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Evalutaion of certain exponential sums of quadratic functions over a finite fields of odd characteristic

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Evalutaion of certain exponential sums of quadratic functions over a finite fields of odd characteristic
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English
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Draper, Sandra D
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Subjects / Keywords:
Artin-Schreier Theorem
Gauss sum
Law of quadratic reciprocity
Legendre symbol
Quadratic form
Dissertations, Academic -- Mathematics -- Masters -- USF
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theses   ( marcgt )
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ABSTRACT: Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)x where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)).^^ We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.
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Thesis (M.A.)--University of South Florida, 2006.
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by Sandra D. Draper.
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EvaluationofCertainExponentialSumsofQuadraticFunctionsoveraFiniteFieldofOddCharacteristicbySandraD.DraperAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofArtsDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:Xiang{DongHou,Ph.D.BrianCurtin,Ph.D.StephenSuen,Ph.D.DateofApproval:June22,2006Keywords:Artin-SchreierTheorem,Gausssum,lawofquadraticreciprocity,Legendresymbol,quadraticformcCopyright2006,SandraD.Draper

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ContentsListofTablesiiAbstractiii1.Introduction12.BackgroundfromNumberTheory33.QuadraticFormsonFnp,theMultivariableApproach54.QuadraticFormsonFnp,theSingleVariableApproach105.ComputationoftheNullity116.Sf+bxFollowsfromSf147.FromSf;ntoSf;qsn,q6=p;2168.FromSf;ntoSf;2sn;s>0219.FromSf;ntoSf;pn2610.When21=22==2k3111.TheFormulaforSax1+p3512.TablesofNumericalResults37References45i

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ListofTables1Valuesoflmfwithpn=3,4381Continued391Continued402Valuesoflmfwithpn=5,3412Continued422Continued432Continued44ii

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EvaluationofCertainExponentialSumsofQuadraticFunctionsOveraFiniteFieldofOddCharacteristicSandraDraperABSTRACTLetpbeanoddprime,anddenefxasfollows:fx=kXi=1aixpi+12Fpn[x]where01<2<
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1.IntroductionLetpbeaprimeandnapositiveinteger.LetFpndenotetheniteeldwithpnelements.Whenmjn,thetracefromFpntoFpmiswrittenasTrn=m.WedenotetheTrn=1asTrn.Leteny=e2iTrny=pfory2Fpn.In1979,Carlitzwroteapaper[3]evaluatingthesumXx2Fqenax3+bx;a;b2Fq;whereq=2n.In1980,Carlitzwroteanotherpaper[4]evaluatingthesumXx2Fqenaxp+1+bx;a;b2Fq;whereq=pnagain,butpisanoddprime.AsimilarsumXx2Fpnenaxp+1;a2Fpn;0;wasalsoimpliedbytheresultsofBaumertandMcEliece[2].Alsosee[7].In2005,Houwroteapaper[8]forthep=2case,generalizingtheresultbyCarlitz.Moreprecisely,in[8],thepolynomialax3+bxin[3]wasreplacedbyapolynomialoftheformfx=Pki=1aix2i+1+bx2F2n[x].Possibleextensionoftheresultsof[8]forthecaseofp=2tothecaseofoddppresentedtheopportunityforthisthesis.Thistopic,evaluationofexponentialsums,hasapplicationsinmanyareaswithinandoutsidemathematics.Innumbertheory,exponentialsumsareapowerfultooltostudythenumberofsolutionsofpolynomialequationsoverniteelds,see[13,Chapter6].Exponentialsumsarealsowidelyusedincodingtheory,informationtheory,cryptographyandcombinatorics.Infact,thesumPx2Fpnenaxp+1aroseinthestudyofweightsofirreduciblecycliccodesseeBaumertandMcEliece[2]andthestudyofthecross-correlationfunctionbetweentwomaximallinearsequencesseeHelleseth[7].Inthisthesis,wealwaysletpbeaoddprime.Wedenefxasfollows:fx=kXi=1aixpi+12Fpn[x]where01<2<
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Hereisabrieyoutlineofthethesis.Section2containssomenecessarytheoremsandinformationfromnumbertheory.Sections3and4discussthequadraticformsonFpnusingboththemultivariableandsingle-variableapproach.Wemayincludealineartermtofx;theresultingsumwillbeSf+bx;n,b2Fpn.Infact,Sf+bx;n,asafunctionofb,istheFouriertransform"ofTrnfx.InSection5,weshowthatSf+bx;nfollowsfromSf;ninastraightforwardway.InSection6,themethodforthecomputationofthenullityisshown.InSections7through9,wederiverelative"formulasforSf;mnintermsofSf;n.Here,Sf;mnistheexponentialsumofthesamepolynomialf2Fpn[x]overanextensioneldFpmn.Threecasesrequiredierentmethods.InSection7,weassumem=qswhereqisaprimedierentfrompand2.WeuseacongruencerelationresultingfromasuitableactionbytheGaloisgroupGalFqsn=FpntodeterminethetypeofTrqsnf.InSection8,Weassumem=2s.InadditiontothecongruenceinSection7,aspecialpartitionofFp2nisneeded.InSection9,weassumem=p.Inthiscase,thecongruencemethodintheprevioustwosectionsdoesnotwork.However,theArtin-SchreierTheoremfortheextensionFppn=FpnallowsustocomputethetraceofFppn=Fpnratherexplicitly.WeareabletoexpressSf;pnintermsofSf;nundertheconditionthatminfpi:i1kg1,wherepisthep-adicorder.InSection10,welookatthecasewhere21=22==2k<2n.AnexplicitformulaforSf;nisobtained.Section11handlesthespecialcaseofSax1+pwhichcanbeevaluatedasaresultoftheprevioussection.TablesofnumericalvaluesofcertainnullitiesareincludedinSection12.2

