equations over finite fields an elementary approach .pdf

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LectureNotesinMathematicsEditedbyA.DoldandB.Eckmann536WolfgangM.SchmidtEquationsoverFiniteFieldsAnElementaryApproachSpringer-VerlagBerlin-Heidelberg9NewYork1976 AuthorWolfgangM.SchmidtDepartmentofMathematicsUniversityofColoradoBoulder,Colo.,80309/USALibraryofCongressCataloginginPublicationDataSehnddt,WolfgangMEquationsoverfinitefields.(Lecturenotesinn~then~tics;536)BibliOgraphy:p.i.Diophantineanalysis.2.Modularfields.I.Title.II.Series:Lecturenotesinmathe-matics(Berlin~;536.QAS.Le8v01.536[QA242]510'.8s[512.9'4]76~6612AMSSubjectClassifications(1970):10A10,10B15,10G05,12C25,14G15ISBN3-540-07855-XSpringer-VerlagBerlin9Heidelberg9NewYorkISBN0-387-07855-XSpringer-VerlagNewYork9Heidelberg9BerlinThis.,,yorkissubjecttocopyright.Allriclhtsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallythoseoftranslation,re-printing,re-useofillustrations,broadcasting,reproductionbyphotocopyingmachineorsimilarmeans,andstorageindatabanks.Unders54oftheGermanCopyrightLawwherecopiesaremadeforotherthanprivateuse,afeeispayabletothepublisher,theamountofthefeetobedeterminedbyagreementwiththepublisher.9bySpringer-VerlagBerlin-Heidelberg1976PrintedinGermanyPrintingandbinding:BeltzOffsetdruck,Hemsbach/Bergstr. PrefaceTheseLectureNoteswerepreparedfromnotestakenbyM.RatliffandK.SpackmanoflecturesgivenattheUniversityofColorado.IhavetriedtopresentaproofassimpleaspossibleofWeil'stheoremoncurvesoverfinitefields.Thenotionsof"simple"or"elementary"havedifferentinterpretations,butIbelievethatforareaderwhoisunfamiliarwithalgebraicgeometry,perhapsevenwithalgebraicfunctionsinonevariable,thesimplestmethodistheonewhichoriginatedwithStepanov.HenceitisthismethodwhichIfollow.ThelengthoftheseNotesisperhapsshocking.However,itshouldbenotedthatonlyChaptersIandIIIdealwithWeil'stheorem.Furthermore,thestyleis(Ibelieve)leisurely,andseveralresultsareprovedinmorethanoneway.IstartinChapterIwiththedsimplestcase,i.e.,withcurvesy=f(x)AtfirstIdothesimplestsubcase,i.e.,thecasewhenthefieldistheprimefieldandwhendiscoprimetothedegreeo~f.Thisspecialcaseisnowsoeasythatitcouldbepresentedtoundergraduates.Thegeneralequationf(x,y)=0istakenuponlyinChapterIll,butareaderinahurrycouldstartthere.Thesecondchapter,oncharactersumsandexpo-nentialsums,isincludedatsuchanearlystagebecauseosthemanyapplicationsinnumbertheory.ChaptersIV,VandVldealwithequationsinanarbitrarynumberofvariables.PossiblesequencesarechaptersIbyitself,orI,IllforWell'stheorem,or fVI.i,IIIforareaderwhoisinahurry,orI,IIforcharactersumsandexponentialsums,orI,If,IV,orI,III,IV.3andV~OriginallyIhadplannedtoincludeBombieri'sversionoftheStepanovmethod.IdidincludeitinmylecturesattheUniversityofColorado,butIfirsthadtoprovetheRiemann-RochTheoremandbasicpropertiesofthezetafunctionofacurve.AproofofthesebasicpropertiesintheLectureNoteswouldhavemadetheseundulylong,whiletheiromissionwouldhavemadetheBombieriversionnotselfcom-plete.HenceIdecidedaftersomehesitationtoexcludethisversionfromtheNotes.RecentlyDeligneprovedfarreachinggeneralizationsofWeil'stheoremtonon-singularequationsinseveralvariables,therebycon-firmingconjecturesofWell.Itistobenoted,however,thatDeligne'sproofrestsonanassertionofGrothendieckconcerningacertainfixedpointtheorem.Tothebestofmyknowledge,aproofofthisfixedpointtheoremhasnotappearedinprintyet.ItisperhapsneedlesstosaythatatpresentthereisnoelementaryapproachtosuchageneralizationofWell'stheorem.Butitistobehopedthatsomedaysuchanapproachwillbecomeavailable,atleastforthosecaseswhichareusedmostofteninanalyticnumbertheory.November,1975W.M.Schmidt NotationFisthemultiplicativegroupofafieldF.isthealgebraicclosureofafieldF.FnistheproductFX...XF,i.e.,thesetofn-tuples(Xl,..-,xn)withxiEF(i=l,...,n).IF1:F2]denotesthedegreeofafieldextensionF1~F2.denotesthetraceand~thenorm.Fwilldenotethefinitefieldwithqelements.qpwillbethecharacteristic~isthefieldofrationalnumbers,Rthefieldofreals,Cthefieldofcomplexnumbers,Ztheringof(rational)integers.denotesisomorphismoffieldsorgroups.Quiteoften,x,y,z..,willbeelementswhichlieinagroundfieldorarealgebraicoveragroundfield,X,Y,Z,...willbevariables,i.e.,willbealgebraicallyindependentoveragroundfield,and~,~,...willbealgebraicfunctions,i.e.,theywillbealgebraicallydependentonsomeofX,Y,...Thusf(XI,...,Xn)isapolynomial,andf(xl,...,xn)isthe~valueofthispolynomialat(Xl,...,xn)9F(x)orF(X)orF(X,Y)orF(X,~),orsimilar,willbethefieldobtainedbyadjoiningxorXorX,YorX,~toagroundfieldF.ThusF(X)isthefieldofrationalfunctionsinavariableXwithcoefficientsinF.R[X]denotestheringofpolynomialsinXwithcoefficientsintheringR. V~Ifa,bareinZ,wewritealb(ora+b)ifadoes(ordoesnot)divideb.Occasionallyweshallwritedlq-1insteadofthemorepropernotationdl(q-1).Again,weshallwritef(X)lg(X)ifthepolynomialf(X)dividesg(X).Further(f(X))(or(f(X),g(X)))willbetheidealgeneratedbyf(X)(orbyf(X)andg(X))!eIdenotesthenumberofelementsofafiniteset~GivensetsA~B,thesettheoreticdifferenceisdenotedbyB~A. TableofContentsChapterPageIntroduction.....................1dI~Equationsy=f(x)andyq-y=f(x)1.FiniteFields...................3d2.Equationsy=f(x)...............83.Constructionofcertainpolynomials.......164.ProofoftheMainTheorem.............215.Removalofthecondition(m,d)=1.......226.Hyperderivatives.................2727.Removaloftheconditionthatq=porp....318.TheWorkofStark.................329.Equationsyq-y=f(x)..............34IIoCharacterSumsandExponentialSums1.CharactersofFiniteAbelianGroups........382.CharactersandCharacterSumsassociatedwithFiniteFields...................413.GaussianSums...................464.Thelowroad...................505.Systemsofequationsyldl=fl(x)'....'Yndn=f(x).......................52n6.AuxiliarylemmasonW1+...+w~........577.Furtherauxiliarylemmas.............608.ZetaFunctionandL-Functions...........629.SpecialL-Functions................65i0.Fieldextensions.TheDavenport-Hasserelations.72ii.ProofofthePrincipalTheorems..........77 VIIIChapterPage12.KloostermanSums.................8413.FurtherResults..................88III.AbsolutelyIrreducibleEquationsf(x,y)=0i.Introduction...................922.Independenceresults...............973.Derivatives....................1054.Constructionoftwoalgebraicfunctions......1075.Constructionoftwopolynomials..........1146.ProofoftheMainTheorem.............1167.Valuations....................119S.Hyperderivativesagain..............1259.Removaloftheconditionthatg=p.......131IV.EquationsinManyVariablesi.TheoremsofChevalleyandWarning.........1342.Quadraticforms..................1403.Elementaryupperbounds.Projectivezeros.....1474.Theaveragenumberofzerosofapolynomial....1575.AdditiveEquations:AChebychevArgument.....1606.AdditiveEquations:CharacterSums........1667.Equationsfl(Y)xldl+...+fn(Y)xndn=0......173V.AbsolutelyIrreducibleEquationsf(xl,...,xn)=0i.EliminationTheory................1772.Theabsoluteirreducibilityofpolynomials(I)..1903.Theabsoluteirreducibilityofpolynomials(II)..1944.Theabsoluteirreducibilityofpolynomials(III).204 IXChapterPage5.Thenumberofzerosofabsolutelyirreduciblepolynomialsinnvariables...........210VIoRudimentsofAlgebraicGeometry.TheNumberofPointsinVarietiesoverFiniteFields1.Varieties......,..............2162.Dimension.....................2283.RationalMaps...................2354.BirationalMaps..................2445.LinearDisjointnessofFields...........2506.ConstantFieldExtensions.............2547.CountingPointsinVarietiesOverFiniteFields..260BIBLIOGRAPHY.........................265 IntroductionGauss(1801)madeanextensivestudyofquadraticcongruencesmoduloaprimep.Heals0obtainedthenumberofsolutionsofthecubiccongruence3ax-by3=_1(modp)forprimesp=3n+i,andofthequarticcongruence4ax-by4~1(modp)forprimesp=4n+1.Hestudiedthecongruence4ax-by2~1(modp)forarbitraryprimesp.2Artin(1924)consideredthecongruencey_=f(x)(modp)wheref(X)isapolynomialwhoseleadingcoefficientisnotdivisiblebypandwhichhasnomultiplefactorsmodulop,andmadethefollowingconjecture:ThenumberNofsolutionssatisfies!N-Pl~2~/~ifdegf=3,IN+I-Pl~2~ifdegf=49ThisconjecturewasprovedbyHasse(1936b,c.).Infact,letFqbethefinitefieldwithqelements,andletNbethenumberof22solutions(x,y)EFoftheequationy=f(x),wheref(X)isqapolynomialwithcoefficientsinFandwithdistinctroots~ThenqIN-q!~2~/~ifdegf=3,IN+I-ql~2~ifdegf=49Supposef(X,Y)isapolynomialoftotaldegreed,withcoefficientsinFandwithNzeros(x,y)withcoordinatesinqFSupposef(X,Y)isabsolutelyirreducible,i.e.,irreducibleqnotonlyoverFq,butalsoovereveryalgebraicextensionthereof. Well(1940,1948a)tprovedthefamoustheorem(the"RiemannHypothesisforCurvesoverFiniteFields")that(i)IN-ql~2g~/q+Cl(d)wheregisthe"genus"ofthecurvef(x,y)=0andwhereCl(d)isaconstantdependingondItcanbeshownthatg~~(d-l)(d-2),hencethatIN-ql~(d-l)(d-2)~/q+Cl(d)Well'sproofdependsonalgebraicgeometry,inparticularonCastelnuovo'sinequality.AsomewhatsimplerproofwasgivenbyRoquette(1953);seealsoLang(1961),Eichler(1963).Morerecently,Stepanov(1969,1970,1971,1972a,1972b,1974)gaveanewproofofspecialcasesofWell'sresultwhichdoesnotdependonalgebraicgeometry,butwhichisrelatedtoThue's(1908)methodindiophantineapproximation.Thismethodconsistsintheconstructionofapolynomialinonevariablewithrathermanyzeros.Theconstructionisbythemethodofundeterminedcoefficients.Inparticular,Stepanovprovedthat(2)IN-ql~c2(d)~/~iff(X,Y)isofsomespecialtype,forinstanceiff(X,Y)=yd_f(X)wheredandthedegreeoffarecoprime.LaterBombieri(1973)andSchmidtproved(2)forabsolutelyirreduciblef(X,Y)bytheThue-Stepanovmethod.Itfollowsfromthetheoryofthezetafunctionthat(2)implies(i).IntheseLecturesweshallprove(2)bytheStepanovmethod.~The1940paperisonlyanannouncement. 3d~.Equationsy=f(x)andyq-y=f(x)References:Stepanov(1969,1970,1971,1972a),Mitkin(1972),Stark(1973).wi.FiniteFields(Galoisfields).LetFbeanyfield.ThereisasmallestsubfieldkgF(theintersectionofallsubfieldsofF),calledtheprimesubfieldofF,andeitherk=~ork=Fp,theintegersmoduloaprimeP9InthefirstcaseFisofcharacteristic0,inthesecondcaseofcharacteristicp.InthecasewhenFisfinite,k=Fp,and[F:Fp]isfinite.If,say,[F:Fp]=NKthenIF1=pHenceifFisafieldwithqelements,qthenq=p,pprime.LetFbeafinitefieldandletFhethemultipli-qqeativegroupofFqThenIF:I~=q-1.Ifx~Fq'thenxq-1=i;hence,forx~Fqwehavexq-x=0.There-fore,Xq-X=U(X-x).SoFisthesplittingfieldqxsqofXq-XoverF,andFisanormalextensionofFPqPMoreover,asasplittingfield,Fisuniqueuptoqisomorphisms.Conversely,letFbethesplittingfieldofXq-XoverFp,whereq=pLetXl,...,xqbetherootsofthispolynomialinF.TheserootsaredistinctsincethederivativeD(Xq-X)=-1I0.Nowx.+x.isarootofijXq-X,since, (xi+xj)q-(Xi+xj.)=xqi+xqJ-xi-xj=0andsimilarlyforx.-x..Alsox.x.isaroot,sinceiJiJ(xx.)q=xqxq.=xixjiJiJ'andsimilarlyxi/xjisarootifxj~0.Theserootsclearlyformafield,so,infact,F=(Xl,X2,.,.,Xq)9Thusafieldwithqelementsdoesexist.Consideringtheabove,wehave:THEOREMIA~IfFisafinitefieldoforderq,then--qKq=p,pprime.Foreverysuchq,thereexistsexactlyonefieldFThisfieldisthesplittingfieldofXq-XqoverFp,andallofitselementsarerootsofXq-XoTHEOREMIB.ThemultiplicativegroupFiscyclic.qFortheproofofthistheoremweneedLEMMAIC.LetGbeafinitegroupoforderd.Supposeforeverydivisoreofd,thereareatmosteelementsxEGwithxe=1.ThenGiscyclic.eThetheoremfollowsimmediately,sinceX-1hasatmosterootsinFItonlyremainstogiveaqProofofLemmaIC.EveryelementofGisofsomeordere,whereeId.Let~(e)bethenumberofelementsofGwhoseorderise.Either~(e)=0or~(e)~0.Suppose(e)~0,andletyEGhaveordere.Thentheelements2eey,y,...,y=1aredistinctandallsatisfyx=1.Sincethereareeoftheseelements,byhypothesistherecanbeenootherelementsxEGsatisfyingx=1. iNowletz6Gbeanyelementofordere;thenz=yi(i~ige).Noticethatz=yhasorderepreciselyif(i,e)=I.Hence~(e)=~(e),where~istheEulerrfunction.So,ingeneral,$(e)~~(e),takingintoaccountthepossibilitythat~(e)=0.Butd=~*(e)~~~D(e)=deldeldHence,foreverydivisoreofd,~(e)=~(e);inparticular,(d)=~(d)~0.Thatis,thereexistsanelementoforderd;hence,Giscyclic.COROLLARYID.Letq=pThenF=F(x)forsomex.qPProof.LetxbeageneratorofFqLetFFbefinitefields;thenr=q~.Considerthemappingqr0):F~Fsuchthat~(x)=xq.Thismappingisone-one.rrForsupposexq=yq,then0=xq-yq=(x-y)qwhencex-y=0andx=yoThemapping~isthenone-oneofafinitesettoitself,henceisonto.Moreover,~isanautomorphismofF,sincerO~(x+y)=(x+y)q=xq+yq=~(x)+~(y)and~0(xy)=(xy)q=xqyq=~0(x)~0(y)Infact,u0isanautomorpbismof"FoverF"(leavingFrqqfixed),sinceifxEF,~(x)=xq=x.Inotherwords,q isamemberoftheGaloisgroupofFoverFThemaprq9TT~iscaliedthe"Frobeniusautomorphism.K2K-IIfr=q,thenl,w,w,...,~areautomorphismsofFroverFq,andtheyaredistinctbecauseif~0i=~J(0~i,j~K-i),theni(x)=~J(x)forallxEF,riqjxq=xforallxs,rqiqjsox-x=0forallxEFrixqjButthedegreeofthepolynomialXqislessthanqK=r,sotheabovecannotholdidenticallyforallxEFr~ixqjunlessXq-isidenticallyzeroandi=j.SincetheorderoftheGaloisgroupisK,thesearetheonlyauto-morphismsofFoverFWehaveshown:rqTHEOREMIE.EveryautomorphismofFoverFisofrqtheformi(0~i~K-i),wherew(x)=xqThatis,theGaloisgroupofFroverFqiscyclicwithgenerator~.Recallthatthetraceofanelementisthesumofitsconjugates.ForthecaseF_cF,thetraceofanelementqrxEFisr2~-i~(x)=x+xq+xq+...