisfact,wecanreadoffconclusionsofthefollowingkind:Ifpolynomialsf(X,Y)andg(X,Y)areobtainedfromhomoge-neouspolynomialsF(X,Y,W)andG(X,Y,W)bytakingW=1,thenthereexistintegersrandssuchthatthepolynomialWrF(X,Y,W)+WsG(X,Y,W)ishomogeneousandsuchthatf(X,Y)+g(X,Y)isobtainedfromitbytakingW=1.Aswementionedabove,P2Khasmorestructurethansimplythestructureofaset.AboutanypointinP2Kwecanintroducevarioussystemsof“affinelocalcoordinates.”Theideaistoimitatewhathappensinthedefinitionofamanifold:thewholemanifoldiscoveredbycharts,eachgivinganinvertiblemappingofasetinthemanifoldtoanopensubsetofEuclideanspace.Hereasinglesystemofaffinelocalcoordinatesplaystheroleofachart;itputsA2Kintoone-onecorrespondencewiththecomplementofthezerolocusofalineinP2K.Let8beamemberofthematrixgroupGL(3,K).Then8mapsthesetK3ofcolumnvectorsinone-onefashionontoK3andpassestoaone-onemapofP2KontoP2Kcalledtheprojectivetransformationcorrespondingto8.Two8’sgivethesamemapofP2Kifandonlyiftheyaremultiplesofoneanother.ThegroupactionofGL(3,K)onP2KistransitivebecauseGL(3,K)actstransitivelyonK3−{(0,0,0)}.IfListheprojectivelinewhosecoefficientsaregivenbytherowvector(abc)andif8isisinGL(3,K),thentherowvector(abc)8−1definesanewprojectivelineL8,andtheKrationalpointsofL8areg #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 548 Context: rem,andSection5givesanumberofsimpleapplications.1.HistoricalOriginsandOverviewAswasmentionedinChapterVIII,modernalgebraicgeometrygrewoutofearlyattemptstosolvesimultaneouspolynomialequationsinseveralvariablesandoutofthetheoryofRiemannsurfaces.ChapterVIIIdiscussedtheimpactofthefirstofthesesources,andthepresentchapterdiscussestheimpactofthesecond.520 #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 21 Context: GuidefortheReaderxxiouttobeacohomologygroupindegree2.ThisdevelopmentrunsparalleltothetheoryoffactorsetsforgroupsasinChapterVIIofBasicAlgebra,andsomefamiliaritywiththattheorycanbehelpfulasmotivation.ThecasethattherelativeBrauergroupiscyclicisofspecialimportance,andthetheoryisusedintheproblemstoconstructexamplesofdivisionringsthatwouldnothavebeenotherwiseavailable.ThechaptermakesuseofmaterialfromChapterXofBasicAlgebraonthetensorproductofalgebrasandoncomplexesandexactsequences.ChapterIVisabouthomologicalalgebra,withemphasisonconnectinghomo-morphisms,longexactsequences,andderivedfunctors.Allbutthelastsectionisdoneinthecontextof“good”categoriesofunitalleftRmodules,Rbeingaringwithidentity,whereitispossibletoworkwithindividualelementsineachobject.Thereaderisexpectedtobefamiliarwithsomeexampleformotivation;thiscanbeknowledgeofcohomologyofgroupsatthelevelofSectionIII.5,oritcanbesomeexperiencefromtopologyorfromthecohomologyofLiealgebrasastreatedinotherbooks.KnowledgeofcomplexesandexactsequencesfromChapterXofBasicAlgebraisprerequisite.Homologicalalgebraproperlybelongsinthisbookbecauseitisfundamentalintopologyandcomplexanalysis;inalgebraitsrolebecomessignificantjustbeyondthelevelofthecurrentbook.Importantapplicationsarenotlimitedinpracticeto“good”categories;“sheaf”cohomologyisanexamplewithsignificantapplicationsthatdoesnotfitthismold.Section8sketchesthetheoryofhomologicalalgebrainthecontextof“abelian”categories.Inthiscaseonedoesnothaveindividualelementsathand,butsomesubstituteisstillpossible;sheafcohomologycanbetreatedinthiscontext.ChaptersVandVIareanintroductiontoalgebraicnumbertheory.ThetheoryofDedekinddomainsfromChaptersVIIIandIXofBasicAlgebraistakenasknown,alongwithknowledgeoftheingredientsofthetheory—Noetherianrings,integralclosure,andlocalization.Bothchaptersdealwiththreetheorems—theDedekindDiscriminantTheorem,theDirichletUnitTheorem,andthefinitenessoftheclassnumber.ChapterVattacksthesedirectly,usingnoadditionaltools,anditcomesupalittleshortinthecaseoftheDedekindDiscrimin #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 550 Context: t.TheanalysisaspectmaybeviewedasusingthetheoryofellipticdifferentialoperatorstoproveexistenceofenoughnonconstantmeromorphicfunctionsfortheRiemannsurfacetoacquireanalgebraicstructure.Forthepurposesofthisbook,wecanjustacceptthiscircumstanceandnottrytoextenditinanyway;however,wewillsketchinamomenthowthealgebraicstructurecanbeobtainedconcretelyforourexample.Thealgebraicaspectmaybeviewedasminingthisalgebraicstructuretodeduceasmanydimensionalityrelationsaspossibleamongthefunctionspacesofinterest.Thisisthetheorythatweshallwanttoextend;wereturntoourmethodforcarryingoutthisprojectafterproducingthealgebraicstructureforourexamplebyelementarymeans. #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 14 Context: xivPrefacetotheFirstEdition•Morethan250problemsattheendsofchaptersilluminateaspectsofthetext,developrelatedtopics,andpointtoadditionalapplications.Aseparatesection“HintsforSolutionsofProblems”attheendofthebookgivesdetailedhintsformostoftheproblems,completesolutionsformany.Itisassumedthatthereaderisalreadyfamiliarwithlinearalgebra,grouptheory,ringsandmodules,uniquefactorizationdomains,Dedekinddomains,fieldsandalgebraicextensionfields,andGaloistheoryattheleveldiscussedinBasicAlgebra.NotallofthismaterialisneededforeachchapterofAdvancedAlgebra,andchapter-by-chapterinformationaboutprerequisitesappearsintheGuidefortheReaderbeginningonpagexvii.Historicallythesubjectsofalgebraicnumbertheoryandalgebraicgeometryhaveinfluencedeachotherastheyhavedeveloped,andthepresentbooktriestobringoutthisinteractiontosomeextent.Itiseasytoseethattheremustbeacloseconnection.Infact,onenumber-theoryproblemalreadysolvedbyFermatandEulerwastofindallpairs(x,y)ofintegerssatisfyingx2+y2=n,wherenisagivenpositiveinteger.Moregenerallyonecanconsiderhigher-orderequationsofthiskind,suchasy2=x3+8x.Eventhissimplechangeofdegreehasagreateffectonthedifficulty,somuchsothatoneisinclinedfirsttosolveaneasierproblem:findtherationalpairssatisfyingtheequation.Isthesearchforrationalsolutionsaprobleminnumbertheoryoraproblemaboutacurveintheplane?Theansweristhatreallyitisboth.Wecancarrythiskindofquestionfurther.Insteadofconsideringsolutionsofasinglepolynomialequationintwovariables,wecanconsidersolutionsofasystemofpolynomialequationsinseveralvariables.Withinthesystemnoindividualequationisanintrinsicfeatureoftheproblembecauseoneoftheequationscanalwaysbereplacedbyitssumwithanotheroftheequations;ifweregardeachequationasanexpressionsetequalto0,thentheintrinsicproblemistostudythelocusofcommonzerosoftheequations.Thisformulationoftheproblemsoundsmuchmorelikealgebraicgeometrythannumbertheory.Adoubtermightdrawadistinctionbetweenintegersolutionsandrationalsolutions,sayingthatfindingintegersolutionsisnumbertheorywhilefindingrationals #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 551 Context: structureandtoseethatsomebasictoolsintroducedinChapterVIinthecontextofalgebraicnumbertheoryaretheappropriatetoolstousehere. #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 590 Context: 562X.MethodsofAlgebraicGeometryFor(d),theinclusionab⊆a∩bisimmediate.Iffisina∩b,thenfisinaandinb,andhencef2isinab.Thusfisinpab.ApplyingV(·)givesV(ab)⊇V(a∩b)⊇V(pab).SinceV(ab)=V(pab)by(c),V(a∩b)=V(ab).FinallyV(ab)=V(a)∪V(b)byProposition10.1c.