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2.BackgroundfromNumberTheoryInthissection,wecollectsomewell-knownfactsfromnumbertheorytobeusedlater.Letpbeanoddprimeanda2Z.TheLegendresymbola pisdenedasa p=8><>:0ifpja;1ifaisasquareinFp;)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifaisanonresidueinFp:Theorem2.1.iWehave)]TJ/F30 10.909 Tf 8.485 0 Td[(1 p=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1;2 p=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1:iiThelawofquadraticreciprocityLetqbeanoddprimewithq6=p.Thenp qq p=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1q)]TJ/F15 7.97 Tf 6.587 0 Td[(1:FortheproofofTheorem2.1,see[11,x5.2]Letq=pn,wherepisanoddprimeandn2Z+.Leten:Fq!Cbethecanonicaladditivecharacter,i.e.,eny=TrFq=Fpyp;y2Fq;wherep=e2i=p.Let:Fq!CbethequadraticcharacterofFq,i.e.,y=8><>:0ify=0;1ifyisasquareinFq;)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifyisanonsquareinFq:Whenq=p,yistheLegendresymboly p.TheGaussquadraticsumonFq,denotedbyG,isdenedasG=Xy2Fqyey:Lemma2.2.iWehaveG=Xy2Fqey2:iiForeacha2Fq,Xy2Fqeay2=aXy2Fqey2:Proof.i:Wehave.1G=en+Xy2Fqyeny=Xy2Fq2eny)]TJ/F34 10.909 Tf 24.763 10.364 Td[(Xy2FqnFq2eny:3

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Ontheotherhand,.2)]TJ/F30 10.909 Tf 8.485 0 Td[(1=Xy2Fqeny=Xy2Fq2eny+Xy2FqnFq2eny:Adding.1and.2,wehaveG=1+2Xy2Fq2eny=Xy2Fqeny2:ii:IfaisasquareinFq,say,a=b2,b2Fq,thenXy2Fqenay2=Xy2Fqen)]TJ/F30 10.909 Tf 5 -8.837 Td[(by2=Xy2Fqeny2:IfaisanonsquareinFq,thenXy2Fqeny2+Xy2Fqenay2=2Xy2Fqeny=0:So,Xy2Fqeny2=)]TJ/F34 10.909 Tf 11.98 10.364 Td[(Xy2Fqenay2=aXy2Fqenay2:TheGaussquadraticsumiscompletelydetermined.WeusegptodenotetheGaussquadraticsumoverFp,i.e.,gp=Xy2Fpy2p:Itiseasytoshowthatgp=p1 2ifp1mod4;p1 2iifp)]TJ/F30 10.909 Tf 20 0 Td[(1mod4:IttookGaussfouryears{1805todeterminethatsignsinbothcasesabovearepositive,i.e.,gp=p1 2ifp1mod4;p1 2iifp)]TJ/F30 10.909 Tf 20 0 Td[(1mod4=i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12p1 2:TheGaussquadraticsumGoverFqfollowsfromtheDavenport-HasseTheoremontheGausssumofaliftedcharacter[5].WehaveG=)]TJ/F30 10.909 Tf 8.485 0 Td[(1n)]TJ/F15 7.97 Tf 6.586 0 Td[(1gnp;whereq=pn.4