+xq LEMMAIF.LetxEFr,withFq_cFThenthefollow-ringthreeconditionsareequivalent:(l)~x)=o.(ii)ThereexistsyEFrwithx=yq-y.(iii)ThereexistpreciselyqelementsyEFwithrx=yq-y.Proof:Exercise.NowletKbeanyfieldofcharacteristicp.Thenthemappingto:x~xpisanendomorphismofK.However,inthiscase,Wneednotbeonto.Example:LetK=F(X),pprime.ThenPto(a0+alX+.o.+atxt)=a0+alXP+...+atxtP.HeretO(Fp(X))=Fp(Xp).Itisclear,however,thattoisontowheneverKisalgebraicallyclosed.Letk[X]denotetheringofpolynomialsoverk9Letbethedifferentiationoperatordefinedasusual:atXt)t-1D(a0+alX+...+=al+2a2X+...+tatXTHEOREM1G.Letkbeafieldofcharacteristicp,andletMbeaninteger,M~poSupposea(X)~k[X]andforsomexEk0=a(x)=Da(x)=D2a(x).....DM-la(x) Thena(X)hasazeroatxoforderM;i.e.,(X-x)Mdividesa(X),orinsymbolsj(X-x)MIa(X).Proof:Writea(X)=co+el(X-x)+c2(X-x)2+...+ct(X-x)t.Then,D~a(X)=~'.p.+(~+l)~c~+l(X-x)+....+(~)ct(X-x)t-~]Substitutingx,for0~~lOOdmwherem=degfoIfNisthenumberofzerosofthepolynomial,then/N-q/~4d3/2mJq~ 11Note.Noparticularimportanceisattachedtothespecificvalues100dm2and4d3/2m.Thistheoremwasprovedbutwithdifferentvaluesofthecon-stants)inanelementarywaybStepanovin(1969)ford=2,moddandqaprime,thenin(1970)for(m,d)=1andqaprime,finallyin(1972a)ford=2,moddandqanarbitraryprimepower.AsomewhatsharperestimatewillbederivedinwofCh.II.TheellipticcaseofthetheoremwasfirstprovedbyHasse~936b,c).ThetheoremisaspecialcaseofWell'sfamoustheorem(1940,1948~onequationsf(x,y)=0,whichwillbeprovedinChapterIII.TheproofofTheorem2Awillbecarriedoutinthenextsectiong.LEMMA2B.SupposeX'=aX+bY+cY~=dX+eY+fisanon-singularlinearsubstitution;ioe.,dI0,withIa~1coefficients,a,b,c,d,e,finsomefieldk9Letf(X,~beapolynomialwithcoefficientsink.Thenf(X,Y)isirreducibleoverkifandonlyiff(aX+bY+c,dX+eY+f)isirreducibleoverkoProof:Exercise.LEMMA2C.Supposethepolynomiald_f(X)hascoefficientsinafieldk9Thenthefollowingthreeconditionsareequiva-lent:(i)yd_f(X)isabsolutelyirreducible.(ii)yd-cf(X)isabsolutelyirreducibleforeveryc~O,CEk.dIds(iii)Iff(X)=a(X-xI)...(X-xs)isthefactori-zationoffink,withx.~x.(i~j),then(d,dl,d2,~=1o 12Proof:Eachpartoftheproofwillbebycontrapositiono(i)~(ii)~Suppose(ii)isnottrue.Thenyd-cf(X)isreducibleover~forsomec~0,whenceYd(X))isreducibleoverk.ByLemma2B,yd-f(X)isreducibleoverk,contradicting(i).(ii)~(iii).Suppose(iii)isnottrue.Lett=(d,dl,..~s)>i.dl/tds/tPutg(X)=(X-xI)...(X-xs)Thenyd_lf(X)=Yd_g(X)tay(t-l)d(t-2)=(yd/t_g(X))(+Yg(x)+...+g(x)t-1).1ydSowithc=~~0,-cf(X)isreducibleink,con-tradicting(ii).(iii)~(i).Consideryd_f(X)asapolynomialintheringL[Y],withcoefficientsinthefieldL=k(X).WethenhaveafactorizationoverL:yd-f(X)=(Y-~i)...(Y-~d),where~i,.~dare"algebraicfunctions";Joe.,elementsof.Infact,wemayset~i=~l~'''''~d=Ca~'where~isanyrootofyd_f(X)in~,andwhere~I....'~dareelementsofkdefinedby 13dY-1=(Y-~i)...(Y-~d)9Supposethatyd_f(X)isreducibleoverk~Thenthereexistsaproduct(y-~il~).~(Y-~ih~)sk[X,Y]whereh1.SotI(d,dl,...,ds),t>1,andthelemmaisestablished.COROLLARY.Supposedegf=m.Thenyd_f(X)isabsolutelyirreducibleif(m,d)=i.Note:Ratherthanthemoregeneralconditionofabsoluteirreducibilityadoptedhere,Stepanovalwaysassumed(m,d)=1.LEMMA2DoLetCbeacyclicgroupoforderh.Foranyintegerd>0,letCdbethesubgroupofdthd'powersofelementsofC.Letd'=(h,d).ThenCd=C_,andconsistspreciselyofthosexECwith 14(2.1)xh/d'=i.ForanyxsCd,thereareexactlyd'elementsysCwithdy=x.Proof:WriteC={g,g2,...,gh=i}.Supposex6Cdidhenceisoftheformx=g,forsomei.Thensinced'Id,9id.hConversely,supposexh/d*=i9Wemusti~how_thereisadidlyE___Cwithy=x.Letx=gTheng=1;itifollowsthat~-.isaninteger,say,i=d'i.Ify=gJ,oweneedjdiod'g=x=gJorjd5id'(modh)oThiscongruencehasasolutionjsince(d,h)=d'dividesid'eMoreover,thenumberofsolutionsj(modh)equals(d,h)=d'.Since(29i)dependsonlyond',wehaveC__d=CC_d',andthelemmaisproved9dGivenanequationy=f(x)inFq,weareinterestedinthenumberN=N(d)ofsolutions(x,y)withcomponentsinFLetNObethenumberofsolutionswithy=09thenqNOisthenumberofx6Fwithf(x)=e.qNowconsiderthenumberofsolutionswithy~0.Forsuchasolution,f(x)E(F*)d,sobyLemma2D,q 15q-If(x)-~v-=12whered'=(q-l,d).q-Id'LetN1bethenumberofxEFwithf(x)=1.Forqsuchanx,thereared'elementsywithyd=f(x)HenceN=NO+dinIIThisexpressiondependsonlyupondI,soN=N(d~)with-outlossofgenerality,wemaythereforeassumethatdI(q-i);thenN=NO+dN1,q-idwhereN1isthenumberofxsuchthatf(x)=1~Finally,letN2bethenumberofxEFwithqIq-lld-1/q-lld-2+...+f(x)d+i=0.ButwehaveZqZ=Z(Z--~1Z-~(d-l)(d-2)d--+Z+...+Z+i.Now,sinceeveryz~Fsatisfieszq-z=0,andqZq-Zisaseparablepolynomial,everyelementofFisaqrootofoneandonlyoneofthefactorsofZq-Z,whenceq=NO+N1+N29Forfuturereference,wesummarize:LEMMA2E:LetNbethenumberofsolutions(x,y)EFXFqqofyd=f(x),wheredI(q-I)ThenN=NO+dNI,where 16NOisthenumberofxEFqwi.thf(x)=0,andN1isq-idthenumberofxEFwithf(x)=1.Further,qNO+N1+N2=q,whereNisthenumberofxsatisfying2(2~w3.Constructionofcertainpolynomials.InordertoproveTheorem2A,wemayclearlysuppose(3.1)m>i,d>19WeassumedI(q-1),and,forthemoment,that(d,m)=1,2wherem=degf.Alsoassumetemporarilythatq=porp,pprime.Forconvenienceletq-Id(3.2)g(x)=f(x)LEMMA3A:Supposeh0(X),hl(X),...,hd_l(X)arepoly-nomialsofthetypeh.(X)=kio(X)_+Xqk(X)+...+xqKkiK(X)iilfor0~iKd-i,andwheredegkij--0,i(i)(i)z(i)sz=c0+cI+.o.+ce_izInparticular,forxEFsatisfyinga(g(x))=O,wehaveqxq=xandi(i)g(x)=71ctg(x)t=OThenforsuchanx,D~r(x)=f(x)M-~(~)(x)g(x)tstt=Owhered-iK(~)st(x)=22c(i)k(~)(X)Xjtiji=oj=0SocertainlyD~r(x)=0forx6F,a(g(x))=0,qprovidedthepolynomialsst(L)(x)(o_(q-md)d(M+m+1)~qgq7M+-~-(m+i)-ms(2M),sinceM>-m+1.IfitisthecasethatBI00dm2,M>-~-4>_~--~~m+1Thereforesll 22Firstchoosea(Z)=Z-I;heres=1.Observethat|isthesetofxEFwitheitherg(x)=1orf(x)=OoqThus:1_3_qq2,I~I=N1+NO<~+4md2whence(4ol)N=dN1+NO~d/~1~q+4md3/2ql/2d-ISecondly,choosea(Z)=Z+...+Z+i.Herea=d-i.Now,={xEF:g(x)d-I+...+g(x)+1=0orf(x)=O}.qTherefore,i~1=N2+NO~d-1q+4rod~zq2lBut1lq4rndVq~N1=q-NO-N2>-~-whence(4.2)N~dN1aq-4md3/2ql/2Finally,combining(4.1)and(4.2),/N-ql~4md3/2ql/2Thisdoesnot,however,completetheproofofTheorem2Ainitsgenerality.Ithasonlybeenprovedunderthetwo2assumptionsthat(m,d)=1andq=porpWeshallproceedtoremovetheseconditions.w5.Removalofthecondition(re,d)=i.Theconditionthat(m,d)=1wasonlyrequiredinthe 23proofofLemma3AoThetaskbeforeusistoprovethislemmaundertheconditionthatyd-f(X)isabsolutelyirreducible.Remark:Recallthath.(X)wasapolynomialofthetype1h.z(X)=ki0(X)+Xqkzl(X).+...+XqKkiK(X)wheredegkijKq-moItiseasytoseethatforcEFq,hi(X-c)isapolynomialofthesametype.Hence,wemaymakeasubstitutionX~X-c,andreplacethepolynomialf(X)byf(X-c)oIfq>m,wemaychoosec6Fsuchthatf(-c)f0.Thereforeqwithoutlossofgenerality,weassumef(O)f0.First,weconsiderthecased=2.Assumethat~-f(X)isabsolutelyirreducibleandsuppose(5.1)h0(X)+hI(X)g(X)=0q-12orh0(X)=-hl(x)f(X)Squaring,weobtainh2(X)f(X)=hI(X)f(X)qThen,forsomepolynomial~(X),222(X)f(0)+xqz(x)ko0(X)f(X)=k0(X)f(O)q+xq~(x)=kloHeredegk20(X)f(X)~-q-2m+m=q-m2,westate,withoutproof,thefundamentaltheoremonsymmetricpolynomials.LEMMA5A:Supposea(Xl,...,Xd)isasymmetricpolynomial(i.e.,invariantunderanypermutationofthevariables)withcoefficientsinanyfield.Thenthereexistsapolynomialb(Ul,.o.,Ud),withcoefficientsinthesamefield,suchthata(XI,...,Xd)=b(Sl(X1,.oo,Xd),...,sd(X1,...,Xd)),wheresI=-(X1+X2+...+Xd),s2=XIX2+...+Xd_iXd,osd=(-i)dXlx2...xdMoreover,(a)Ifa(Xl,...,Xd)isofdegree5ineachXi,thenbisoftotaldegree89 25(b)Ifa(Xl,~d)isoftotaldegrees,theneachiimonomialU11..oUddofhwithnon-zerocoefficientshasthepropertythatiI+2i2+~+did=EFormapolynomiald-ia(Y;HO,~176I)=H0+HIY+.~+Hd_IYLet__CI,...,~dbeelementsofF--withqdX-1:(X-_.<1)...(X-~d)andputdb(Y;H0,...,Hd_I)=--~a(~iY;H0,..~I)i=lThenbisapolynomialsymmetricin~iY,...,~dY.ByLemma5A,bmustbeapolynomialintheelementarysymmetricfunctionssl,o.~dof~iY,...,~dY.Butinourcase,dsI=..~=Sd_1=0andsd=-ysothatb(Y;H0,...,Hd_I)=c(Yd9H0Hd_I)Herec(W;H0,...,Hd_I)isapolynomialofdegreed-1inW,andofdegreedinH0~...,Hd_1.Nowsetd(U)VjH0)..o,Hd_I)=vd-lc(u/V;Ho,...,Hd_I)Thenisaformofdegreed-1inU,V;andofdegreedinH0,...,Hd_1. 26Wenowassumeyd-f(X)tobeabsolutelyirreducible.Suppose(5.2)h0(X)+hl(X)g(X)+~+hd_l(X)g(X)d-1=0.Withtheabovenotation,a(g(X);h0(X),.,.,hd_l(X))=0,andweobtainc(g(X)d9h0(X),hd_l(X))0q-1Recallingthatg(X)=f(X)d,weobtaing(X)d=f(x)q/f(X)andd(f(X)qf(X);ho(X),hd_l(X))0CollectingalltermswithnofactorofXq,(5.3)d(f(0),f(X))ko0(X),...,kd_l,0(X))+XqZ(X)=0,forsomepolynomial~.Now,(5.4)d(f(0),f(X)jk00(X),...,kd_l,0(X))isofdegreed-1inf(0),f(X),andofdegreedink00,...,kd_l,0Butdegkij~~q-m,sothepolynomial(5.4)isofdegree~_(d-l)m+dlq-ml----0~...,it~->0iI+o9+it=2Proof.Itwillsufficetoprovethecaset=2,sincethegeneralcasefollowsbyanobviousinductionont.Thuswehavetoshowthat(i)(L-i)(6.1)E(~)(f(X)g(X))=E(f(X))E(g(X))i=OBythelinearityofE(j),wemaysupposethatf(X),g(X)aremonomials;sayf(X)=Xag(X)XbThen(6l)isequivalentto----~i~-ii=O 29Butthisidentityisanimmediateconsequenceofthedefinitionofasthenumberofsubsetswith~elementscontainedinasetofa+belements.COROLLARY6B.E(~)(X-c)t=(~l(X-c)t-s(~)t(i1)(it)Proof.E(X-c)=~(E(X-c))...(E(x-c)).iI>O,.oo,lt--=">0il+~(i)(i)NowE(X-c)=1andE(X-e)=0ifi>=2.Henceintheabovesum,weneedonlyconsidersummandswitheachi.either0ori.ThenumberofsuchsummandsisJ;),andeachsummandis(X-c)t-~COROLLARY6C.Suppose0~~~t.Then(6.2)E(~)(a(X)f(X)t)=b(X)f(X)t-~,whereb(X)isapolynomialwithdegb=dega+~((degf)-l)Proof.In(io)a(i1)(it)E(~)(a(X)f(X)t)=~,(E(X))(Ef(X))...(E(f(x))io>0,=...,itS0io+-..+it=~everysummandisdivisiblebyf(X)t-~.Henceaformulasuchas(6.2)holds.Furthermore, 30degb=deg(E(~)(aft))-(t-sdegf=dega+tdegf-s-(t-~)degf=dega+~(degf-1).THEOREM6D.SupposeE(~)(f(x))=0for~=O,1,...,M-1.Then(X-x)Mdividesf(X)oProof.Wemaywritef(X)=a0+al(X-x)+..~+ad(X-x)d.ByCorollary6B,E(s=a~+a~+l(x-x)+..,+ad(x-Thehypothesisofthelemmaimpliesthata~=0for%=O,1,.~andtheconclusionfollows.LEMMA6E.Supposekisofcharacteristicp>0~Letr(X)=h(X,X)forsomepolynomialh(X,Y).Thenfor~<~,E(~)r(X)=E(~)h(X,Xp~)Xwhere~X-(~)isthe"partial"hyperderivativewithrespecttoXofh(X,Y)9Proof.Bylinearity,itsufficestotakethecasewhenh(X,Y)=xayb.ThenbyLemma6AitsufficestoshowthatforOdm2HoM.Stark(1973)obtainedthesharperbounds1/2IN-q/<=(m-1)q,ifq=pandiff(X)hasmdistinctroots.Setifmisd,gImp2ifmisevenThenumbergiscalledthe"genus"oftheequation,,ThusStarkobtainsIN-ql-<2gql/2'ifmisodd,(s.1)_3;hence~>-1.Let'~~+IK-Ig(X)=f(X)q+f(X)q+..o+f(X)q,h(X)=f(X)+f(X)q+...+f(X)qLEMMA9C:Letw~Fbefixed.LetMbedivisiblebyq,qE-%)-Iand0r_:_~+3Nw=r-NvqqvSw~+3and,inparticular,r<~+3/No-CIqByLemma9Bagain,K~+4[~]+4//l~-rl100dm2,wehave(2.1)1~")((f(x))t<5md3/2ql/2xEFqThisresultwillturnouttobeaconsequenceofTheorem2AofCh.I.Weshallalsoprove 43THEOREM2B~SupposeXisacharacteroforderd>1.dthSupposef(X)EFIX]isofdegreemandisnotapower,qi.e.notofthetypef(X)=c~(X))dwithcEFand~(X)EFIX]9qq2Thenifq>100dm,wehaveagain(2.1)Lateronweshalltakethe"highroad"andprovethefollowingsharperresults.THEOREM2C.SupposeX~Xoisamultiplicativecharacterofexponentd.Supposef(X)EF[X]haspreciselymdistinctonesqamongitszeros,andsupposethatya_f(X)isabsolutelyirreducible.Then(2.2)I~x(f(x))l<(m-1)ql/2xEFqTHEOREM2CI.LetXbeoforderd>1.Supposef(X)hasdthmdistinctonesamongitszeros,anditisnotapower.Thenagain(2.2)holds.WenowturntoadditivecharactersofFSuchanadditiveqcharacterissimplyacharacteroftheadditivegroupofFqIfq=pwherepisthecharacteristic,thenthisadditivegroupisthedirectsumof~copiesofCWrite~forthetrace=PfromFtoFqPLEMMA2D.ForeveryaEFq,thefunction~awitha(X)=e~(ax)/p)isanadditivecharacterofFEveryadditivecharacterofFqqisofthistype. 44Proof.Wehave~(a(xl+x2))=~(axI)+~(ax2),whence~a(Xl+X2)=~a(Xl)~a(x2)9Thus~aisanadditivecharacter.ByTheorem1C,thenumberofadditivecharactersisq;butsoisthenumberofelementsaEFqSince,asiseasilyseen,~aI~a''ifa~a~,itfollowsthatasarunsthroughF,then~arunsthroughalladditivecharacters.qAdditivecharacterswillalwayshedenotedbytheletter~.Thecharacter4owith~o(x)=1forallxistheidentityelementofthegroupofadditivecharacters.THEOREM2E.Suppose~d~oisanadditivecharacter.Letg(x)beapolynomialinF[X]ofdegreen.Supposethateitherq(i)ni0063m2n2,thenumberNofsolutions(x,y1,-..,yn)sFn+loftheequationsinthetitlesatisfiesqIN-ql<5mnd65/2qI/2Proof.Writed!=g.c.d.(d.,q-1).Byanargumentusedin11Ch.I,w,thenumberofsolutionsoftheequationsinthetitleisthesameasthenumberofsolutionsofdtd/1n(5.4)Yl=fl(X)"'"'Yn=f(x)'ndIMoreover,writedi=dZieiandlet~....,~satisfy~ii=f.(X)Ie(i=i,...,n),andlet~I'''"'~nhave~ii=~i'(i=i,...,n)Then(5.2)andhence(5.3)holds.Wehave[>q(X,m~,...,re'n):~(x)]~d'...d'qin'I/[Fq(X,~l,...,~]n):Fq(X,~I,"'"'~-n)]gel"'"enHenceinviewof(5.3),[>q(x,<~,...,~):~q(X)]=%.'..d'n:d',say.Thereforethesystemofequations(5.4)alsosatisfiesthehypothesisofthetheorem.Wemaythereforesupposewithoutlossofgeneralitythat(5.5)dil(q-i)(i=i,...,n) 54LetXbeacharacteroforder5,andletXibethecharacter5/d.1X.