§AnaffinevarietyisanyaffinealgebraicsetoftheformV(p),wherepisaprimeideal2ofA.Thatis,anaffinevarietyisthelocusofcommonzerosofanyprimeidealofA.Forexample,iffisanirreduciblepolynomialinA,thenfisprimebecauseAisauniquefactorizationdomain,andconsequentlytheprincipalideal(f)isprime.ThusthezerolocusinA2ofanirreduciblepolynomialfink[X,Y]isanexampleofanaffinevariety.Thisparticularkindofaffinevarietyiscalledanirreducibleaffineplanecurve.3,4Moregenerally,iffisirreducibleinA=k[X1,...,Xn]withn∏2,thenthezerolocusoffinAniscalledanirreducibleaffinehypersurface.5AnotherexampleofanaffinevarietyisanytranslateofanyvectorsubspaceofAn.Examplesofaffinevarietiesotherthanirreduciblehypersurfaces,translatesofvectorsubspaces,andvarietiesbuiltfromothervarietiesinsimplewaysoftentakesomeworktoestablish.Thereasonisthatitisusuallynoteasytoshowthataparticularnonprincipalidealisprime.Hereisoneexamplethatismanageable.EXAMPLE.ThetwistedcubicinA3isthezerolocusV(p)oftheidealpink[X,Y,Z]givenbyp=(Y−X2,Z−X3);thatis,V(p)={(x,x2,x3)|x∈k}.ThesubstitutionhomomorphismϕthatfixeskandsendsXtoX,YtoX2,andZtoX3carriesk[X,Y,Z]intok[X].Itisontok[X]becauseanypolynomialinXaloneissenttoitselfbyϕ.Thekernelofϕmanifestlycontainsp.Toseethatitequalsp,wearguebycontradiction.ChooseapolynomialfinkerϕnotinpwhosedegreeinZisassmallaspossibleandwhosedegreeinYisassmallaspossibleamongthoseofminimaldegreeinZ.IfZoccurssomewhereinf,thenbyreplacingalloccurrencesofZinfwithX3,wereplacefbyanothermemberoff+poflowerdegreeinZ,contradiction.ThusfhasnoZinit.Arguing2Warning:ThebooksbyFultonandHartshorneintheSelectedReferencesusethenarrowdefinitionofvarietythatisreproducedhere.Somebooksbyotherauthorsallowallaffinealgebraicsetstobecalledvarieties.Volume1ofShafarevich’sbooksdoesnotusetheword“variety. #################### File: Advanced%20Algebra%20-%20Anthony%20W.%20Knapp%20%28PDF%29.pdf Page: 22 Context: xxiiGuidefortheReaderknowledgeoflocalizations,andtheindispensableCorollary7.14inSection3concernsDedekinddomains.ThemostimportantresultistheNullstellensatzinSection1.TranscendencedegreeandKrulldimensioninSections2and4aretiedtothenotionofdimensioninalgebraicgeometry.Zariski’sTheoreminSection5istiedtothenotionofsingularities;partofitsproofisdeferredtoChapterX.ThematerialoninfiniteGaloisgroupsinSection6hasapplicationstoalgebraicnumbertheoryandalgebraicgeometrybutisnotusedinthisbookafterChapterVII;compacttopologicalgroupsplayaroleinthetheory.ChaptersVIII–Xintroducealgebraicgeometryfromthreepointsofview.ChapterVIIIapproachesitasanattempttounderstandsolutionsofsimulta-neouspolynomialequationsinseveralvariablesusingmodule-theoretictools.ChapterIXapproachesthesubjectofcurvesasanoutgrowthofthecomplex-analysistheoryofcompactRiemannsurfacesandusesnumber-theoreticmethods.ChapterXapproachesitssubjectmattergeometrically,usingthefield-theoreticandring-theoretictoolsdevelopedinChapterVII.AllthreechaptersassumeknowledgeofSectionVII.1ontheNullstellensatz.ChapterVIIIisinthreeparts.Sections1–4arerelativelyelementaryandconcerntheresultantandpreliminaryformsofBezout’sTheorem.Sections5–6concernintersectionmultiplicityforcurvesandmakeextensiveuseoflo-calizations;thegoalisabetterformofBezout’sTheorem.Sections7–10areindependentofSections5–6andintroducethetheoryofGr¨obnerbases.Thissubjectwasdevelopedcomparativelyrecentlyandliesbehindmanyofthesymbolicmanipulationsofpolynomialsthatarepossiblewithcomputers.ChapterIXconcernsirreduciblecurvesandisintwoparts.Sections1–3definedivisorsandthegenusofsuchacurve,whileSections4–5provetheRiemann–RochTheoremandgiveapplicationsofit.ThetoolforthedevelopmentisdiscretevaluationsasinSectionVI.2,andtheparallelbetweenthetheoryinChapterVIforalgebraicnumberfieldsandthetheoryinChapterIXforcurvesbecomesmoreevidentthanever.SomecomplexanalysisisneededtounderstandthemotivationinSections1and4.ChapterXlargelyconcernsalgebraicsetsdefinedaszerolocioveranalge-braicallyclosedfi 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