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3.QuadraticFormsonFnp,theMultivariableApproachAquadraticformonFnqisafunctionF:Fnq!Fqdenedbyahomogeneouspolynomialinx1;:::;xn.Whenqisodd,aquadraticformonFnqcanbewrittenasFx1;:::;xn=x1;:::;xnAx1;:::;xnT;whereAisannnsymmetricmatrixoverFq.ThematrixAiscalledthematrixofF.TwoquadraticformsFx=xAxTandGx=xBxT,wherex=x1;:::;xn,arecalledequivalentifthereexistsP2GLn;FqsuchthatFx=GxP,orequivalently,A=PBPT.So,theclassicationofquadraticformsonFnqqoddunderequivalenceisthesameastheclassicationnnsymmetricmatricesoverFqundercongruence.Congruenceofsymmetricmatricesisdenotedby=.Theclassicationofquadraticformoverniteeldsiswell-known.Whenqisodd,theclassicationissimpleandisgiveninthenexttheorem.Whenqiseven,theclassicationisslightlyinvolved,see[1,6].Fortherestofthissectionqisapowerofanoddprimep.Theorem3.1.Letx=x1;:::;xnandAbeannnsymmetricmatrixoverFnq.TheneveryquadraticformFx=xAxTonFnpisequivalenttox21++x2r)]TJ/F15 7.97 Tf 6.587 0 Td[(1+dx2r;where0rnandd2Fq.TheintegerriscalledtherankofFandisdenotedbyrankF.TheimageofdinFq=Fq2,i.e.dFq2,iscalledthediscriminantofF.SometimeswesimplysaythatthediscriminantofFisd.Ifr=0,thediscriminantofFisdenedtobe1.Twoquadraticformsareequivalentifandonlyiftheyhavethesamerankanddiscriminant.TheproofofTheorem3.1willfollowtwolemmas.Lemma3.2.EveryelementofFqisasumoftwosquares.Proof.ThemultiplicativegroupFqcontains1 2q)]TJ/F30 10.909 Tf 10.664 0 Td[(1squaresand1 2q)]TJ/F30 10.909 Tf 10.664 0 Td[(1nonsquares.So,Fqhas1 2q+1squaresincluding0.WeclaimthatthesetofsquaresinFqisnotclosedunderaddition.Otherwise,thesetofsquaresofFqwouldbeasubgroupofFq;+,whichisimpossiblesince1 2q+1doesnotdivideq.Thus,thereexistsanonsquared2Fqsuchthatd=a2+b2forsomea;b2Fq.Nowletx2Fqbearbitrary.Ifxisasquare,say,x=y2,thenx=y2+02.Ifxisanonsquare,sincejFq=Fq2j=2,wemusthavex=dy2forsomey2Fq.Thenx=a2+b2y2=ay2+by2.5