=X(i=i....,n).1ThenXiisoforderd.ThecharactersofexponentdareIid-i02iXi=Xo,Xi,Xi,"'",XidiByLemma2A,thenumberofYEFwithy=wequalsqdi-iXofexp.dj=OiHencedl-idn-iJlN=9""XOnn(fn(x))xEFqJl=0On=0J16/dI"'"fn(x)JnS/dn))X(fl(x)J1=0Jn=OxFqThemaintermisforJl.....Jn=0,anditequalsq.Theothersummandsarecharactersums~,X(g(x))xEFqwithJlS/dIJn6/dnJl5Jn5g(X):fl(X)''"fn(X)=~l"''~n9.6thhavingJl'"'.Jn~O,...,0.Ifg(X)wereapowerinJlJn[x]then~I"''~n~~[x].Butinviewof(~.S),theq'q 55Jl3nelements~i9"'~nwith0~Ji0.COROLLARY5B.Lettbeafixedpositiveinteger9Foraprimep,letL=Lt(P)bethenumberofx(modp)suchthatx+l,x+2,...x+tare(non-zero)quadraticresiduesmodp.Thenforlargep,L=--%Pt+o(pl/2)2DeductionoftheCorollary.InthefieldF,considerthePsystemofequations22(5.7)Yl=x+l,...'Yt=x+t. 56InthenotationofTheorem5A,m=1and,sinced=2forl1giKt,wehaved=2t.Let~l'"'"'~]tbequantitieswith22~i=x+l.....~t=x+t.InordertoapplyTheorem5Atothisease,weneedthat[Fp(X,~I,...,~t):Fp(X)]=2tThisistrueifp~t,PI2,asmaybeshownasanexercise.Infact,thereadermightwanttodothefollowingExercise.LetDbeauniquefactorizationdomainofcharacteristic2withquotientfieldK.LetPl'"'"'PtbedistinctprimesinD9Then....El:2t(SeealsoBesicovitch(1940)).IfNisthenumberofsolutionsofthesystem(5.7),thenbyTheorem5A,IN-Pl=O(pl/2)IfNtisthenumberofsolutionswithx+l,...,x+tallnon-zerothenIN-NIl=O(i),sothatI'-pl:O(p1/2).tSinceN'=2L,theCorollaryfollows. 57wAuxiliarylemmason~i+...+t0ZGivenacomplexvaluedfunctionf(~)andarealvaluedfunctiong(~)>0,theVinogradovnotationf(v)<O9If(6.1)0~+...+W_<0, 58c>0.If...B~(~=1,2,...)thenlwjl~l-~q|%""+;~I~,forgivene>0.FortheproofweshallneedDirichlet'sTheoremonSimultaneous~pproximations:LEMMA6D.Le__~teI,...,9shereal.Thereexist~+l)-tuplesofintegersv,ml,...,mwitharbitrarilylargeV>0andwithmil-(1/~)(e.5)lei-%-I<~-(i=i,...,~)9Proof.Writec~=[~]+{~},where[~]istheintegerpartofc~,i.e.theintegerwith~-1<[oL]gff,andwhere{o~}isthefractionalpartofoz,i.e.,thenumberwith0g{~}<1suchthat~-{~}isaninteger. 59NowsupposeN>0isaninteger.Thepoints(6.6)({u@I})1....,~uo~withu=0,i,...,~are~+1pointsinthehalfopenunitcube0~xI0,wemayset=tb,m1=tal,...,m=tawitht=1,2,.... 60ProofofLemma6C.Observethatforreal@,Write~j=IWjle(ej)withreal@jTherewillbeinfinitelymany~,andintegersml,...,ms,havingl~@j-mjl=l~jl"~>,whence(6.4).wFurtherauxiliarylemmas.LEMMA7A..Let~,mbepositiveintegers.Writing(~,m)=g.c.d.(~,m),wehavethepolynomialidentity(7.l)v(1-e(mu/~)X)=(1-XT~-j~)(.,m)u=lProof.Inthecase(~,m)=1,theidentityreducesto(1-e(mu/~)X)=1-Xvu--1Itiscorrectinthiscase,sincebothsidesarepolynomialsofdegree~withconstanttermiandwithrootse(-mu/v)(u=I,....v)Ingeneral,put~=Vl(~,m),m=ml(~,m)Asurunsthrougharesiduesystemmodulo~,itruns~,m)timesthrougharesiduesystemmodulo~iThus(7.1)isobtainedbyraising 61(7.2)(i-e(mlU/~l)X)=1-X1u=ltothe(~,m)thpower.But(7.2)iscorrectbythespecialcasealreadyconsidered,since(~l,ml)=1.dd-iLEMMA7B.Leth(X)=X+alX+...+adbeanirreduciblepolynomialinF[X].TheninF[itsplitsintor:(,,d)....qvqirreduciblepolynomialsofdegreed/r:(7.3)h(X)=hl(X)...hr(X)Ifwenormalizeh.(X)suchthatitsleadingcoefficientis1,1thenh(X)EF[X](i=I,..r)Theelements~ofthe-ir"'q,7GaloisgroupofFr/Fqpermutethepolynomialshl,...h'rqGivenhi,hj,thereisaeintheGaloisgroupwitho~h=h1jProof.ConsiderthefieldsFFdq/qFrq1FqTherootsofh(X)arealgebraicofdegreedoverF,henceqlieinFdTheyarealgebraicofdegreed/roverFrHenceqqinFr'thepolynomialh(X)hasthefactorization(7.3),whereqeachh.isofdegreed/randisirreducibleoverFSince1rqt)welet~operateonthecoefficientsofthepolynomials. 62(d/r,~/r)=1,therootsarestillofdegreed/roverF,qandhencethepolynomialsh.(X)arestillirreducibleover1F~)Theelements(5oftheGaloisgroupG=G(Fr/Fq)leaveqqhinvariant;hencetheypermutehl,...,hrGiveniin1~i~r,thepolynomial~hiisinvariantunderG,henceliesinFIX].Ithasrootsinqcommonwiththeirreduciblepolynomialh,henceequalsh=h1.~hrSoas(yrunsthroughG,thenuh.runsthroughhl,~,hr.1wZetaFunctionandL-Functions.Throughout,h=h(X)willdenoteamonic(i.e.withleadingcoefficientl)polynomialwithcoefficientsinFIfh(X)isqofdegreed,putd(h)=qForcomplexs=~+it,putC(s)=~1h~l(h)sHerethesumisovermonicpolynomialsh6F[X].q 63THEOREM8A.(i)Thesumfor~(s)isabsolutelyconvergentfor>1,infactuniformlyconvergentforq>qo>i.(ii)For~>1,~(s)=~(1-~(h)-S)-1hirred.wheretheproductisoverirreduciblemonicpolynomialsinF[X]q(iii)~(s)=--1-s1-qProof.(i)N(d)q(8.1)~(s)=ds=~'d---Oqd=OqwhereN(d)isthenumberofmonicpolynomialsofdegreed.Thesumontherightisclearlyabsolutelyconvergentif~>1,anduniformlysoif~>(~o>i.(ii)Sinceeverypolynomialmayuniquelybewrittenasaproductofpowersofirreduciblepolynomials,wehave,for(~>1,11C(s)=~-(i+--+--+...)hirred.~(h)s9~(h2)s-s-1(1-~(h))ghirred.=(iii)followsimmediatelyfrom(8.1).Remark.Wecall~(s)aZetaFunction.Itisalmost(butnotquite)theZetaFunctionofthe"functionfield"F(X)ForaqreaderfamiliarwithZetaFunctionsoffunctionfields,weremarkthe 64following.TheprimedivisorsoftherationalfunctionfieldFq(X)consistofprimedivisorswhichcorrespondtoirreduciblemonicpolynomials,plusthe"infinite"primedivisor.OurZetaFunctiondiffersfromtheZetaFunctionofthefieldFq(X)inthatintheproduct(ii)thefactorcorrespondingtotheinfiniteprimedivisorismissing.Thisiswhywehave~(s)=(i-ql-s)-i,whiletheZetaFunctionofthefunctionfieldis(i-q-S)-l(l-ql-S)-iLetGbethegroupofrationalfunctionshl(X)/h2(X),wherehi,h2aremonicinF[X].LetGbeasubgroupofGsuchthatq(8.2)ifhlh26G,thenhI,h2EforpolynomialshI,h2.Let~beacharacteronGWeextendthedefinitionofSbysetting~(h)=0ifhisapolynomialnotinGoThenstilll(hlh2)=~(hI)~(h2)formonicpolynomialshI,h2.Fors=~+it,putL(s,I)=~~(h)~(h)-s,h=wherethesumisovermonicpolynomialshEF[X].qTHEOREM8B.(i)ThesumforL(S,~)isabsolutelyconvergentfor~>1,infactuniformlyconvergentfor~>~o>1.(ii)For~>i,L(sl)==h-S-ihirred.Proof.Everythingworksalmostthesameasinparts(i),(ii)ofTheorem8A.Thedetailsareleftasanexercise.Remark.TheexpertswillseethatourfunctionsL(s~I)areL-FunctionsassociatedwiththefunctionfieldF(X).q 65wSpecialL-Functions.Letf(X)beafixedmonicpolynomialinF[X]9InF[X]qqaIamitfactorsinto(X+~i)..~(X+~m),say.LetGbethesubgroupofGconsistingofrationalfunctionsr(X)=hl(X)/h2(X)havinghl(%{i)h2(~i)~0(i=1.....m)t)ThenGsatisfies(8.2).Forr(X)EG,putaa(9.1){r}=r(~l)1...r(~m)mIfr(X)=(X+~i)...(X+O~u)(X+81)-l...(X+~v)-Ithen-i{r}=f(~l)...f((~u)f(~l)-I...f(~v)Always{r}EFand{rlr2}={rl}{r2}.ThusifXisamultiplicativeqcharacterofF,thenqX({rlr2})=X({r1})X({r2})ThereforeX({r})isacharacteronthegroupG.LetHbethesubgroupofGconsistingofr(X)=hl(X)/h2(X)withhl(Ti)=h2(~i)%0(i=l,...,m).LEMMA9A.X({r})=1forr6H9Proof.Obvious.Letg(X)beafixedpolynomialinFq[X],ofdegreenandwithconstanttermzero.Givenr=r(X)EG,put[r]=0ifr(X)=1,andT)Put(r}=1iff(X)=1.*)Weallown=O,g(X)=0. 66(9.2)[r]=g(o~l)+...+g(~u)-g(S1)....-g(Sv)-i)-lifr(X)=(X+al)...(X+~u)(X+~i)...(X+~vwith~i'"''%'B1.....~vin}Then[r]eFand[rlr2]=qq[rl]+[r2]Thusif*isanadditivecharacterofF,thenq([rlr2])=9([rl])~([r2])9Thus$([r])isacharacteronthegroupG.LetHbethesubsetofGconsistingofrationalfunctionsr(X)=h1(X)/h2(X)having(9.3)hI(X)=Xu+aIxu-I+...+ah2(X)=Xv+bIXv-I+...+bu'vwith(9.4)a1=bI,a2=b2,...,a=bnnForexample,polynomialsXulieinH,andsodopolynomialsuu-n-iX+an+iX+...+awithu>n.ItiseasilyseenthatHisausubgroupofG.LEMMA9B.~([r])=1ifrEH9Proof.In(9.2),g(~l)+...+g~u)isasymmetricpolynomialofdegreenin~i'"'"'~uHenceitisapolynomialinthefirstnelementarysymmetricpolynomialsin~i'"'"'~u'i.e.,inthecoefficientsal,...,ain(9.3):ng(~l)+...+g(~u)=~l(al,...,an)withapolynomial%1"Similarly, 67g(~l)+"'"+g(~v)=~2(bl'"'"'bn)Sincethetwosymmetricfunctionsg~l)+...+g(~u)andg(~l)+...+g(~v)haveconstanttermzero,andsincetheyare"thesame",exceptperhapsforthenumberofvariables,thetwopolynomials~iand22arethesame.Thus(9.4)impliesthat[r]=0,whence~([r])=1.NowputX(r)=({r})~([r])Then~willbeacharacteronthegroupG.LetHbetheintersectionH=HN~9ThenHisasubgroupofG,andwehavetheCOROLLARY9C.I(r)=1ifrEH9LEMMA9D.Suppose~~0.TheneverycosetofHincontainspreciselyqpolynomialsofdegreen+m+Z.Proof.Itwillsufficetoshowthatifr(X)isinG,thentherearepreciselyq~polynomialsk(X)=Xn+m+~+blXn+m+~-I+...+bwithk(X)/r(X)EH9Ifr(X)hastheexpansionn+m+~r(X)=Xu+alxu-I+a2xU-2+...,thenthisconditionmeansthat(9.5)bI=al,...,bn--anandthat(9.6)k(~i)=r(Ti)(i=I,...,m)Thecoefficientsbl,...,bnaredeterminedby(9.5).Pick 68bn+l,.;,,bn+~arbitrary.Thentherelations(9.6)arem(non-homogeneous)linearequationsinthemremainingcoefficientsbn+~+l,...,bn+~+m9Thematrixofthissystemofequationsis(y~)(1Ki%m,0gjgm-l).ThedeterminantisaVanderMondedeterminant.SinceYl'"'"'Ymaredistinct,thedeterminantisnon-zero.Thuswecansolvethesystem(9.6)uniquely.Henceourfreedomconsistspreciselyinpickingbn+l,...,bn+~Thisgivesqpossibilities.LEMMA9E.SupposethateitherX~Xoisofexponentdandyd_f(X)isabsolutely(9.7)irreducible,or~~~oandeither(i)(n,q)=1or,moregenerally,(ii)Zq-Z-g(X)isabsolutelyirreducible.ThenthecharacterXisnotprincipal,i.e.,~(k)~1forsomekE~.Proof.SupposeX(k)=i.ThenX({k})~([k])=I.SincedththX({k})isarootofunityand~([k])isaprootofunitywith(d,p)=1,itiseasilyseenthatX({k})=,([k])=1.HenceinthefirstcaseofthelemmaitwillsufficetofindakwithX({k})~1,andinthesecondcaseitwillsufficetofindakwith~([k])i1.IfX~Xo,supposeittobeoforderewitheld.SincedY-f(X)isabsolutelyirreducible,notalltheexponentsin 69aIamf(X)=(X+~i)...(X+~m)aremultiplesofe.(SeeLemma2CofCh.I)9Sayefa1Givenc2,9..,eminFq,wecantherefore9aIa~al.c~m)pickcIEFwithcI...cmmff(F)e,whencewith~(cI../1.qBytheargumentofLemma9D,thereisapolynomialk(X)EGwithk(~i):ci(i=l,...,m)aIamThen{k}=cI...CmandX({k})j1.If~J~o,supposefirst~(i)~that(n,q)=1andf(X)=I.say,g(X)=aXn+gl(X)whereglisofdegreen+m,byLemma9F.HenceL(s,X)isapolynomialinUofdegree1anddlq-1.TherearedcharactersXofexponentd.ForeachsuchcharacterX,wemaydefinethesumsS=SandS=SX~XWethenhaveforX~Xo,(Zl.3)SXv=-reX1-''"-C~TakingthesumoverXIXoofexponentd,weobtainm-i(n.4)XdXoX~Xoi=lofexp.dofexp.dOntheotherhand,forX=Xo,(Ii.i)yields(11.5)S=qXo,O~,LEMMAIIA.Forgivenw6F,thenumberofYEFwithqqdy=wequalsv~x(w)=~x~(w)).'9'9XXofexp.dofexp.dProof.Wefirstnotethatthemapw-~~(w)isagrouphomomorphismF-~FForeachzEF,thenumberofwEFwithqqqq~-l(w)wl+q+...+q~-~Z 79is~l+q+...+~'q~-i=(q'---1)/(q-i)=It*'I/IF'I~*","hencethenumberqofthesewisexactlythisnumber,andourhomomorphismisonto..*dTherestrictionofthemapto(F*~)disamap(F*))d-*(Fq)andqqcomparingeardinalitiesweseethatitisontoagain.AccordingtoLemma2A,thesuminLemmaIIAisdor0ori(F*)d~)drespectively,if~(w)Eqor~(F,~0or=0.Inthefirstcase,bywhatwejustsaid,wE(F*)d,andtherearedqd(-F*)delementsywithy=w.Inthesecondcase,w~,i0Mqdandtherearenosolutionsywithy=w.Inthethirdcase,w=0,andthereisthesinglesolutiony=09WritingNforthenumberofsolutionsx,yinFofqMdY=f(x),weimmediatelyobtainLEMMAlIB.MXofexp.dx~Xofexp.dXMNowweknowfromTheorem2AofCh.Ithatifyd_f(X)isabsolutelyirreducible,then(11.6)N_q~<2ofthetypeXuu-2X2.Itis_+a2X+...+au_2+aueasilyseenthatHisasubgroupofG.AsananalogofLemma9B,wenowobserve 87LEMMA12B.~(r)=iifrEH9Proof.Ifr6~{,then~i+"''+~u-~i-"''-~v=al-bl=0,ai11iu-ibv-i=0,--+'''+---~-'''---=ab~i~u1~vuvsothat[r]=0.TheanalogofLemma9DisLEMMA12C.Suppose~>0.TheneverycosetofHincontainspreciselyq~(q-l)polynomialsofdegreeA+3.Theproofofthisisleftasanexercise.CarryingouttheobviousanalogtotheargumentinwoneseesthattheL-Function~-sL(s,)isapolynomialinU=qofthetypeL(s,U)=1+ClU+c2u2=(i-miU)(1-~2U)withCl=x~F*r-1)qql/2Thusitsufficestoshowthatl~il~(i=1,2).ThisisaccomplishedbyshowingthatthenumberNofsolutionsx,zinMFofx/0,zq-z=ax+bx-I,satisfies(67.1_._..1SinceqclearlyaX2-(zq-Z)X+hisabsolutelyirreducible,thisfollowsfromTheoremIAofCh.III. 88wFurtherResults.Let~=4obeanadditivecharacter.Letg(X)beapolynomialofdegreenwith(n,q)=1andwithconstanttermzero.WeknowfromTheorem9GthatifX(r)=9([r]),where[r]isdefined-Sasinw,thenwithU=q,L(s,~)=1+ClU+...+Cn_lUn-IWenowprove(n-I)/2T~OREM13A.len_ll=qProof.Wehave~(h)an_I=hmonicdegh=n-1sothatICn_112=~(hl/h2)hhI2degn-idegn-1Now~(k)dependsonlyonthecosetCofkmodulothesubgroupHofG.Thus~13.1~/Cn_ll2=Zy~c~z~c~CwherethesumisovercosetsCofHinG,andwhereZ(C)isthenumberofpairsofmonicpolynomialshI,h2ofdegreen-1withhl/h26C. 89WewriterI~r2(mod~ifrl/r2sH,i.e.,ifrl,r2lieinthesamecosetC9Ifweexpandtherationalfunctionsasuu.-Iu.-2i1lri(X)=X+ailX+ai2X+...(i=1,2)thenrI~r2(modH)ifandonlyifall=a21,...,aln=a2nThusifC(Vl,Vn)consistsofrationalfunctionsr(X)XuXu-I...,=+aI+.~withaI=Vl,...a=vthenthesetsC(Vl,...,Vn)arejust'nn'thecosetsofHinG9n-1n-2Nowhl/h2withhI=X+alX+...+an_1,h2=xn-in-2+blX+...+bn_1liesinC(Vl,...,Vn)preciselyifa1=b1+v1a2=b2+blV1+v2,(13.2)an_1=bn_1+bn_2Vl+...+blVn_2+Vn_1,0=bnlVl+...+blVn1+vThusZ(C(Vl,...,Vn))issimplythenumberofsolutionsinal,...,an_1,bl,...,bn_1inFof(13.2).qLEMMA13B.In-2q~ifVl,...,Vn_1arenotO,...,0n-1Z(C(Vl,...,Vn))=qifv1.....Vn_1=v=0n0i...f_fv1.....Vn_1=0,vn#0. 90Proof.Thenumberofsolutionsal,...,an_1,bI,...,bn_IinFqof(13.2)isjustthenumberofsolutionsbl,...,bn_linFofthelastequation(13.2).qInviewof(9.8),wehaveLEMMA13C.X(C(0,...,0,v))=~((-l)n+in~V),whereaistheleadingcoefficientofg(X).TheproofofTheorem13Aisnowcompletedasfollows.By(13.1),Lemmas13B,13C,andsinceX(C(0,...,0))=1=~(0)weobtain]en-iI2=~...