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Lemma3.3.EverynnsymmetricmatrixofrankroverFqiscongruentto266666666641...1d0...037777777775:Proof.LetA=26664a11a12a1na12a22a2n.........a1na2nann37775beannnsymmetricmatrixoverFq.WemayassumeA6=0.BysuitablepermutationsofrowsandcolumnsofA,wemayassumethattherstrowofAisnotall0.First,weshowthatAiscongruenttoadiagonalmatrix.TherearetwocasesforA,a116=0ora11=0.Case1:Assumea116=0.LetP=266641)]TJ/F17 7.97 Tf 9.681 4.489 Td[(a12 a110......)]TJ/F17 7.97 Tf 9.68 4.489 Td[(a1n a110377752GLn;Fq:ThenPAPT=26664a11000...A1037775;whereA1isann)]TJ/F30 10.909 Tf 10.791 0 Td[(1n)]TJ/F30 10.909 Tf 10.791 0 Td[(1symmetricmatrixoverFq.Usinginductiononn,wemayassumethatA1iscongruenttoadiagonalmatrix.So,Aiscongruenttoadiagonalmatrix.Case2:Assumea11=0.Thenoneofa12;:::;a1nisnonzero,say,a126=0.LetP=26664111...1377752GLn;Fq:ThenPAPT=266642a12.........37775;where2a126=0.Therefore,weareincase1.SowehaveprovedthatAiscongruenttoadiagonalmatrix.Wecanuserowandcolumnpermutationstomoveallnon-zerodiagonalentriesofAtotherstentriesinthediagonal.6

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ThenAhastheform:266666664a11...arr0...0377777775;a11;:::;arr2Fq:Therearetwocasesforeachaii:itiseitherasquareoranonsquareinFq.Withoutlossofgenerality,wemayassumethata11;:::;assaresquaresandas+1;s+1;:::;arrarenonsquares.So,aii=b2iif1is;db2iifs+1ir;whereb1;:::;br2Fqandd2Fqisanonsquare.LetP=266666664b1...br1...1377777775:Then.1PAPT=26666666666666641...1d...d0...03777777777777775:ByLemma3.2,d=a2+b2,a;b2Fq.So,dd=ab)]TJ/F33 10.909 Tf 8.485 0 Td[(ba11a)]TJ/F33 10.909 Tf 8.485 0 Td[(bba=11:Therefore,thediagonalmatrixin.1iscongruentto2666666641...10...0377777775or266666666641...1d0...037777777775dependingonwhetherr)]TJ/F33 10.909 Tf 10.909 0 Td[(sisevenorodd.7

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ProofofTheorem3.1.LetFx1;:::;xn=x1;:::;xnAx1;:::;xnTbeaquadraticforminx1;:::;xnoverFq,whereAisannnsymmetricmatrixoverFq.ByLemma3.2,thereexistsQ2GLn;FqsuchthatQAQT=266666666641...1d0...0377777777759>>>=>>>;r;d2Fq:ThenFx1;:::;xn=Fx1;:::;xnQ=x1;:::;xnQAQTx1;:::;xnT=x21+x22++x2r)]TJ/F15 7.97 Tf 6.586 0 Td[(1+dx2r:LetGx1;:::;xnbeanotherquadraticformoverFqwithrankr0anddiscriminantd0.Ifr0=randd=d0,ofcourse,G=F.Ontheotherhand,ifG=F,bythenexttheorem,wehavedgrppn)]TJ/F17 7.97 Tf 6.587 0 Td[(r=Xx1;:::;xn2FnpFx1;:::;xnp=Xx1;:::;xn2FnpGx1;:::;xnp=d0gr0ppn)]TJ/F17 7.97 Tf 6.586 0 Td[(r0:Itfollowsthatr=r0andd=d0.LetFbeaquadraticformonFnqwithdiscriminantd.Thend2f1giscalledthetypeofF.So,typeofF=1ifthediscriminantofFisasquare;)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifthediscriminantofFisanonsquare:Theorem3.4.LetFbeaquadraticformonFnpwithtypetandrankr.ThenXx1;:::;xn2FnpFx1;:::;xnp=tgrppn)]TJ/F17 7.97 Tf 6.586 0 Td[(r:8