~I(C(vI,...,Vn))Z(C(vI....,Vn))vIvnn-2:qZ...ZI(C(Vl,"",vO)vIvn+(qn-1-qn-2)~(C(O,...,0))_qn-2v~i0~(C(O,...,0,v))Herethefirstsummandiszero,sinceC(Vl,...,Vn)rangesthroughallthecosetsofHinG.Combiningthesecondandthirdsummand,weobtainn-in-2~-Iqq~_~--*((-1)n+ln~-V)r..~/vTheproofofTheorem13Aiscomplete.NowweknowthatinL(s,X)=(i-Wl~...(i-~n_l~,theabsolutevaluesl~jlKql/2(j=i,...,n-l)ButinviewofTheorem13A,wenowhave 91(13.3)l~jl=ql/2(j=1....,n-l).COROLIARY13D.Letg(X)beofdegreenwith(n,q)=1,andlet~4obeanadditivecharacterofFThenqs,=~~,~(g(x))xEF%9qisoftheformS=-toI-"""-~n-1whereel'"'"'~n-ihave(13.3).iInparticular,neithertheexponent~northeconstantfactorn-1inTheorem2Emayheimproved.Infact,byLemma6C,wehaveCORO~RY13E.LetSbeasabove.ThereareinfinitelymanypositiveintegerswithIS%91>(n-l)J/2(i-2~-I/(n-l)).1Similarly,neithertheexponent~northefactor(q-I)(n-i)in(11.14)maybeimproved.Theargumentsofthissectionmaybecarriedover,withsuitablechanges,tomultiplicativecharactersumsandhybridsums. III.AbsolutelyIrreducibleEquationsf(x,y)=0References:Stepanov(1972b,1974),Schmidt(1973).wIntroduction.ThischapterisdevotedtoaproofofTHEOREMIA.Supposef(X,Y)6Fq[X,Y]isabsolutelyirreducibleandoftotaldegreed>0LetNbethenumberofzerosoffinF2Ifq>250d5,then--q--(i.1)IN-ql<~/~d5/2ql/2Asiswellknown,thisestimatefollowsfromtheRiemannHypothesisforcurvesoverfinitefields,whichwasfirstprovedbyWell(1940,19486).Infact,theRiemannHypothesisgivesthestrongerestimateIN-ql=<(d-i)(d-2)qI/2+c(d)forsomeconstantc(d)SpecialcasesofTheoremIA(butwith~2d5/2replacedbysomeotherconstantdependingond)wereprovedbyStepanovbyelementarymethods;hismostgeneralresultwasin(1972b,1974).Stepanov'smethodwasextendedbySchmidt(1973)toyieldTheoremIA,andalsobyBombieri(1973).Inordertoprovideeasyexamplesofabsolutelyirreduciblepolynomialsf(X,Y),wenowstateTHEOREM1B.Letf(X,Y)=g0Yd+gl(X)yd-l+...+gd(X) 93wheregoisanon-zeroconstant,beapolynomialwithcoefficientsinafieldkPut1(f)=maxTdeggiisi=~a+~b=~d.Henceinfactdegu(X,Y)=~aanddegv(X,Y)=~bItfollowsthatdeguj(X)2d,(whichisthecaseinTheorem1A),wecanchoosecEFqsothata.(c)~0and(c)~0S0d~0J0dTheninthepolynomialh(X,~,Yoccurswithanon-zerocoefficient;moreover,hisseparableinY.Dividingbyanappropriateconstant,weachievethefollowingReduction:Withoutlossofgenerality,wemayassumethatf(X,Y)=yd+gl(X)yd-i+...+gd(X),deggi(X)~i,andthatf(X,~isseparableinYo 97wIndependenceresults.Webeginwithasimpleremark.Supposef(X,Y)isapolynomialwithcoefficientsinafieldKandofdegreed>0inY.Supposef(X~isirreducibleoverK.Thenifweregardf(X,~asapolynomialinYwithcoefficientsinthefieldK(X),itisstillirreduciblet).Henceif~satisfiesf(X,~)=0,then[K(X,~):K(X)]=d9LEMMA2A.Supposef(X,Y)andg(Z,U)arepolynomialswithcoefficientsinafieldK,bothabsolutelyirreducibleoverK9Supposefisofdegreed>0inYandgisofdegreed'>0inU.Let~,Ubequantitieswithf(x,~)=0,g(z,a)=0.(sothat[K(X,~):K(X)]=dand[K(Z,U):K(Z)]=d'.)Then[K(X,Z,~,~):K(X,Z)]=dd'Remark:Theabsoluteirreducibilityoffandgisessential.Bywayofexample,takeK=Qandt)Forsupposetothecontrarythatf(X,Y)=gl(X,Y)g2(X,Y),wherethegiarepolynomialsofpositivedegreeinYwithcoefficientsinK(X).Givenanypolynomialg(X,Y)inYwithcoefficientsinK(X),wemayuniquelywriteg(X,Y)=(u(X)/v(X))~(X,Y),whereu(X),v(X)arecoprimepolynomialswithleadingcoefficient1,andwhere~(X,Y)=c0(X)+Cl(X)Y+...+ct(X)ytwithcoprimepoly-nomialsc0(X),...,ct(X).Writer(g)=u(X)/v(X).SinceK[X]hasuniquefactorization,itcanbeshownthatr(glg2)=r(g1)r(g2)9(ThisissimilartoGauss'Lemma.)Nowifthepolynomialf(X,Y)aboveisirreducibleoverK,wehaver(f)=i,whencer(gl)r(g2)=i.Thusf(X,Y)=r(gl)r(g2)gl(X,Y)g2(X,Y)=gI(X,Y)g2(X,Y)withpoly-nomialsgl'g2'contradictingtheirreduciblityoff. 9842f(X,Y)=Y-2X,g(Z,U)=U4-2Z2Ifandareasabove,thenwehavethefollowingdiagram:rdee2//~x~eg2(x,z,~)~(x,z,t~)deg2~~g2Q(x,z,Jff)Ideg2~(X,Z)Hence[~(X,Z,~,U):~(X,Z)]=8~16.Proofofthelemma:Weneedtoshowthat(2.1)[K(X,Z,~,U):K(X,Z,~)]=d'and(2.2)[K(X,Z,~):K(X,Z)]=dToshow(2.1)itwillsufficetoshowthatg(Z,U)remainsirreducibleoverK(X,~)Otherwise,g(Z,U)=gl(Z,U)g2(Z,U),wheregi(Z,~(i=1,2)hascoefficientsinK(X,~)andisofdegreelessthand'inU.Writegi(Z,U)=~CijkZJ~(i=1,2),j,k 99where(0)r(1)-1)d-1Cijk=rijk(X)+ijk....(h)..withrationalfunctionsr...(X).1JKPickxeKsuchthatthedenominatorsofther(h!(x)are1jKnon-zeroandsuchthatiff(X,Y)=ao(X)yd+aI(X)Yd-I+...+ad(X),thena0(x)~0.PickyEKsuchthatf(x,y)=0.Thenthepair(x,y)satisfiesanyequationoverKwhichissatisfiedby(x,~)*).Put-(0)(x)r(d-l)(x)yd-1Cijk=rijk+"'"+ijkandgi(Z,U)=~CijkZJuk(i=1,2)j,kThencijkEKandg(z,u)=~l(Z,U)~2(z,u)contradictingtheabsoluteirreducibilityofg(Z,U)Thiscompletestheproofof(2.1).Theproofof(2.2)issimilarbutsimpler.LEMMA2B:Supposef(X,Y)EK[X,Y]isofdegreed>0inY,irreducibleoverKandseparableinY.Letf(X,~)=0andf(Z,~)=0.Thenfisabsolutelyirreducibleifandonlyif[K(X,Z,~,U):K(X,Z)]=d29t)Forif~(X,Y)isapolynomialwith~(X,~)=0,then~(X,Y)isdivisiblebyf(X,Y). 100Proof:The"onlyif"partfollowsfromLemma2A.The"if"partwillbegivenlate~)intheselectures;wedonotneeditnow.LetKbeafieldofcharacteristicp;letq=pEIff(X,Y)=~aij@Xly(aijEK),i,jdefineiji)jSincethemappingx~xqisanautomorphismofK,itfollowsthatiffisabsolutelyirreducible,thensoisf[qJCOROLLARY2C:Supposef(X,Y),f[q](x,Y)areasabove.Supposefisofdegreed>0iny.LetX,Zbevariables)andlet~,~besuchthatfCX,~)=O,f[q](z,u)=0Then[K(X,Z,.~,~):K(X,Z)]=d2.LEMMA2D:LetKbeafieldofcharacteristicp.Supposef(X,Y)=yd+gl(X)yd-i+...+gd(X)isapolynomialinK[X,Y],absolutelyirreducibleandwithdeggi(X)Ai(iKi~d)Letf(X,~)=0Ift~Ch.V,w 101a(X,Y,Z,W)i0isapolynomialwith(i)degXa~(q/d)-dJ(ii)degya~d-1,(iii)degwa~d-1,thena(X,~,xq,~q)~0Beforecommencingwiththeproof,wegivesomeheuristicargu-ments.Sincefisirreducible,theelements~i(0~i~d-l)arelinearlyindependentoverK(X).Ontheotherhand,sincethereared2ofthem,theelements~i~qk(0~i,k~d-I)arelinearlydependentoverK(X).Hencethelemmaisnottrivial.However,thepowersofXina(X,Y,X~Yq)arerestricted.WehaveonlythepowersXqj+v(0~v~(q/d)-d;j=0,i,...).dthSoroughlyonlyoneofallpossibleexponentsinXcanoccur.Thatiswhythelemmahasachanceofworking.Proofofthelemma:ThemethodissimilartothatofChapterI,wPut 102da(X,Y,Z;WI,.o.,Wd)=~a(X,Y,Z,Wi)i=lThisisapolynomialind+3variables,symmetricinWl,...,Wd.ByLemma5AofChapterI,a(X,y,Z;Wl,...,Wd)=b(X,Y,Z;sI(wl,...,Wd),~176d(W1,...,Wd))whereSl,.~daretheelementarysymmetricpolynomialsinWl,...,Wd9Bythesamelemma,thetotaldegreeofb(X,y,Z;Vl,..o,Vd)inVI,.~disatmostd-1.Nowsincedgl(x)~d-1=-.....gd(X),wehaveforanypositiveintegert,~d-l+t(t)(x)~d-l~+..+g_(t)(2.3)(X),=gldwhereitiseasilyverifiedbyinductionthat(t)deggi(X)~(t-1+i)(i-oSincedegxc+Xy=f2M-2(~+1)(X,~)a(~+1)(X,~)ysay.Itisthenclearthatdega(~+1)1--d.Definego(X)--iandeI(X,Y,Y')=Y-y',dgd_j(X)(yj-i+yj-2y,+....+y,j-l)e2(X,Y,Y')=j=l 109Thenf(X,Y)-f(X,Y')=el(X,Y,Y')e2(X,Y,Y')9IfxE~andyE%1(x)U%2(x),then0=f(x,y)=(f(x,y))q=f(x,yq),whence0=f(x,y)-f(x,yq)=(y-yq)e2(x,y,yq)IfyE~:1(x),thenyEF,soy-yq=0;andbecauseyisqasimplerootoff(x,Y),e2(x,y,yq)f0.IfyE~2(x),thenyq~y,hencee2(x,y,yq)=0.Hencefork=1or2,~:k(x)isthesetofywithf(x,y)=0andwithel(x,y,yq)=0.Notation:Setal=i,E2=d-I.Thenekhastotaldegree~k(k=1,2).LEt,~IA4A:Supposek=ior2.LetMbeapositiveintegerwithdiM,M>-d2,2(d-1)(M+8)2~q9Thenthereexistsapolynomiala(X,Y)suchthat(i)a(X,.~)i0,(ii)ifa(~)(X,Y)isdefinedasinLemma3A9then(sa(x,y)=0(0~~-(q/d)-d-2(d-i)-j>(q/d)-3d-j. 113Summingoverj,0~j~K-k,thenumberofavailablecoefficientsisatleast(q/d)(K-k+1)-3d(K+l)-~(K-k)(K-k+l)=((q/d)-3d)(K+l)-~(K-k)(K-k+l)-(q/d)k.Summingoverk,0(q-3d2)(Kd+d)-89-l)-~~d2>(q-3d2)(s2-d)---~qd(d-i)-892>~kqM+q(~d2-89-89E~I-6e)Md2-2E)Md2,sinceM~=d2byhypothesis.Inorderthatthepolynomialsd(~)(X,Y,Y')vanish,wehavetosolveahomogeneoussystemofBlinearequationsinAvariables.Inordertogetanon-zerosolution,itissufficientthatB250d5,then2Md~2d~=~2dq<89sothat4-4-degc<(ek/d)qM+q(d-1).5.Constructionoftwopolynomials.LEMMA5A:SupposeMsatisfiestheconditionsofLemma4A.Letk=1or2befixed.Thenthereexistsapolynomialr(X)10with(i)D~r(x)=0forxE~and0~~N1>qd-d3-N2qd-d3-(d-l)q-d(d-i)(q/M)=q-d(d-i)(q/M)-d3.Therefore(6.1)IN-q/250d5.Theassumptionthatq>250d5alsoguaranteesthat5,~2-d3/2<19J.q-g3iBymakingthesimpleobservationthatif0-min{v(a),v(b)),withtheadditionalconventionthatv(0)=+~.LetK0bethesetofa6Kwithv(a)>_0.ItiseasytoseethatK0isasubringofK,andthattheunitsofK0arepreciselytheelementsaEK0withv(a)=0.LetK1bethesetofa~Kwithv(a)>_1.ItisclearthatK1c%,andthatK1isclosedunderadditionandsubtraction.Infact,K1isanidealinK0,sinceifa~KO,b6K1,thenv(ab)=v(a)+v(b)>~0+i=1,sothatabEK19Moreover,anyproperidealinK0mustnotcon-tainaunit,somustnotcontainanyelementawithy(a)=0,hencemusthecontainedinK1.Thatis,K1isamaximalidealinK0;infact,K1istheuniquemaximalidealinK0.Wesummarizein 120LEMMA7A:Let....vbeavaluationofafieldK.LetK0bethesetofaEKwithv(a)m0,andK1thesetofaEKwithv(a)~1.ThenK0isasubringofK,andK1istheuniquemaximalidealinK0.HenceK0/K1isafield.Example:LetE=~,andpanyprime.Anynon-zerorationalnumbercanbewrittenintheform(a/b)p~,p~ab,wherevisunique.Putv((a/b)p~)=~.Thenitiseasytocheckthatvisavaluation.NowQOistheringconsistingofzeroandofelements(a/b)p~withv~0,andQ1istheuniquemaximalidealinQO'consistingofzeroandofelements(a/b)p~with~~1.AcompletesetofrepresentativesofrmoduloQ1is{0,1,2,..o,p-I)9Forif(a/b)p~EQ0'picktheintegerxin{0,1,...,p-l)withap~bx(modp)a~ap-bxThen~p-x=bEQ1'sothatxliesinthesamecosetmoduloQ1as(a/b)p~.ItfollowsthatQ0/Q1isafieldwithpelements,whence~0/@1~FpLEMMA7B:SupposeKisafieldwithavaluationv,andisahomomorphismfromK0ontoafieldFwithkernelK1LetXbeavariable.Thenthereexistsanextensionv~ofvto 121K(X)withvt(X)=0,andanextension%Orof%owhere%O:(K(X)-~F(X),suchthat(X)=X,risonto,andthekernelof%0'is(K(X))l"Proof:Firstdefine%0'onKo[X]by%Ol+...=...(a0+alX+atxt)%o(aO)+%0(al)X++%o(at)XtItisclearthat%O~isahomomorphismandthat%0textends%O9Next,definevIonK[X]byvt(a0+alX+...+atXt)=min(v(aO),..~Clearly,ttv(f(X)+g(X))~min(v'(f(X)),v(g(X))).Weclaimthat(7.1)v'(f(X)g(X))=v'(f(X))+v'(g(X))ThereexistsanelementpEKwithv(p)=1,sincevisonto.Put(x)=p-V'(f)f(x)I-v(g)(x)=pg(x)^^Thenvt(f)=v~(g)=0,anditsufficestoshowthatv~(fg)=0,sincethenv'(fg)=vI(f)+vI(g)+v'(fg)=v'(f)+v'(g)Wemaythereforeassumewithoutlossofgeneralitythatv'(f)=vl(g)=0.WewishtoshowvI(fg)=0.ButsincevI(f)=0,f(X)EKo[X],andsimilarlyg(X)EKo[X];thereforef(X)g(X)E~[X], 122andv~(fg)>-0.Supposewehadv'(fg)>~i.Thenf(X)g(X)EKI[X]and~p(f)~(g)=W~(fg)=0.Soeither~p/(f)=0orcp'(g)=0,henceeitherv~(f)>~iorIv(g)>-1,whichisacontradiction.Thereforev'(fg)=09Theproofof(7.1)iscomplete.Henceifingeneralv~isdefinedbyv,/f(x)~g-UffY/~=v'(f(x))-v'(g(x)),thenv'becomesavaluationofK(X).Tofurtherextend~,noticethateveryelementof(K(X))0isoftheform(f(X)/g(X)),wherevt(f)20andv'(g)=0.(Ifnecessary,multiplybothfandgbyasuitablepowerofpEK,wherev(p)=1).DefineW~on(g(X))0by,/f(x)~=~'(f(x))Itiseasytocheckthat~'isawell-definedhomomorphismfrom(K(X))0ontoF(X)withkernel(K(X))1"Example:a3"oLetK=Q.writeeverynon-zerorationalaswhere3~ab,anddefineThen,forexample,v,/5X+6~O-0=0,,[sx+6=fieEd=/xT+Jx+l 123LEMMA7C:LetvbeavaluationofafieldK.Let~0beahomomorphismofK0ontoF,withkernelK1.Let~bealgebraicoverF.Thenthereexistsanelement~whichisalgebraicoverK,suchthat~isseparableoverKif~isseparableoverF.Thereexistsavaluationv~;ofK~),withv;~5)=0,extendingv;andthereisahomomorphism~0~ofK~ontoF(~)extending~,suchthatthekernelofIIisK~)1K--K(~)~0~KS)0~"v1~v"~oProof:Letf(X)betheirreducibledefiningpolynomialofoverF.Wemaychoosef(X)tohaveieadingcoefficient1.Letf(X)beapolynomialinK0[X]withthesamedegreeasf,leadingcoefficient1,andwith~0t(f(X))=f(X),wheretistheepimorphismconstructedinLemma7B.Weclaimthatf(X)isirreducibleoverK.Suppose,bywayofcontradiction,thatf(X)=fl(X)f2(X)isaproperfactorization.Wemayassumethatv~(fl)->0andv~(f2)>-0.(Otherwise,multiplybyappropriatepowersofanelementpofKwithv(p)=i.)Thenfl'f2EK0[X],andf=~0~(f)=~(fl)~01(f2)=flf2providesaproperfaetorizationoff,whichgivesacontradiction.Pickaroot,say~,off(X).Itisclearthatif~isseparableoverF,then~isseparableoverKo 124Nowdefine~"onKO[~]byH(a0+al~+ooo+at~t)=(0(aO)+~(al)~+o.o+~(at)~t~"isahomomorphismontoF[~]=F~).Alsodefine~ionK~)by^vl/(a0+al~+.oo+ad_l~d-1)=min{V(ao),V(al),O.o,V(ad_l)},whered=degreeof~overF=degreeof~overK.ItiseasilyverifiedthatvisavaluationofK~),extendingv.Theproofthatfor~,~EKd),V#(~p)=VII(00+Vu(p),goesastheproofof(7.1)inLemma7B.TherestofLemma7CnowfollowsafternotingthatKd)0=K0[~]_.@Example:LetK=Q,andpaprime~Wedefineasbefore,vp=~ifp~ab.Wehaveseenthatthereisahomomorphism~0from~0ontoFpwithkernelQ1ThefieldFwhereq=p,isofthetypeqF=F~),with~separablealgebraicofdegree~.LetqPbechosenasinthelemmaandwriteN=Q~)Thenthereisavaluationv"ofthefieldN=Q~)extendingv.Alsothereisahomomorphism~ifromNOontoFwithkernelN1qvlIv"~(J{==}=~U{=}F~FPq 125Remark.ItisclearthatNisanumberfieldofdegree<.Also,expertsinalgebraicnumbertheorywillsaythatpis"inertial"inNTheassertionsofthefollowingexerciseswillnotbeneededinthesequel.Exercise1.Showthateveryfieldofcharacteristicp~0isthehomomorphicimageofanintegraldomainofcharacteristic0.(Forgeneralfields,anappealtoZorn'sLemmaisnecessary~ItisnotnecessaryforfieldswhicharefinitelygeneratedoverFp).Exercise2.LetvbeavaluationofafieldK.Givenamonicpolynomialf(y)=yd+alYd-1+...+adwithcoefficientsinK,put~(f)=min(i/i)v(a.).Showthatformonicpoly-1l~i~dnomialsf,g,wehave~(fg)=min~(f),4(g))9Deducethatifdegf=dand~(f)=m/dwith(m,d)=1,thenfisirreducible.(IfK=F(X)andifv(a(X)/b(X))=degb(X)-dega(X),theseresultsreducetoTheoremIB,LemmaIC.IfK=~andifv((a/b)pM)=v,ourirreducibilitycriterionyieldsEisenstein'scriterion.)wHyperderivativesagain.InwofChapterIwedefinedhyperderivativesforpolynomials.Inthepresentsectionweshallmoregenerallydefinehyperderivativesforalgebraicfunctions.Foranotherapproachtohyperderivatives(Hassederivatives)seeHasse(1936a),TeichmUller(1936).LetFbeafinitefieldofcharacteristicp.Wehaveaqvaluationof~givenbyvpX)Associatedwiththisvaluationvof~isahomomorphism~fromQ0ontoFpwithkernel@l"WecanthenbyLemma7CfindafieldN~Qsuchthatvcanbeextendedtoavaluationv'ofN,andbyLemma7Bfurtherextendedtoavaluation~ofN(X).Moreover, 126canbeextendedtoahomomorphism~'fromNOontoFwithqkernelN1,andq'canbeextendedtoahomomorphism~zfromN(X)0ontoFq(X)withkernelN(X)I.Supposef(X,Y)sFq[X,Y]isanirreduciblepolynomialwhichisseparableinY.Let~heanalgebraicfunctionwithf(X,~)=0.ThenthereisbyLemma7Canelement~whichisseparablealgebraicoverN(X),suchthatwemayextendv#toavaluationvHjofN(X,~),and~ztoahomomorphismi~ifromN(X~)0ontoF(X,~)havingkernelN(X,~)I.'qQO~NO~(N(X))o=(N(X'~))OFp~Fq~Fq(X)EFq(X,~)Hereafter,v,vI,,arealldenotedbyv,and~,~,,~arealldenotedby~.Elementsinfieldsofcharacteristiczerowillhewrittenas~,~,a(X),etc.LetDbethedifferentiationoperztoronN(X).DmaybeextendedtoaderivationonN(X,~),sincetheextensionN(X,~)(~)overN(X)isseparable.WeintroduceanoperatorEonN(X,~)byE~)~)=1D~(~)9Oneverifiesimmediatelythat(S.1)S(~)(al~"'"at)=(E(Ul)(~l))...(E(ut)(gt)).Ul+...+ut=~ 127LEIVI]WJ~8A:Forany~EN(X,~),v(E(~)~))~v~)(l=0,1,2,...)Proof:Theproofisbyinductionon%.Thecase%=0istrivial.Togofrom~-1to~,weconsiderthreecases.(i)Thelemmaisobviousif~ENIx].(ii)Suppose~tN(X).Let~=f(X)/g(X),sothat}(X)=g(X)~.By(8.1),(8.2)E(~)(f(X))=(E(%-J)g(X))E(J)~.j=0Sincef(X),g(X)6N[X]andbyinductionon%,thelefthandsideof(8.2)andeverysummandontherighthandSideof(8.2),exceptpossiblythesummandg(X)E(~)~,hasavaluation=>v(f(X))=v(g(X))+V(~)Hencealsov(g(X)E(~)~)>-v(g(X))+v~),whichyieldsv(E(~)~)~v~).(iii)AnyEN(X,~)maybewrittenas^^^d-i=r0(X)+(X)+...+rd_l(X)~withr0(X),rl(X),...,~d_l(X)EN(X).Sincev~)=min{v(~0(X)),...,v(rd_l(X))},itsufficestoshowthatfor0~i~d-l,v(E(~)(r^i(x)~i))av(ri(X)~i)=v(ri(X))Applying(8.1)totheproductri(X)~I=ri(X)~..~~,itbecomesclearthatweneedonlyshowthatv(E(~)(~))->0. 128Letf(X,Y)=yd+gl(X)yd-1+...+gd(X)Now~wasconstructedastherootofapolynomial(X,Y)ydyd-I=+gl(X)+"'"+gd(X)where~0(i(X))=gi(X)(i<--id)Wehaved(8.3)o=~ix,~))~~~d-~(x)~'i=Oandby(8.1),E(~)~^i=E~)(%-i(x)~)(~d-i(x)~)...~))(uo)(uI)(ui)^UO+...+u.=~1~(Uo,...,ui)Uo+...+ul=~say.CollectingthetermswhereoneofUl,...,uiequals%,weobtaini}d_i(x)~i-lE(%)~)+^Z~(Uo,Ul,-..,ui)Uo+...+Ul=%Ul,U2,...,ui<~Henceby(8.3),do=E(~)(~)}y(x,~)+~~(Uo....,ui>i=0Uo+...+U.l=~Ul,-.o,Ui<% 129Butbyinductionhypothesis,everysummand,exceptpossiblythefirstone,hasavaluation=>0.Hencealsothefirstonehas,i~e.,v(E(~)~))+V(~y(X,~))>-0.SincefhascoefficientsinNO,V(~y(X,~))~0.But~p(}y(X,~))=fy(X,~)~0,andhence(8.4)=o,sinceotherwise}y(X,~)EN(X,~)I(=kernelof~0),acontradiction.Itfollowsthatv(E(~)@))>-0,andtheproofofthelemmaiscomplete.WearegoingtodefineoperatorsE(~)onFq(X,~)SupposeUEF(X,~)Thenthereexist~EN(X,~)0with~0~)=~ByqLemma8A,v(E(~)~))~v(a)>-0,whenceE(%)~)EN(X,~)0DefineE(%)onF(X,~)by.qE(%)(a)=~o(E(~)(~))ThenewoperatorsE(gj)arewell-defined,becauseif~i)=~0~2)=~,then12)=0,whencev(E(~)~l)-E(~)~2))=v(E(~)~l-U2))~V~l-U2)-~I,sothat~(E(%)~i)-E(%)~2))=0, 130whence%0(E(~)~i))=~(E(~)~2))Animmediateconsequenceofourdefinitionandtheformula(8.1)forE(~)inN(X,~)is(u1)(ut)(8.5)E(Z)(a1...at)=~(ECal))...(E(at))u1+...+ut=Remark:InthedefinitionoftheoperatorsE(~)on~q(X,~),weconstructedthefieldN(X,~),whichisnotuniquelydeterminedbyFq(X,~)Conceivably,theoperatorsE"~'(~coulddependon(~)thisconstruction.Infact,theoperatorsEareindependentoftheconstruction.Asketchoftheproofisasfollows.Weproceedbyinductionon.Inthestepfrom~-1to~weconsiderthreecases,whichareanalogoustothoseintheproofofLemma8A.(i)~EFq[X].InthiscaseitiseasilyseenthatourhyperderivativescoincidewiththosedefinedinwofChapterI.Incidentally,wenoteforlaterthatTheorem6DofChapterIisvalid.(ii)a~F(X).Say~=f(X)/g(X).By(8.5)andincompleteqanalogywith(8.2),E(~)f(X)=(E(Z-j)g(X))E(J)~.j=OSincef(X),g(X)EFq[X],andbyinductionon~,thelefthandsideandeverysummandontherighthandside,exceptpossiblythesummandg(X)E(Z)a,isindependentofourconstruction.Hencealsothissummand,whencealsoE(~)a,isindependentofourconstruction. 131(iii)LtEFq(X,~).Theargumentisanalogoustothatinpart(iii)ofLemma8A.LEMMA8B:Let1iEF(X,~).Suppose0-v(~D~)>-0.Since0<~0.Thereforev(E(~)~P~)>0jsothatE(~)(LLpp)=~(E(~)~~)=0.wRemovaloftheconditionthatq=p.WeprovetheanaloguetoLemma3A:LEMMA9A:Letf(X,Y)and~begivenasusual.LetMbeapositiveintegeranda(X,Y)apolynomial.Thenfor0K~~M,(9.i)E(~)(f2M(x,9)a(X,9))=f2M-2~y(X,9)a(~)(X,9),(~)where(X,Y)isapolynomialwith(~)dega(X,Y)Kdega(X,y)+(2d-3).6. 132Proof:Findapolynomiala(X,Y)inN0[X,Y],ofthesamedegreeasa(X,y),with~0(~(X,Y))=a(X,Y)t).Lemma3AdidnotdependonthegroundfieldFIfweapplythislemmatoqD~(~2M(X,~)a(X,~))anddivideby~:,weobtain(9.2)E(~)(?~M/x,g)a(x,~))=~2M-2~(x~)a(~)(x,~)Y'wheredega(Z)(X,Y)<-deg~(X,Y)+(2d-3)~.Wemaysupposethata(~)(X,Y)isofdegreeatmostd-1inY,becausewemayusetherelationf(X,~)=0toexpress~d~d+l...,etc.,aslinearcombinationsofi,~,...,~d-IThisprocessdoesnotincreasethetotaldegreeofthepolynomial.Wehavev(?M>obutV(fy(X,~))=0by(8.4),whencev(~(~)(X,~))>0.Let^a(~)(X,~)=b0(X)+bl(X)~+...+bd-l(X)~d-1thenbyourdefinitionofvonN(X,~),v~i(X))>_0(0~igd-l)Thus~(~)(X,Y)liesinN0[X,Y]_..Wemaythereforeapplyto^(~)a(X,Y);leta(~)(X,Y)=W(~(Z)(x,Y)).Applying%0to(9.2),weobtain(9.1).WewishtoprovetheanalogueofLemma4A,wherethehigherderivativesD~arereplacedbytheoperatorsE(~)Wesett)ClearlymaybeextendednotonlytoN0[X],butalsoinanobviouswayto~o[X,Y]. 133Kd-Ih(X,Y,Z,W)=~_]bjk(X,Y)zJwk,j=0k=0j'+k~Kandputa(X,Y)=h(X,Y,Xq,Yq)Weareinterestedin(~)(f2M(X,~)~(X,~)))=f2M-2sYt"X,!~J~~"a(9')(X,~)ButKd-io.(X,9)xqJ~qka=2?.Jkj=0k=Oj+k':K=Kthisfollowsfrom(8.5)andthefactthatifmp-I~1.Remark:TheoremsICandIDarenolongertruewhend=n.Foranypositiveintegernandanyprimepowerq,letWl,...,wnbeabasisofFoverFLetnqqn-iqjg(X)==-]-!-(elXl+"'"+~q3Xnn)j=0Observethatg(X)isapolynomialinnvariablesoftotaldegreen.ByTheoremIEofChapterI,theelements~J(0~jKn-i)aretheconjugatesofw.Sinceg(X)isevidentlyinvariantundertheiGaloisgroupofFoverFq,ithascoefficientsinFqMoreover,nqnifx=(Xl,...,xn)EFqand~=WIX1+oo.+WnXn,theng(x)=isnthenorm9~(~)of~.HenceifxEFandx~0,then~~0,q~whenceg(~)=~(WlX1+...+~nXn)=~(~)~0.Thereforeg(X)hasonlythetrivialzero.SoN=1andN~0(modp) 137THEOREMIE:(Warning'sSecondTheorem)(Warning(1935)).UnderthehypothesisofTheoremIC,ifN>0,thenn-dN>qGivenasubspaceSofFnandanelementtEFnletq=q'W=S+tbethesetofpointss+twithsES.SuchasetWwillbecalledalinearmanifold.ThesubspaceS(butnott)isdeterminedbyW,andwemaysaythatWisobtainedfromSbyatranslation.ThedimensionofWisbydefinitionthedimensionofS.TwolinearmanifoldsofthesamedimensionaresaidtobeparalleliftheyareobtainedfromthesamesubspaceS.nInwhatfollows,Vwillbethesetofx=EFqwithfl(~).....ft(x)=0LEMMAIF:IfWIandW2aretwoparallellinearmanifolds,thenIWlnVI~IW2nVI(roodp).Proof:SincethecasewhereWI=W2isobvious,wemayassumethatW1%W2.Moreover,afteralinearchangeofcoordinates,wemaysupposethatWI={(xI....,Xn):0=xI=x2.....Xn_d}andW2={(Xl,....Xn):i=xI,0=x2.....Xn_d}.Nowwriter(x)=xq-l-1="1"1-(x-a),aEFqand 138g(X)=(-l)n-dr(x2)...r(Xn_d)~(X1-a)a~0,ia6FqItmaybeseenthatg(X)isapolynomialoftotaldegree(n-d)(q-i)-1,=withthepropertythat-iif~EW1,g(~)=1if_~EW2,I0otherwise.Putq-i_f~-l(x))g(X)h(~)=(1-~i(~))...(i==h(X)isapolynomialinnvariablesoftotaldegree(n-d)(q-i)-1+d(q-i)=n(q-i)-1q+..+q+l>q 140THEOREMIG9(J.Ax(1964))MakethesamehypothesesasinTheorem1C.Letbbeaninteger,b_2elementsXl,.o.,xninK,notallzero,with22f(x)=f(xl,...,Xn)=alxI+...+ax=0.=nnWithoutlossofgenerality,wemayassumexI~0.PutYl=Xl(l+t),Y2=x2(l-t)'''''Yn=Xn(l-t),withtEKtobedetermined.Then22f(Yl'''''Yn)=2t(alxI-a2x2-...-anX)2=4talxI*2NowifaEKandifwesett=a/(4alxl),weobtainf(yl,...,yn)=aThusfrepresentsa.Wenowreturntoourgeneralthemebyfocusingattentiononquadraticformsoverafinitefield.SinceitwasnecessarythatwerequirecharK~2inthissection,weconsiderfinitefieldsFqwithqodd.Supposed6FWeintroducethenotation:q 144I1ifdE(F~)2,-1if1~(F*)2.qSupposefl(~)andf2(~)areequivalentnondegeneratequadraticformsoverFwithrespectivedeterminantsdandd2Thenq1dl/d2E(Fq)2:,whence[qd__!l)=(~__22).Thatis,thesymbol(d)isinvariantunderequivalence.LEMMA2D:Letf(Xl,....Xn),n~3,beanondegeneratequadraticformoverF,whereqisodd.Thenqf(Xl,...,Xn)~XlX2+h(X3,...,Xn).Proof:ByChevalley'sTheorem(TheoremID),f(X)hasanon-=trivialzeroinF;i.e.,f(X)representszero.ByLemma2C,q=2f(X)=represents16FqByLemma2A,f(X)=~X1+g(X2,...,Xn)2forsomeformg.HenceX1+g(X2,...,Xn)representszero,sothereexistXl'''''XnEFq'notallzero,with2x.~+g(x2,...,xn)=0.2IfxIi0,thengrepresents-xI,hencegrepresents-1.IfxI=0,thengrepresentszero,andtherefore,byLemma2C,gagainrepresents-1.ByLemma2A,2g(X2,...,Xn)~-X2+h(X3,...,Xn),whence22f(X1,...,Xn)~X1-X2+h(X3,...,Xn)NXlX2+h(X3,...,Xn) 145nNowletNnbethenumberofzerosinFoff(Xl,...,Xn)qn-2andletNn_2bethenumberofzerosinFqofh(X3,...,Xn).InordertofindtherelationbetweenNnandNn_2,weobservethatf(Xl'''''Xn)andXIX2+h(X3,...,Xn)musthavethesamenumberofzeros,sincetheyareequivalent.Wefirstcountsolutionsofx1~+h(x3,...,x)=0nwithh(x3,...,Xn)=0,hencewithXlX2=0.Thenumberofpossibilitiesforx3,...,xnisNn_2thenumberofpossibilitiesforXl,X2is2q-1,sothataltogetherweobtain(2q-l)~n-2"Wenextcountsolutionswithh(x3,...,xn)~0.Thenumberofn-2possibilitiesforx3,...,xnisq-Nn_2,andforgivenx3,...xn,thenumberofpossibilitiesforXl,X2isq-1,sothatweget(q-1)(qn-2-Nn-2)suchsolutions.Addingthesetwonumbers,weobtainn-1n-2(2.1)~=q-q+q~n-2"THEOREM2E:Letf(~)=f(Xl,...,Xn)beanondegeneratequadraticformofdeterminantdoverF,qodd.ThenthenumberNofq,zerosoff(X)inFisgivenby~_~q 146q,ifnisodd,In-i/2d)(2.2)N=q+(q-i)---,ifniseven.2Proof:Supposenisodd.Ifn=1,f(X)=aX,andN:1.Ifn~=3,wemaysupposethatf=XlX2+h(X3,...,Xn)n-3Ifthetheoremholdsforn-2,thenNn.2=qandby(2.1),n-1n-2N=q-q+qNn_2n-in-2n-3=q-q+q.qn-1=qNowsupposeniseven.Ifn=2,f(Xl,X2)isequivalenttoanondegeneratediagonalform2222alX1+a2X2=al(X1+(a2/al)X2),and=q~If=-1,then=-1,whence(-.(a2/al))=-1.If(Xl,X2)wereanon-trivialzerooff(X1,X2),q/22thenxI=-(a2/al)x2,whichisimpossible.Thereforef(Xl,X2)hasonlythetrivialzero;i.e.N=1,whichagreeswith(2.2).If~I-dl"=+1,theninasimilarwayJ'/-(a2/al)/=+1,andwesee/qJthatx12=_(a2/al)x22has2(q-1)non-trivialsolutions(Xl,X2)ThereforeN=1+2(q-l)=2q-1,againagreeingwith(2.2)Ifn~4,wemaysupposethatf=XlX2+h(X3,...,Xn).Observethatthedeterminantofhisminusthatoff.Nowsuppose 147thetheoremholdsforn-2.Thenn-in-2Nn=q-q+qNn_2n-in-2n-3+(q_1)q(n-4)/2(-1)(n-2)/2(-d)=q-q+qq{qI)n-11)q(n-2)/2=q+(q-wElementaryupperbounds.Projectivezeros.LEMMA3A:Letf(Xl,...,Xn)beanon-zeropolynomialoverFqoftotaldegreed.ThenthenumberNofzerosoff(XI,...,Xn)ninFsatisfies--qn-iN%dq!~f(xI.....xn)ishomogeneous,thenthenumberofitsnon-trivialzerosisatmostd(qn-1-l).Proof:Ifd=0,fisanon-zeroconstantandhasnozeros.Ifd=1,thenf(XI,...,Xn)=alX1+o..+anXn+c9n-iandN=qIffishomogeneousofdegreed=1,thenc=0n-iandthenumberofnon-trivialzerosoffisq-1.Ifn=1,thenclearlyN~d.Ifn=1andfishomogeneous,thenfcanhavenonon-trivialzeros.Wehaveshownthatthelemmaholdsifd~1orifn=1.Weproceedby"doubleinduetlon.Supposen>1,d>1,andthelemmaistrueforpolynomialsinatmostnvariablesofdegreeless 148thand,andthelemmaistrueforpolynomialsinlessthannvariablesofdegreeatmostd.Wemustprovethelemmaforapolynomialf(XI,...,Xn)innvariablesofdegreed.Therearetwocases.Casei:f(Xl,...,Xn)isnotdivisiblebyXI-xforanyxEFThenforanyx6F,f(x,X2,...,Xn)isanon-zeroqqpolynomialofdegreeatmostdinn-1variables.Bythen-iinductivehypothesis,thenumberofzeros(x2,...,xn)6Fqofn-2f(x,X2,...,Xn)isatmostdqButwehaveqchoicesforxEF,sothatN~qdqn-2=dqn-IqBythesamereasoning,thenumberofzerosoff(x,X2,999,Xn)withxi~0isatmost(q-l)dqn-2Iff(Xl,...,Xn)ishomogeneous,thensoisf(0,X2,..,Xn),andthenumberofnon-trivialzerosoff(O,X2,...,Xn)isatmostd(qn-2l)byinduction.Thereforethetotalnumberofnon-trivialzerosoff(Xl,...,Xn)isd(q-l)qn-2+d(qn-2-i)=d(qn-I_i)Case2:f(Xl,...,Xn)isdivisiblebyXI-xforsomex6FqThenf(X)==(XI-x)g(X)=,wheregisanon-zeropolynomialinatmostnvariablesofdegreeatmostd-1.Weimmediatelyseethatn-l_qn-1=dqn-1N0,say=iiif(X)=f(X1,...,X)=~a..XllX22...Xnn=nIi,12,...,iniI+...+ingdAssociatewithf(X)theform,iiif(X0'Xl'''"Xn)=ZaX00XII...Xnn,il,i2,...,ini0+iI+...+in=d 150Wemaysaythattheequationf(x)=0definesa"hypersurfaceinn-space".Thezerosoff(X)arethe"points"ofthishypersurface.Theequationf(Xo,Xl,...,Xn)=0definesa"hypersurfaceinn-dimensionalprojectivespace".Inthiscase,weconsideronlynon-trivialzeros(x0,xl,-..,xn)i(0,0,...,0),andtwozerosareconsideredidenticaliftheircoordinatesareproportional.Theseare,1..1,called"pointsontheprojectivehypersurface",orprojectlvezeros.Suppose(Xo,Xl,...,Xn)representsazerooff.Therearetwopossibilities:(a)x/0.Thezeromaythenberepresenteduniquelyby0an(n+i)-tuple(l,Yl,...,yn)Sincef(l,Yl,...,yn)=0,wehavef(yl,...,yn)=0.Conversely,if(yl,...,yn)isazerooff,then(1,Yl,.o.,yn)isazerooffThesepointsofthe,,..,tprojectivehypersurfacearecalledflnlte.Thereisthusai-icorrespondencebetweenfinitepointsontheprojectivehypersurface,f=0andpointsontheaffinehypersurfacef=0.(b)x0=0.Thesepointsarecalled"pointsatinfinity"ofthehypersurface.22Example:Letf(Xl,X2)=X1-X2-1.Theequationf(xl,x2)=0definesahyperbola.Thishyperbolahasthetwoasymptotes*222x2=xIandx2=-xI.Inthisexample,f(X0,XI,X2)=X1-X2-X0*Thepointsatinfinityarethezerosoffwithx0=0.Thereare,ifcharK~2,twopointsatinfinity,representedby(0,i,i)and(0,i,-i).Theymaybeinterpretedas"pointsinfinitelyfaroutonthetwoasymptotes".Whetherornotthereexistpointsatinfinitymaydependontheunderlyingfield. 15122Example:Letf(Xl,X2)=X1+X2-i.Theequationf(xl,x2)=0*222definesacircleofradius1.Sinceheref(X0,X1,X2)=X1+X2-X0,thepointsatinfinityarethoseelements(0,Xl,X2)satisfying22xI+x2=0.Ifthefieldunderconsiderationisthefieldofreals,thereisnopointatinfinity.IfourfieldisthefieldCofcomplexnumbers,therearetwopointsatinfinityrepresentedby(O,l,i)and(O,l,-i).LEMMA3B:Letf(X)beapolynomialofdegreedwithcoefficientsninFLetNbethenumberofzerosoffinFLetNbe--q--__qthenumberofprojectivezerosasdefinedabove.Then*n-2n-3N~Nele2~eI+e2.Let 153vj(Xl,...,Xn)=uj(Xl,X2+c2XI,...,Xn+CnXI)e=pj(c2,.'',cn)xlj+.@.e1Wewishtochoosec_,...,zCnEFsothatthecoefficientofXIqe2invI(X)=andofXIinv2(X)arenotzero.Nowpjisapolynomialofdegreeatmoste.,andisnotidenticallyzero.By3Lemma3A,thetotalnumberofzerosofp_inFn-IisatmostJqn-2ejqThereforethetotalnumberofzerosofbothPlandP2isqn-2n-1(eI+e2)0(i=1,2,...,n)iqinTHEOREM5A.ThenumberNofsolutionsof(5.1)inF--qsatisfiesn-1(n-1)/2(1-1)-n/2IN-qI-0(i=1,2,...,n).Let5=l.c.m.r[dl,d2,...,dn]9Then(5.7)IN-qn-11=2,ann-tupleal,...,awillben 170countedbyAnpreciselyif0c.LetR(X0,...,Xk_I,UI,...,U,VI,...,Vc,As)iibeobtainedfromRbyomittingalltermswheresomeU.orV.11withi>coccurs.Thens>ll11*tt9Risclearlytheresultantofthetwopolynomialsf=UIfI(Xo,...,Xk)+...+%fc(Xo,-..,Xk)-*Xd-*=aok+"'"+ad'g=Vlfl(X0,...,Xk)+...+Vefc(X0,...,Xk)-*d-*=b0Xk+...+bd,whenconsideredaspolynomialsinXk.Ifwewrite,forl~j~r,iikfj(X0.....Xk)=~A(j)i0...ikX00...xki0+.9.+ik=dthenumberofsummandsinfisnotmorethan(d+i)k.SotheJnumberofsummandsinforgisboundedby(d+l)kc~2d(d+l)kK(2d)k+l 188Thereforethenumberofsummandsineachaorb.isalsoi1k+l-*-*boundedby(2d)Buteachcoefficientinaiorbiiseither0or1,sothat]]a.*tlK(2d)k+l,t]b;ll~(2d)k+l(i=0,...,d)1Theresultantoffandgisofdegree2dina0,...,ad,h0,...,bd.Thisresultantisa(2d2d)-determinant,sotheresultantrsatisfies11rli~(2d>'ByLemmaIE,R=r(a0,...,ad,b0,...,bd)hasI1R*tl-<:(2d):((2d)k+l)2dHencellRu,v0andisgivenby(2.l)f(Xl'''''Xn)=ail...iXI...Xnn1nil+...+i~dnTHEOREM2A:(E.Noether(1922))Thereexistformsgl,.~invariablesAi1...i(il+"'"+in~d)suchthattheabovenpolynomialf(X1,...,Xn)isreducibleoverKorofdegreed.n 192andhencegdividesf.Wehaveshownthatgdividesfifandonlyif(2.2)hasanon-trivialsolutioninthevariablesc,{Ckl...k}.ThenumberofnJivariablesisk+iwithk=In+d-i].Thereforetheconditionnlthatgdividefisthatallthe(k+1)X(k+l)determinants,sayAI,...,Aofthesystemoflinearequations(2.2)vanish.r'ButeachA.isaforminthecoefficientsb.ofdegreek,zJl...Jnandthenumberofthesecoefficientsisalsok.Weknowfromeliminationtheory,specificallyTheorem1A,thatthereexistformshl''''hsinthecoefficientsofAI,...,Ar,suchthattheequations==0haveanon-trivialsolution(inthebs)A1....Arjl...jnifandonlyifh.....h=0.AlsobyTheorem1A,1s2k-I2k(2.3)degh.=2k-Ik-i~k(1~i~s).1IfcharK=0,itfollowsfromTheoremIDthat224k-42k-1k2k.4)I]h.l[~k~2(1~i~S)91NowletgibeobtainedfromhbysubstitutingforthecoefficientsioftheformsAI,...,Artheirexpressionsintermsoftheoriginalcoefficientsaoff.Eachsuchcoefficientislinearinthei199.inawithnormatmostk'.Combining(2.3),(2.4)withLemmasIB/i.ooi1nrE,weobtain2k<*cdeggi=degh.=k(1-~i=0withrationalintegralcoefficients.Supposefisabsolutelyirreducible(i.e.irreducibleover~)Letpbeaprimewithp>(4Hfl!1k2k,wherek=(n+d-i).Thenthereducedpolynomialmodulopisnagainofdegreedandabsolutelyirreducible(i.e.irreducibleoverFp)9Proof:Letfbegivenby~.i),wherethecoefficients{ail'''in}arenowintegers.Sincefisofdegreedandabsolutelyirreducible,inthenotationofTheorem2A,notallthenumbersgi({ail...in})arezero.Letussaygl({a.ll...in})i0.Wehavetheestimate2k2ko(4llfll)k,thenthenumbergl({ail'''in})isstillnon-zero 194modulop.Itfollows,againbyTheorem2A,thatthereducedpolynomialmodulopisofdegreedandabsolutelyirreducible.COROLLARY2C:Letf(X,Y)beapolynomialwithrationalintegercoefficientswhichisabsolutelyirreducible.IfN~)denotesthenumberofsolutionsofthecongruencef(x,y)"0(modp),thenforlargeprimesp,N~)=p+o(pl/2)Proof:CombineCorollary2BwithTheoremIAofChapterIII.wTheabsoluteirreducibilityofpolynomials(II)LetKandLbetwofieldswithK~L9ThealgebraicclosureofKinL,denotedbyK~,isdefinedasthesetofelementsofLwhicharealgebraicoverK.ClearlyK~isafieldandK~K~~L.THEOREM3A:Supposef(Xl,...,Xm,~isapolynomialwithcoefficientsinafieldK,irreducibleoverK,andofdegreed>0inY.FurthersupposethatfisnotapolynomialinonlyX~,...'Xm'pYPi~fKhascharacteristicp~0.Let~beaquantitysatisfyingf(XI,...,X,~)=0,andletL=K(XI,...,Xm,~).mLetKObethealgebraicclosureofKinL.!hen[K~K]isadivisorofdandK~isseparableoverK.Moreover~thepolyn~ialf(XI,...,X,Y)isabsolutelyirreducibleifandonlyifmoK=K. 195Theoremsofthistypearewellknowntoalgebraicgeometers.See,e.g.,Zariski(1945)SeealsoCorollary6CinCh.Vl.24Example:Considerthepolynomialf(X,Y)=2X-YoverthefieldK=~ofrationalnumbers.Clearlyf(X,Y)isirreducibleover~.Choose~sothat~4=2X2andletL=~(X,~).Ifweput~=~2/X,then2=2,so~/2hoThismeansthatQisonotalgebraicallyclosedinL,or~~Q.ByTheorem3A,f(X,Y)isnotabsolutelyirreducible;infact,weseedirectlythatf(x,,)=x-x+isafactorizationoff(X,Y)overQ(~2)ProofofTheorem3A.Webeginwiththefollowingremark:IfK~isalgebraicoverKofdegreedthenKO(x,...,Xm)isalgebraic'1overK(XI,...,Xm)ofdegreed,andviceversa.IfK~isseparable(orinseparable)overK,thenK~)isseparable(orinseparable)overK(XI,...,Xm),andconversely.ThisfollowsfromtheargumentusedinLemma2AofChapterIIl.Nowobservethato(3.1)K(X1,...,Xm)cK~)GK(X1,...,Xm,~)=K(X1,...,Xm,~)SinceK(XI,...,Xm,~)isanextensionofK(XI,...,Xm)ofdegreed,itfollowsthat[K~):K(XI,...,Xm)]dividesd,whence[K~K]dividesdbytheaboveremark.Iffisabsolutelyirreducible,thenfisirreducibleoverK~oHence~isalgebraicofdegreedoverK(XI,...,Xm);thatis,[K~:K~.....Xm)]=d 196oFrom(3.1)itfollowsthatK(XI,...,Xm)=K(XI,..~m),sothatoK=KFortheremainderoftheproof,weshalltacitlyassumethatcharK=p~0.ActuallythecasewhencharK=Oissimpler,andseveralstepsmaybeomitted.LetfI(XI,...,Xm,Y)beanirreduciblefactoroff(XI,...,Xm,Y)overKsuchthat(3.2)fI(XI.....Xm,~)=0~Wenormalizeflbyrequiringthattheleadingcoefficient(insomelexicographicorderingofthemonomials)is1.Theneverypoweroffalsohasthisproperty.LetKbethefieldobtainedfromK11byadjoiningthecoefficientsoffLetabethesmallestpositive1aintegersuchthateverycoefficientofflisseparableoverK.Ifbbisapositiveintegersuchthatfhascoefficientswhichare1separableoverK,thena~b~Forifb=at+rwith0~r1andlet[K(X,~):K~=v,sothatuvdInthechainK(X,Z)cKO(x,z)~K~c_K~=K(X,Z,~,1I),thefieldextensionsareofrespectivedegreesu,v,v,sothat[K(X,Z,O,I/):K(X,Z)]=uv2<(uv)2=d2,whichcompletestheproof.InwofChapterIVweintroducedanequivalencerelationforquad-raticforms.Wemakeaslightad2ustmentofthatdefinitiontodefineanequivalenceforpolynomialsinnvariablesoverafieldK.Wesaythatf(X)~g(X)ifthereisanon-singular(nXn)matrixTandavectort,bothhavingcomponentsinK,suchthatf(x)=g(TX+t)==___Thisisclearlyanequivalencerelation. 199LEMMA3B:Supposef(X)~g(X).IffisirreducibleoverK(orabsolutelyirreducible),thensoisg.Moreover,thetotaldegreesoff(X)andg(X)areequal.Proof:Exercise.NoticethatthefirstpartofthelemmaisageneralizationofLemma2BofChapterI.Letf(XI,...,Xn)beapolynomialoverK.For1~Z%n,wewillwritef(Xl''""~Z'X~+I.....Xn)whenthepolynomialistobeinterpretedasapolynomialinthevariablesX~+I,..~n,withcoefficientsinthefieldE(XI,...,X~)LEMMA9C:Iff(XI,...,X)isirreducible(overK),thennf(Xl,...~,Xs1....,Xn)isirreducible(overK(XI,...,X~)).Proof:ThisfollowsfromtheuniquefactorizationinK[XI,...,X~].Thedetailsareleftasanexercise.Weremarkthatiff(XI,...,Xn)isabsolutelyirreducible(i.e.irreducibleoverK),itdoesnotfollowthatf(~l~..~,X~+I,999,X~isabsolutelyirreducible(i.e.irreducibleoverK(XI,...,X~)).Infactif~=n-1,thenewpolynomialisapolynomialinonevariable,whichcannotbeabsolutelyirreducibleunlessitsdegreeisone.Asanotherexample,thepolynomial22f(XI,X2,X3)=X2-XIX3isabsolutelyirreducible,whilef(l,X2,XS)hasthefactorization 200f(x,x2,x3:%-x3)%+x3overK(X1).THEOREMSD:Supposef(Xl,...,Xn)isapolynomialoveraninfinitefieldK.Supposefisabsolutelyirreducibleandofdegreed~0.Leti~~gn-2.Thenthereisapolynomialg~fsuchthatg(Xl,...,xs,X~+l,...,X)nisabsolutelyirreducibleandofdegreed(inX~+l,...,X).nWeshallneedLEMMA3E:LetJ~LhefieldssuchthatLisafiniteseparablealgebraicextensionofJ.ThenthereareonlyfinitelymanyfieldsJ'withj_cg'cL.Proof:LetNbeafiniteseparablealgebraicnormalextensionofJwithL~N.LetGbetheGaloisgroupofNoverJ,andletHbetheGaloisgroupofNoverL.ThenH~G.FromGaloistheory,weknowthatthereisaone-onecorrespondencebetweenfieldsJ'withJ~J'cLandgroupsH'withHcH'~G.ThenumberofsuchgroupsH'isfinite,sothenumberoffieldsJ'isfinite.Remark:SeparabilityisessentialinLemma]E.ForletFbeanalgebraicallyclosed(henceinfinite)fieldofcharacteristicp.Take 201J=F(X,Y)cL=j(x1/P,yl/p)andifcEF,letJ'=J((X+cy)l/P)=j(X1/p+cl/Pyl/p).CClearlyj~j'cL,butfordifferentchoicesofcEFwegetcdifferentfieldsJ',sothatthecollectionofintermediatefieldscisinfinite.WebegintheProofofTheorem3D:WeshalltacitlyassumethatcharK=p/0,theproofforthecasecharK=0beingeasier.Firstobservethatf(XI,...,Xn)isnotapolynomialinXI,.P..,Xpn'forifitwerethenpipif(Xl''"'Xn)=~ail'''inX11~Xnn11,-..,inx:n)ii...i..Xnil'''"'nncontradictingtheassumptionthatf(XI,...,X)isabsolutelynirreducible.Wechangenotationandwritef=~(xI....,x,Y)wherem=n-1.AfteralineartransformationofvariablesJ(X.=X.+c.Y;i=1,2,...,m)wemaysupposethatfisofdegree111dinYandseparableinY~Let~beaquantitysatisfying 202f(xI....,xm,~)=0,andletL=K(X1,...,Xm,!~)ForcEK,put(c)=X+cXX11mConstructthefieldsK(X(c))and,thelatterbeingthealgebraicclosureofK(X~c))inL.LEMMA3F:ForsomeeEK,K(XI)=K(X.Proof:ForeveryeEKwehaveK(X1....,Xm)c(K(X:e)))o(X2,..-,Xm)_cL.NotethatLisaseparableextensionofK(XI,..~ofdegreed.ByLemma3E,thereareonlyfinitelymanysubfieldsofLcontainingK(XI,...,Xm)Hencethereexisttwodistinctelementse,e'sKsuchthat,...,X),mor(Xm)(X2'''''Xm-1)=lK(xlC'))jo[r~(Xm)(X2....'Xm-1)ButsinceX2,...,Xm_1arealgebraicallyindependentoverK(X1,Xm)itfollowsthat(X)9m(c)ForbrevityweshallwriteX=X1andZ=X~(ct)ByTheorem3A, 203K(x~C))~isafiniteseparableextensionofK(XI(c)),andhencethereexistsanelement~suchthatSimilarly,thereisa8with=K(Z=K(Z,.Let~havethedefiningequationhl(X,~)=0,whereh1isirreducibleoverE;let8havethedefiningequationh2(Z,8)=0,whereh2isirreducibleoverK.NowbyTheorem3Aandtheabsoluteirreducibilityoff,K=K~,sothatKisalgebraicallyclosedinL.ItfollowsthatKisalgebraicallyclosedinK(X,~)andinK(Z,8).ThenbyTheorem3Aagain,hIandh2areabsolutelyirreducible.Henceif~isofdegreedIoverK(X)andifisofdegreed2overK(Z),then[K(X,Z,t,~:K(X,Z)]=dld2byLemma2AofChapterIII.Butwehavemm'sothatK(X,Z,~)=K(X,Z,~=K(X,Z,~,8)ThesethreefieldsareextensionofK(X,Z)ofrespectivedegreesdl,d2anddld2,sothatdI=d2=dld2,andthereforedI=d2=1.HenceK(Xco=K(XcandK(Xco=K(Xc,whichprovesthelemma. 204WenowconcludetheproofofTheorem3D.Wemaywritef(XI,...,XmY)=g(X~c),X2.....Xm,Y)wherecEKisobtainedfromLemma3Fandwhereg(X,X2,...,Xm,Y)=f(X-CXm,X2,...,Xm,Y)9Clearlyg(XlC),x2,...,Xm,~)(=0andgisirreducible.Butg(Xl'~,X2,...,Xm,Y)isabsolutelyirreducible(i.e.,irreducibleoverK(X~c)))because(K(x~C)))~=K(x~C)).Byachangeofnotation,g(XI'X2'''''Xm'uisabsolutelyirreducible.Thisne~polynomialisclearlyequivalenttofandisofdegreedinY.Thisprocessmustnowberepeatedbysetting.x2(c)=X2+CXmwithc6K,etc.,toobtaintheresult.NotethatinthelaststepX~(c)=X~+cXmhencethatwecertainlydoneedtheconditions<_-m-l=n-2.w4.Theabsoluteirreducibilityofpolynomials(IIl).LetKbeafield.WehavedenotedbyKnthen-dimensionalvectorspaceoverKconsistingofn-tuples(Xl,...,xn)withcomponentsinK.SupposeMisanm-dimensionallinearmanifoldinK,where1~m~n.ThenMhasaparameterrepresentation=Y0+~1~i+'"§~m~'nwhereY=O'~I....'~m~K,with~l,...y=mlinearlyindependent,andwhereUI,...,%areparameters.Wewrite~=L(~).SupposeMhasanotherparameterrepresentation 205ThenU=TU'+t,whereTisanon-singular(mm)-matrixoverKandt=EKn,henceL(TUI=+t)==L'(UI)=9Iff(XI,...,Xn)isapolynomialwithcoefficientsinKandMisalinearmanifoldwithparameterrepresentationL(~,putIfL'isanotherparameterrepresentationofM,thenfL,(U')_=f(L'(U'))==f(L(TU'=+=t))=fL(TUj=+t)=HencethepolynomialfLisdeterminedbyMuptoequivalenceinthesenseofw.Onecanthereforespeakofthe"degreeoffonM"andoftheirreducibilityorabsoluteirreducibilityoffonM.LEMMA4A:Supposef(Xl,...,Xn)hascoefficientsinaninfinitefieldK,isofdegreed>0andisabsolutelyirreducible.Letn~3andsupposethatmissuchthat2Km0whichisabsolutelyirreducible.Letn~3andletAbethenumberof2-dimensionallinearmanifoldsM(2).LetBdenotethenumberofmanifoldsM(2)enwhichfisnotofdegreedorisnotabsolutelyirreducible.Let~=2dk2kwherek=(d+21t7ThenB/A~~/q.Proof:EverylinearmanifoldM(2)hasaparameterrepresentationx__={o+~i{i+"2~'whereY=O'Yl'Y=2EFqn~andY--1andY--2arelinearlyindependentIfAtisthenumberofsuchparameterrepresentations,thenAt=qn(qn_1)(qn_q)>_~1q3nButeachlinearmanifoldM(2)hasD=q2(q2_1)(q2-q) 209differentparameterrepresentations,whenceA=A~/D9Nowona(2)manifoldM,fL(X)=f(Y0+U1Y=I+5Y--2)isapolynomialinUI,U29ByTheorem2A,thereareformsgl,.o.,gsinthecoefficientsofthispolynomialsuchthatgl....gs=0isequivalenttothepolynomialbeingofdegree0,then(5.1)IN-ql<~(q,d)wheneverq>x(d),whereNisthenumberofzerosoff(X,Y).Withinterpretation(i),thisstatementhasbeenprovedasTheorem]AofChapterIII.Howeverthestatementalsoholdsunderinterpretation(li),asfollowsfromthestudyofthezetafunctionofthecurvef(x,y)(Well(1948~,Bombieri(1973)),andasmaybeknown,toamoresophisticatedreader.THEOREM5A;Supposef(XI...,Xn)isapolynomialoverF'qoftotaldegreed>0andabsolutelyirreducible.LetNbethennumberofzerosoffinFThenq(5.2)IN-qn-11M(d),~(q,d)iffisabsol,irred,onM(2),(5.4)1N(M(2))-ql0andirreducible.Supposefisnotequivalenttoapolynomialg(X1,...,Xn_2),whereonlyn-2variablesappear.AsinTheorem4C,letAbethenumberof2_dimensiona!linearmapif01dsM(2).LetC 212bethenumberofmanifoldsM(2)wherefisidenticallyzero.Then32C/A~d/qProof:ConsidertheplanesM"2)(paralleltotheplane*n-2*xI=...=Xn_2=0;thesenumberA=qLetCbethenumberofthoseparallelplanesonwhichfisidenticallyzero.Atypicalplaneofthistypeis(2)M:xI=cI,...,Xn_2=Cn_2Thepolynomialfcan,ofcourse,bewrittenasf(XI,...,Xn)=~Pij(Xl,...X~)Xi.Xj,:'n-zn-Ini,j(2)IffisidenticallyzeroonM,thenPij(Cl,...,Cn_2)=0foralliandj9Ifthesepolynomialsp..haveacommonfactorijg(X1,...,Xn_2)ofpositivedegree,thengdividesfand,sincefisirreducible,f=cg.Butbyhypothesisfisnotapolynomialinonlyn-2variables,hencethePijhavenopropercommonfactor.ByLemma3DofChapterIV,thenumberofco,onzeros(el,...,en_2)ofthepolynomialsPijisatmostd3qn-4ItfollowsthatC~sd3qn-4and**d3/q2c/AThesameargumentholdsforplanesparalleltoanygivenplane,andtheresultfollows.Wenowcontinuethe 213ProofofTheorem5A:Theproofisbyinductiononn.Thecasen=1iscompletelytrivial,andthecasen=2holdsbywhatwesaidabove.Iff~gwheregisapolynomialinn-2variables,thenthenumberofzerosoffisq2timesthenumberN~ofzerosn-2osginFSobyinductionqIN'_qn-31x(d)From(5.3)and(5.4)wefindthatIN"-~~q]~~(~O(q,d)~1+dq1+q1M(2)M(2)))fnotabsol,f=-0onM(2)irred.Inourestablishednotation,itfollowsthatIN-q]~7(q,d)A+dqB+q2C=(A/E)(~(q,d)+dq(B/A)+q2(C/A))~qW(q,d)+d~+d3)2Ontheotherhandifq0.With(5.2)oneneedsthatqiscertainlylargerthan2d~,hencethatqisverylargeasafunctionofd. 215Schmidt(1973)appliedthemethodofStepanovdirectlytoequationsinnvariablesandobtainedn-1(3/2)6N>q-3d3qn-providedq>c0n3dif(5.1)isusedwithw(q,d)givenby(i),andN>qn-1(d-l)(d-2)qn-(3/2)_6d2qn-2providedq>co(sn3d5+~if(5.1)isusedwith~(q,d)givenby(ii)Muchmoreistruefor"non-singular"hypersurfacesbythedeepworkofOeligne(I~73)%)+)ButseetheremarkinthePreface. VI.RudimentsofAlgebraicGeometry.T.heNumberofPointsinVarietiesoverFiniteFields.GeneralReferences:Artin(1955),Lang(1958),Shafarevich(19~7~),Mumford()wVarieties.THEOREMIA.Letkbeafield.LetXl,...Xbevariables.'n(i)Intheringk[Xl,X2,...,Xn]jeveryidealhasafinitebasis.(ii)Inthisringtheascendingchainconditionholds,i.e.,if~I1c_~I2_c...isanascendingsequenceofideals,thenforsomem'~mm+l(iii)Everynon-emptysetofidealsinthisringwhichispartiallyorderedbysetinclusion,hasatleastonemaximalelement.Statement(i)istheHilbertBasisTheorem(Hilbert1888).Itiswellknownthatthethreeconditions(i),(ii),(iii)foraringRareequivalent.AringsatisfyingtheseconditionsiscalledNoetherian.AproofofthisTheoremmaybefoundinbooksonalgebra,e.g.VanderWaerden(1955),Kap.12orZariski-Samuel(1958),Ch.IV,andwillnotbegivenhere.Ifk,Karefieldssuchthatk~K,thetranscendencedegreeofKoverk,writtentr.deg.K/k,isthemaximumnumberofelementsinKwhicharealgebraicallyindependentoverk.Inwhatfollows,k,~willbefieldssuchthatk~~,thetr.deg~/k=~,andQisalgebraicallyclosed.Wecallkthegroundfield,and~theuniversaldomain.Forexample,wemaytake 217k=Q(therationals),Q=C(thecomplexnumbers).Ork=F,theqfinitefieldofaqelements,Q=Fq(X1,X2,...),i.e.thealgebraicclosureofF(XI,X2,...)qConsider~n,thespaceofn-tuplesofelementsinQ.Supposeisanidealink[Xl,...,Xn]=k[X].LetA~)bethesetofx=(Xl,...,Xn)6Qnhavingf(x)=0foreveryf(X)6~.EverysetA(~)soobtainediscalledanalgebraicset.Moreprecisely,itisak-algebraicset.Ifwehavesuchanideal~,thenbyTheoremIA,thereexistsabasisof~consistingofafinitenumberofpoly-nomialssayfl(X)...,f(X).ThereforeA(~)canalsobecharacterized'~'masthesetofxE~nwithfl(x).....f(x)=0.No~ethatif==m=C:Jl-~2'thenA~I)_DA~2)"Examples:(1)Letk=~,Q=C,n=2,and~theideal22generatedbyf(X1,X2)=X1+X2-1.ThenA6)istheunitcircle.(2)Againletk=@,Q=C,n=2,andtake~tobethe22idealgeneratedbyf(X1,X2)=X1-X2.ThenA~)consistsofthetwointersectinglinesx2=x1,x2=-x1.THEOREMlB.(i)TheemptysetrandQnarealgebraicsets.(ii)Afiniteunionofalgebraicsetsisanalgebraicset.(iii)Anintersectionofanarbitrarynumberofalgebraicsetsisanalgebraicset.Proof:(i)If~=k[X1,...,Xn],thenA(~)=r.If~=(0),L@,theprincipalidealgeneratedbythezeropolynomial,thenA(~)=Qn.(ii)Itissufficienttoshowthattheunionoftwoalgebraicsetsisagainanalgebraicset.SupposeAisthealgebraicsetgivenby 218theequationsfl(x=).....f~(x)=0,Bisthealgebraicsetgivenbytheequationsgl(=X).....gin(x)=0.ThenAUBisthesetofxEQnwithfi(x)g1(x)=fl(x=)g2(x).....fsgin(x)=0.(iii)LetA,~EI,whereIisanyindexingset,beacollectionofalgebraicsets.SupposethatA=A(~),where~isanidealink[X].Weclaimthat(i.i)crEI(~Iwhere~~5~~-istheidealconsistingofsumsfl(X)=+...+f~(X)__witheachfi(X)in~(~forsome~EI.Toprove(I.I),supposethatxE~A(~).Thenforeach~EI,=xEA~),whencef(x)=0iffE~5~~-9Thereforef(x)==0iffE~.~~.HencexEA(~~I"Conversely,if=xEA(~~),thenf(x=)=0if~EIo~EIfE,~c~e'.Soforany(YEI,iffE~"~5(~,thenf(x)==09Thus,xEA(~)forall~,orx~~A~)Thisproves(i.i).ItfollowsthatNAG=nA~(~)isanalgebraicset.InQnwecannowintroduceatopologybydefiningtheclosedsetsasthealgebraicsets.ThistopologyiscalledtheZariskiTopology.Asusual,theclosureofasetMistheintersectionoftheclosedsetscontainingM.ItisthesmallestclosedsetcontainingMandisdenotedbyM.LetMbeasubsetofQn.Wewrite~(M)fortheidealofallpolynomialsf(X)whichvanishonM,i.e.,allpolynomialsf(X) 219suchthatf(x)=0foreveryx6M.ItisclearthatifM1C~,then~(M1)_D_q(M2).THEOREM1C.M=A~(M)),Proof:ClearlyA(~(M))isaclosedsetcontainingM.ThereforeitissufficienttoshowthatA(~(M))isthesmallestclosedsetcontainingM.LetTbeaclosedsetcontainingM;sayT=A~).SinceT2M,itfollowsthat8G~(T)~~(M),sothatT=A(8)~A~(M))Remark:IfSisanalgebraicset,thenitfollowsfromTheoremiCthatS=A~(S)).If~isanideal,definetheradicalof~,written~,toconsistofallf(X)suchthatforsomepositiveintegerm,fm(x)691.==Theradicalof~isagainanideal.Foriff(X),g(X)6v~,then==thereexistpositiveintegerm,~suchthatfm(x),g~(X)E~9ThusbytheBinomialTheorem~(f(~)~g(~))m+~E~,sothatf(~)~g(~)E~9Also,foranyh(X)ink[X],(h(X)f(X))mE~,sothath(X)f(X)E~9If~isaprimeideal,then~=~,sinceiff(X)~,=thenfm(x)ED,whichimpliegthatf(X)E~9:=THEOREM1D.Let~beanidealink[X].ThenExample:Letk=Q,~=C,n=2,and~theprincipalidealgeneratedbyf(XI,X2)=(X21+X~-l)3ThenA~/)istheunitcircle,and~(AOd))=(X~+X2l>.Thus=+BeforeprovingTheoremIDweneedtwolemmas,nLZ~A1E.Givenaprimeideal~%k[X],thereexistsanx~:with 220Proof.Formthenaturalhomomorphismfromk[X]tothequotientrL~k[X]/~.Since~~k=CO},thenaturalhomomorphismisanisomorphismonk.Thuswemayconsiderk[5]/Vasanextensionofk,andthenaturalhomomorphismrestrictedtokbecomestheidentitymap.Thusourhomomorphismisak-homomorphism.LettheimageofXibe~i(i=l,...,n).Thenaturalhomomorphismisthenhomomorphismfromk[X1,...,.Xn]ontok[~l,...,~n]withkernel~.Since~wasaprimeideal,k[~l,...,~n]isanintegraldomain.Trytoreplace~ibyx.1E~.If,say,~i,...,~darealgebraicallyindependentoverkwith~d+l,...,~nalgebraicallydependentonthem,chooseXl,...,xdE~algebraicallyindependentoverk.Thenk(~l,...,~d)isk-isomorphictok(Xl,...,Xd)Also,~d+lisalgebraicoverk(~l,...,~d),andsosatisfiesacertainirreducibleequationwithcoefficientsink(~l,...,~d).ChooseXd+1in~suchthatitsatisfiesthecorrespondingequationas~d+lbutwithcoefficientsink(Xl,...,Xd).Thenk~l,...,~d+l)isk-isomorphictok(Xl,...,Xd+l).Thereisak-isomorphismwith~i~xi(i=l,...,d+l).Continuinginthismanner,wecanfindXl,...,xnE~suchthatk(~l,...,~n)isk-isomorphictok(Xl,...,Xn).Thereisanisomorphismwith~(~i)=xi(i=l,...,n).Composingthenaturalhomomorphismwiththeisomorphism~weobtainahomomorphism~:k[Xl,~..,xn]~k[~1.....xn]withkernel~.Write~=(Xl,...,Xn).Now~(x)=~,forf(x)=0preciselyif~(f(X))=0,whichistrueiff(X)E~9LEMMAIF.Let~beanon-emptysubsetofk[X]whichisclosedundermultiplicationanddoesn'tcontainzero.Let~beanideal 221whichismaximalwithrespecttothepropertythat~~~=rThenisaprimeideal.Proof:Supposef(X)g(X)E~.butthatf(X)andg(X)arenotin~.Let9..[=~,f(X))"~,sothat~properlycontains~.Since~)ismaximalwithrespecttothepropertythat~~~=r,itfollowsthat~~~;~r.Sothereexistsac(X)=p(X)+h(X)f(X),wherec(X)E~,p(X)E~),h(X)Ek[X]_.Similarly,thereexistsacI(X)=pt(X)+ht(X)g(X),wherec'(X)E~,pl(X)E~),ht(x)Ek[X]_.Thenc!(X)c(X)=(p'(X)+h!(X)g(X))(p(X)+h(X)f(X))6~~!However,since~isclosedundermultiplication,c(x)c(X=)E~:,contradictingthehypothesisthat~N~=rProofofTheoremID:SupposefE~,sothatthereexistsapositiveintegermwithfmE~9ThusforeveryxEA~I),fro(x)=0.Hencef(x)=0foreveryxEAr.Thereforef(X)E~(A(~I)),andd~-~~(A~!))Supposef~~9If~isthesetofallpositiveintegerpowersoff,then~(~~=r;also~doesnotcontainzero.Lettbeanidealcontaining~whichismaximalwithrespecttothepropertythat~n~=~9ByLemma1F,~isaprimeideal.ByLemma1E,thereexistsapointxE~nsuchthat~=~(x)Sincefr~,f(x)~0~Also,(~)=A(~(x))=A(~)~A~),sothatxEA~).Itfollowsthatf~$(A(gJ)).Thus~(A(~))~-~.rTheexistenceofsuchanidealisguaranteedbyTheoremIA. 222SupposeSisanalgebraicset.WecallSreducibleifS=S1US2,whereSI,S2arealgebraicsets,andS~SI,S29Otherwise,wecallSirreducible.Example:Letk=Q,K=C,n=2,andlet~betheidealgeneratedink[XI,X2]bythepolynomialf(XI,X2)=X~-X~.Then222S=A~)isthesetofallx6CsuchthatxI-x2=0.IfS12isthesetofall~6CwithxI+x2=0,andS2isthesetofall~6C2withxI-x2=0,thenS=S1US2,andS1~S~S2.HenceSisreducible.THEOREMIG.LetSbeanon-emptyalgebraicset.Thefollowingfourconditionsareequivalent:(i)S=(~),i.e.Sistheclosureofasinglepointx,(ii)Sisirreducible,(iii)~(S)isaprimeidealink[X],=(iv)S--ACId),where~isaprimeidealink[X].=Proof:(i)~(ii),SupposeS=A[JB,whereAandBarealgebraicsets,andA%S~B.Wehavex6S=AUB.Wemaysupposethat,say,x6A.ThenS=(~)~~=A,whenceS=A,whichisacontradiction.(ii)=(iii).Supposethat~(S)isnotprime.Thenwewouldhavef(X)g(X)6~(S)withneitherf(X)norg(X)in~(S).Let~/=~(S),f(X))(i.e.theidealgeneratedby~(S)andf(X)).Let--~(S),g(X)).=LetA=Ar,B=A~).InviewofS=A~(S))and9/_D~(S),wehaveA~S.ButA~SsincefE~(A)and 223f~~(S).ThusA~S.Similarly,B~S.ButweclaimthatS=AUB.ClearlyAUB~S.Ontheotherhand,ifxES,then=f(~)g(~)=0.Withoutlossofgenerality,letusassumethatf(x)=0.Thenxisazeroofeverypolynomialof~,s@thatxEA.ThereforeSCAUB.ThusS=AUBwithAIS#BThiscontradictstheirreducibilityofS.(iii)=(iv),Set~=~(S).ThenS=A(~(S))=A(~).(iv)=(i).Choose~accordingtoLemmaIEwith~(~)=~.ThenS=A(~)=A(~(~))=(~)=TheproofofTheoremIGiscomplete.AsetSsatisfyinganyoneofthefourequivalentpropertiesofTheoremIGiscalledavariety.(Moreprecisely,itisak-variety.)IfVisavariety,xEViscalledagenericpointofVifV=(~).COROLLARYIH.Thereisaonetoonecorrespondencebetweenthecollectionofallk-varietiesVin~nandthecollectionofallprimeideals~~k[X]ink[X],givenbyv~=3(v)an__AdV~v=A@)BProof:LetVbeavarietyinan,.the,V-~~(V)-~A(~(V))=V.Also,ifisaprimeidealink[X],then~~A~)~~(ACB))=~=~Examples:(1)LetS=~]n.Now~((~n)=(0),aprimeideal.Supposex==(Xl,...,xn)isoftranscendencedegreen,i.e.then 224coordinatesarealgebraicallyindependentoverk.Then~(~)=(0),so(~)=A(~(x))=A((0))=~n.Soanypointof~noftranscendencedegreenoverkisagenericpointof~n.(2)Letk=~,~=C,n=2.Let~betheprincipalideal22generatedbyf(Xl,X2)=X1+X2-1.~isaprimeidealsincefisirreducible.ThusA~),i.e.theunitcircle,isavariety.ChoosexIE~andtranscendentaloverQ.Pickx2E~withx~=1-x~Thenthepoint~=(Xl,X2)belongstoA(~)Infact,x=isagenericpointofA~):Toseethis,itwillsufficetoshowthat~(~)=(X+X22-i)i.e.theprincipalidealgeneratedbyX~+X2_1.Ifg(XI,X2)E~(x)=thatis,ifg(xl,x2)=0,theng(Xl,X2)isamultipleofX~-1+x~,22sincex2isarootofX2-1+xI,whichisirreducibleoverQ(Xl).Moreprecisely,gc~l,X2)=(-1+~)h(Xl,X2),whereh(XI,X2)isapolynomialinX2andisrationalinX1SincexIwastranscendental,wegetg(xl,x2~=r+2-l~hrInviewoftheuniquefactorizationin@[X1],itfollowsthath(X1,X2)isinfactapolynomialinXl,X2Thus~(x__)=(X2+X~-1).(3)Letk=Q,~=C,n=2.Let~betheprincipalideal2generatedbyf(Xl,X2)=X1-X2.ThenA(~)isirreducibleandisaparabola.ChoosexIE~andtranscendentaloverQ,andput2x2=x1.Then~=(Xl,X2)liesinArAnargumentsimilarto 225theonegivenin(2)showsthatxisagenericpointofArForexample,Lindemann'sTheoremsaysthateistranseentalover~,andtherefore(e,e2)isagenericpointofAm).(4)Letk=@,~=C.Let~betheprincipalideal=(X~-X~).Thenaswehaveseenabove,A~/)isreducibleandisthereforenotavariety.(5)ConsideralinearmanifoldMdgivenbyaparameterrepresentationxi=bi+ailtI+...+aidtd(1<=i<=n).Heretheb.andthea..asgivenelementsofk]withthe(dn)-i13matrix(aij)ofrankd.Astl,...,tdrunthrough~,x__=(Xl,...,Xn)drunsthroughMItfollowsfromlinearalgebrathatMdisan9ytalgebraicset.(Itisa"d-dimensionallinearmanlfold.Seealsowaboutthenotionofdimension).InfactMdisavariety:Choose~l'''''~dalgebraicallyindependentoverk.Put~i=bi+ail~l+"'"+aided(Igign)and~=(~l,~2,...,~n)6Qn.Now~=EMd,so(~_)~Md.Conversely,iff(~_)-0,thenf(bI+allTI+...+aldTd,b2+a21T1+..~+a2dTd,...,bn+anlT1+...+andTd)=0,whereT1,...,Tdarevariables.Thusif~EMd,thenf(x)==0.Soeveryx6MdliesinA(~))=(~').ThereforewehaveshownthatMd=(~),orthatMdisavariety. 226(6)Takek=~,~=C,n=2,and~theprincipalidealgeneratedbyf(XI,X2)=X~-2X~.Overk=@,thispolynomialisirreducible.Thus~[isaprimeideal,andA~/)isavariety.However,ifwetakek~=~Q/2),thenf(Xl,X2)isnolongerirreducibleoverk~,sothat91isnolongeraprimeidealinkl[Xi,X2],andArisnolongeravariety.Thispromptsthedefinition:Avarietyiscalledanabsolutevarietyifitremainsavarietyovereveryalgebraicextensionofk.THEOREMiI.Everynon-emptyalgebraicsetisafiniteunionofvarieties.Proof:Wefirstshowthateverynon-emptycollection~ofalgebraicsetshasaminimalelement.Forifweformallideals~(S),whereSE~,thereisbyTheoremIAamaximalelementofthisnon-emptycollectionofideals.Say~(SO)ismaximal.WeclaimthatSOE~isminimal.ForifS1~SOwhereS1E~,then~(SI)~~(SO);butsince~(S0)ismaximal,~(SI)=~(S0).ThusS1=A(~(SI))=A(~(So))=SoSupposethatTheoremIIisfalse.LetbethecollectionofalgebraicsetsforwhichTheoremiIisfalse.ThereisaminimalelementSOof~.IfSOwereavariety,thenthetheoremwouldbetrueforSO.HenceSOisreducible.LetSO=AUB,whereA,Barealgebraicsets,withA%SO~B.SinceSOisminimalandA~SO,B~SO,thetheoremistrueforA,B.Hence,wecanwriteA=V1U...UV,andB=W1U...UWZwhereV.(I0andgiisofdegree2andisnotdivisiblebythecharacteristic.SeeShafarevich(1969),p.8.Let~bearationalfunctiononavarietyV=(x=)andletybeapointofV.Wesaythat~isdefinedaty_ifthereexistsarepresentativer(X)=a(X)/b(x)withb(y)t0.Ifthisisthecase,setw(~)=a(y)/b(y)__.Wehavetoshowthatthisindependentoftherepresentative.Supposethat(0isrepresentedbybotha(X__)/b(x)andbya(X)/5(X)=andthatb(y)~0,S(y)~0.Thedifference(aS-~b)/(bS)representsthezerorationalfunctiononV.Hencea(x__)S(x=)-a(x)b(x)=0,andsincex-~y,wehavea(y)b(y)-~(y)b(y)=0.Weconcludethata(y)/b(y)=~(y)/5(y). 239Examples:(1)Letn=3,k=Q,~=C,andVthesphere222x1+x2+x3-1=0.Letebetherationalfunctionrepresentedby1=1/1.Put~=(1,090).Now~isdefinedatyand~(~)=1.Now~isalsorepresentedby1/(X~+X~+X~).Againthedenominatordoesnotvanishat~.Ifweusethisrepresentation,weagainfind,asexpected]that~(y)=1.Finally~isalsorepresented222by(X1-X1-X2-X3)/(X1-1)Thisrepresentativecannotbeusedtocompute~(~),sinceitsdenominatorvanishesat[.(2)Letn,k,~andVbeasabove.LetWbetherationalfunctionrepresentedbyI/X3.Thisfunction~iscertainlydefinedif~EVandY3~0.Weaskifthereisrepresentativeofwhichallowsustodefine~(~)forsome~withY3=09Leta(~)'b(X)bearepresentative.Then=1a(X)=b(X)=-X3a(X)=X3b(X)-X3b(X__)22vanishesonV.Thusb(X)-a(X_)_X3E(X+X2+X3-i).So222b(X)E(X3,X1+X2+X3-i),andthereforeb(y=)=0,ify=EVandY3=0.Itfollowsthat~isdefinedpreciselyforthosepointsyonthespherewhicharenotonthecircleY3=0,22Yl+Y2-1=0.THEOREM3C.Let~bearationalfunctiononavarietyV.ThesetofpointsyEVforwhichq)isnotdefinedisaproper 240algebraicsubsetofV.Proof:Thesetofpointswhere~isnotdefinedisS=Vn~A((b(X))b(X)=wheretheintersectionistakenoverallb(X)whichoccurasadenominatorofarepresentativeof~.Sincetheintersectionofanarbitrarynumberofalgebraicsetsisanalgebraicset,Sisanalgebraicset.Inaddition,SisapropersubsetofV,sinceagenericpointofVisnotinS.Let~bearationalfunctionofavarietyV,andletWbeasubvarietyofV.Wesay~isdefinedonWif~isdefinedatagenericpointofW.Arationalmap~fromavarietyVtoomisdefinedsimplyasanm-tupleofrationalfunctions(~l,...,~m).Wesay~isdefinedaty~V,ifeach~i(~)isdefinedat~9Ifthisisthecase,put~([)=(~l(~),...,~n(~)).Thesetofpoints~EVforwhichisnotdefinedistheunionofthesetsofpointsforwhich~iisnotdefined(i=1,...,m).InviewofTheorem3C,andsinceafiniteunionofproperalgebraicsubsetsofavarietyisstillaproperalgebraicsubset,thepointswhere~isnotdefinedareaproperalgebraicsubsetofV.Theimageof~isdefinedastheclosureofthesetofpoints~(y),~EVAforwhich~isdefined.THEOREM3D.Theimageof~isavarietyW.IfxisagenericpointofV,then~0(x)isagenericpointofW. 241Proof:LetV=(~)9Ifx~Zandif~([)isdefined,we==_havetoshowthat~(x)-)~_(y)Let~_=(el,...,~m),andsupposethateiisrepresentedbyai(x)/bi=(X)=withb.1(y)=/0.Letf(~(x))=0andsupposethatf(U)=f(UI,...,Um)isofdegreed--=*_--iinU.Putldd1mg(Ul....'Urn'VI'''''Vm)=Vl"'"VmSincef(al(x__)/bl(X=),...,am(X=)/bm(X))=0,itfollowsthatg(al(x),...,am(X),bl(X),...,bm(X))====0.Butx=-~y=,sog(al(y__),...,am(y=),bl(y__),...,bre(y))=0,anddd/ai(y-)am(y-)bl(Y)=1...hm(y)~mf[~,''',~]=0.ddSincebl(Y__)1...bna(y__)r~0,itfollowsthata1(Y=)am(y)=f]=o.Soeverypolynomialfvanishingon~(x)alsovanisheson~(y),and~p(x)~~(y).221Example:LetVbethespherex1+x2+x3=1,andlet~_:V-~f~2havearepresentationas~_=((X+X2)/X3,-1/X).Let~==(~1,~2,~3)beagenericpointofV9Wehave~211-I,-~9~(~-)=k~2'~3 242Thusqo~)__=(~i,~2)satisfies~i+~2+i=09Since~_)hastranscendencedegreei,itisinfactagenericpointofthelinezI+z2+i=0.Thusthislineistheimageof~.Butnoteverypointonthislineisofthetype~(Z).If(Zl,Z2)isonthelineandis~(-i,0),thenifwepickyl,Y2,y3inwithYs=l/J222Yl+Y2+Y3-1=0,weobtain~(y)=(Zl,Z2)But(Zl,Z2)=(-i,0)isnotofthetype~(y)ForifY3~0,then~(Z)I(-i,0),andifY3=0,then~(y)isnotdefined.THEOREM3E.Let~bearationalmapfromVwithimageW.LetTbeaproperalgebraicsubsetofW.ThenthesetL~VconsistingofpointsZwhereeitherisnotdefinedorwhere~(y)sT,isaproperalgebraicsubsetofV.Proof:SupposeWandTliein~m.SupposeTisdefinedbyequationsgl(y).....gt(y)=0,where=y=(yl,...,ym).Letgi(Yl'''''Y-)IILhavedegreedijinY.(i~-i~t,1~-j~-m).3Put~Y)hi(Yl"''Ym'Zl....Zm)=ZdilZimgi.....'.....ZI'mLetr=.~r_I~=(aI(X)/bI(X),...==,am(X)/bm(X)=represent~andputr%=i"(X)==bl(X)...bm(X)==hi(al(X),=...,am(X),bl(X),==...,bin(X))=(I-~i=0andforeveryaEkthereisabEkwithbp=a. 248(ii),Everyalgebraicextensionofkisseparable.Proof.Weclearlymaysupposethatchark=p>0.(i)-~(ii).Apolynomialofk[X]ofthetype(4.4)a0+alXP+...+atXtpequals~0+blX+"'"+btxt)pwherebP=a(i=O,...,t).Thus11anirreduciblepolynomialoverkisnotofthetype(4.4),henceisseparable.(ii)~(i).SupposethenisanaEknotofthetypea=bpwithbEk.Thenthereisabwhichisnotinkbutinanalgebraicextensionofk,witha=bp.Sincepisaprime,itiseasilyseenthati=pisthesmallestpositiveexponentwithbiEk.ThepolynomialXp-a=(X-b)phasproperfactors(X-b)iwith1=1andthatourclaimistrueforsmallervaluesofn-d.WemaysupposewithoutlossofgeneralitythatXl,...,Xd+1havetranscendencedegreedoverk.Then(Xl,...,Xd+l)isthegenericpointofahypersurfacein~d+l.Thishypersurfaceisdefinedbyanequationf(zl,...,Zd+l)=0wheref(Zl,...,Zd+l)isirreducibleoverk.Sincekisperfect,itisclearthatfisnotapolynomialinZ~,...,Zd+p1ifchark=p>0.WemaythensupposewithoutlossofgeneralitythatfisnotapolynomialinZI'"'''Zd'ZPd+I"ThusfisseparableinthevariableZd+1,andXd+1isseparablealgebraicoverk(Xl,...,Xd)Bythetheoremoftheprimitiveelement(seeVanderWaerden,wthereisanx'withk(Xl,...,xd,Xd+1,Xd+2)=k(Xl,...,Xd,Xl).ThusxI=(Xl,...,xd,xI,Xd+3,...,xn)hask(xs)=k(x).ByinductionhypothesisthereisaY~~d+lwithk(xt)=k(y=),hencewith(4.5). 2505.LinearDisjointnessofFieldsLEMMA5A:Supposethat~,K,L,karefieldswithk-CKc-~,k_cL-CQ:Thefollowingtwopropertiesareequivalent:(i)IfelementsXl,...,xofKarelinearlyindependentm--overk,thentheyarealsolinearlyindependentoverL.(ii)Ifelementsyl,...,ynofLarelinearlyindependentoverk,thentheyarealsolinearlyindependentoverK.Proof:Bysymmetryitissufficienttoshowthat(i)implies(ii).Letyl,.~176nofLbelinearlyindependentoverk.LetxI,.-~nofKbenotallzero.Wewanttoshowthat(5.l)xlYI+...+XnYnr0.LetdbethemaximumnumberofXl,...,xnwhicharelinearlyinde-pendentoverk.Withoutlossofgenerality,wemayassumethatXl,...,x~arelinearlyindependentoverk~ThusfordPo'thesetVpisanabsolutevarietyofdimensiond0HerePodependsonlyonn,andthedegreesofthepolynomialsfl,.o.,f~oHenceifp>Po'thenthenumberN(p)ofsolutionsofthesystemofcongruencesfl(x)=-.o.=-f~(x)-=0(modp)satisfiesiN(p)_pdl<=cpd-1/2(iv)TheWell(1949)conjectures(seealsoCh.IV,wimplymuchbetterestimatesthanTheorem7AifVisa"non-singular"varietyofdimensiond>1TheseconjecturesWererecentlyprovedbyDeligne+)ButseetheremarkinthePreface. BIBLIOGRAPHYE.Artin(1924).QuadratischeK~rperimGebietderhoherenKongruenzenI,II.Math.Z.19,153-246.(1955)oElementsofAlgebraicGeometry.LectureNotes,NewYorkUniv.,InstoofMath.Sciences.J.Ax(1964)oZerosofpolynomialsoverfinitefields.Amer.J.Math.86,255-261.E.Bertini(1882).RendicontiR.Ist.Lombardo15,24-28.A.S.Besicovitch(1940).Onthelinearindependenceoffractionalpowersofintegers.J.LondonMath.Soc.15,3-6.E.Bombieri(1966).OnExponentialSumsinFiniteFields.Am.J.Math.88,71-105.(1973).Countingpointsoncurvesoverfinitefields(d'apres!S.A.Stepanov).SeminaireBourbaki,25eannee1972/73,No~430,Juin1973oZ.I.BorevichandI.R.Shafarevich(1966)oNumbertheory.AcademicPress.(Translatedfromthe1964RussianEd.)L.Carlitz(1969).Kloostermansumsandfinitefieldextensions.ActaArith.16,179-193.L.CarlitzandS.Uchiyama(1957).Boundsforexponentialsums.DukeMath.J.24,37-41.J.H~H.ChalkandR.A~Smith(1971)oOnBombieri'sestimateforexponentialsums.ActaArith.18,191-212.C.Chevalley(1935).Demonstration'd'unehypotlCesedeM.Artin.Abh.Math~Sem.Hamburgii,73-75.H.DavenportandH.Hasse(1935).NullstellenderKongruenzzetafunktioneningewissenzyklischenFallen.JoReineAng.Math.172,151-182.P.Deligne(1973).LaconjecturedeWeiloI.Inst.desHautesEtudesScientifiquesPub.Math.No.48,273-308.M.Deuring(1958).Lecturesonthetheoryofalgebraicfunctionsofonevariable.(TataInstitute)(1973SpringerlecturenotesNo.314)oB.Dwork(1960).Ontherationalityofthezetafunctionofanalgebraicvariety~Am.J.Math.82,631-648.M.Eichler(1963).Eiuf~hrungindieTheoriederalgebraischenZahlenundFunktionen.BirkhauserVerlagoC.F.Gauss(1801).DisquisitionesArithmeticae.FischerVerlag,Leipzig. 266H.Hasse(1936a).TheoriederhoherenDifferentialeineinemalgebraischenFunktionenk~rpermitvollkommenemKonstantenk~rperbeibeliebigerCharakteristik.J.ReineAng.Math.175,50-54.(1936b).ZurTheoriederabstraktenelliptisehenFunktionenk~rper--II.J.ReineAng.Math.175,69-88.(1936c).ZurTheoriederabstraktenelliptischenFunktionenk~rper--III.Ibid.193-208.D.Hilbert(1892)."UberdieIrreduzibilit~tganzerrationalerFunktionenmitganzzahligenKoeffizienten.J.ReineAng.Math.ii0,104-129.9/!!.J.R.Joly(1973).Equationsetvarletesalgebrzquessuruncorpsfini.L'EnseignementMath.(2)19,1-113.S.Lang(1958).Introductiontoalgebraicgeometry.Interscience,NewYork-London.(1961).DiophantineGeometry.Interscience,NewYork-London.S.LangandA.Well(1954).Thenumberofpointsofvarietiesinfinitefields.Am.J.Math.76,819-827.D.A.Mitkin(1972).Ontheestimationofarationaltrigonometricsumwithaprimedenominator(InRussian).VestnikMoscowUniv.(Mat.,Mech.)5~50-58.D.Mumford().LectureNotesonAlgebraicGeometry.HarvardUniversity.L.B.Nisnevich(1954).Onthenumberofpointsofanalgebraicvarietyinafiniteprimefield.(InRussian).Dokl.Akad.NaukSSSR99,17-20.E.Noether(1922).EinalgebraischesKriteriumfurabsoluteIrre-duzibilitHt.Math.Ann.85,26-33.A.Ostrowski(1919).ZurarithmetischenTheoriederalgebraischenGraVen.G~ttingerNachr.279-298.G.I.Perelmuter(1962).Oncertaincharactersums.(InRussian).Dokl.Akad.NaukSSSR144,58-61.G.I.PerelmuterandA.G.Postnikov(1972).Onthenumberofsolutionsofmonicequations.(InRussian).ActaArith.21,103-110.A.G.Postnikov(1967).ErgodicAspectsofthetheoryofcongruencesandofthetheoryofDiophantineApproximation.Transl.fromthe1966Russianed.(SteklovInst.ofMath.No.82).Am.Math.Sot.,Providence,R.I.P.Roquette(1953).ArithmetischerBeweisderRiemannschenVermutunginKongruenzzetafunktionenk~rpernbeliebigenGeschlechts.J.ReineAng.Math.191,199-252. 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