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Proof.WemayassumeFx1;:::;xn=x21++x2r)]TJ/F15 7.97 Tf 6.586 0 Td[(1+dx2r,0rn,d2Fq.ThenXx1;:::;xn2FnpFx1;:::;xnp=)]TJ 9.388 1.527 Td[(Xx12Fnpx21p)]TJ 14.096 1.527 Td[(Xxr)]TJ/F16 5.978 Tf 5.756 0 Td[(12Fnpx2r)]TJ/F16 5.978 Tf 5.756 0 Td[(1p)]TJ 14.391 1.527 Td[(Xxr2Fnpdx2rp)]TJ 19.013 1.527 Td[(Xxr+12Fnp1)]TJ 9.862 1.527 Td[(Xxn2Fnp1=)]TJ 6.932 1.527 Td[(Xx2Fpx2p)]TJ 11.932 1.527 Td[(Xx2Fpx2p)]TJ 6.932 1.528 Td[(Xx2Fpx2p)]TJ 11.932 1.528 Td[(Xx2Fpdx2ppn)]TJ/F17 7.97 Tf 6.586 0 Td[(r=)]TJ 6.932 1.528 Td[(Xx2Fpx2pr)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ 6.932 1.528 Td[(Xx2Fpdx2ppn)]TJ/F17 7.97 Tf 6.587 0 Td[(r=gr)]TJ/F15 7.97 Tf 6.587 0 Td[(1pdgppn)]TJ/F17 7.97 Tf 6.586 0 Td[(rbyLemma2.2ii:9

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4.QuadraticFormsonFnp,theSingleVariableApproachInSection3,quadraticformsonFnqarerepresentedbyhomogeneouspolynomialsofdegree2inx1;:::;xn;thiswayofrepresentingquadraticformsisusuallyreferredtoasthemulti-variableapproach.However,sinceFnqisidentiedwiththeniteeldFqn,thereisanotherwayofrepresentingquadraticformsofFnqwhichiscalledthesingle-variableapproach.Leta2Fqnand0.Let1;:::;nbeabasisofFqnoverFqandletx=x11++xnn,x1;:::;xn2Fnq.LetTr=TrFqn=Fq.ThenTraxq+1=TrhanXi=1xiiq+1i=TrhanXi=1xiqinXj=1xjji=TrhanXi=1qijxixji=Xi;jTraqijxixjisaquadraticforminx1;:::;xnoverFq.Therefore,ifa1;:::;ak2Fqand1;:::;k0,then.1TrkXi=1aixqi+1isaquadraticforminx1;:::;xnoverFq.Ontheotherhand,everyquadraticformonFnqidentiedwithFqncanbewrittenintheformof.1.Infact,ifnisodd,abasisoftheFq-spaceofquadraticformsonFnqisgivenbyTraxqi+1;0in)]TJ/F30 10.909 Tf 10.909 0 Td[(1 2;a2A;whereAisabasisofFqnoverFq.Ifniseven,abasisoftheFq-spaceofquadraticformsonFnqisgivenbyTraxqi+1;0in 2;a2AandTraxqn=2+1;b2B;whereAisabasisofFqnoverFqandBisabasisofFqn=2overFq.See[9]fordetails.10

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5.ComputationoftheNullityLetFx1;:::;xn=x1;:::;xnAx1;:::;xnTbeaquadraticformonFnpwhereAisannnsymmetricmatrixoverFp.Forz;x2Fnp,wehaveFx+z)]TJ/F33 10.909 Tf 10.909 0 Td[(Fx=xAzT+zAxT+zAzT=2zAxT+zAzT:Thus,zA=0ifandonlyifFx+z)]TJ/F33 10.909 Tf 10.909 0 Td[(Fxisconstantforallx2Fnp.Son)]TJ/F30 10.909 Tf 10.909 0 Td[(rankF=dimFpfz2Fnp:Fx+z)]TJ/F33 10.909 Tf 10.909 0 Td[(Fx=constantforallx2Fnpg:Thenumbern)]TJ/F30 10.909 Tf 10.909 0 Td[(rankFiscalledthenullityofF.Consider.1fx=kXi=1aixpi+1+bx2Fpn[x];where012k=.InSection4,wesawthateveryquadraticformonFnpidentiedwithFpncanbewrittenasTrnfxforsomef2Fpn[x]oftheform.1.DenotethenullityofTrnfxbylnf,i.e.lnf=dimFpLnf;whereLnf=fz2Fpn:Trnfx+z)]TJ/F33 10.909 Tf 10.909 0 Td[(fx=constantforallx2Fpngwhichiscalledthenullspaceoff.LettnfdenotethetypeofTrnfx.ThenbyTheorem3.4,wehaveSf;n=tnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnf:Thus,Sf;niscompletelydeterminedbylnfandtnf.11

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Thereisaneectivealgorithmtocomputelnf.WehaveTrnfx+z)]TJ/F33 10.909 Tf 10.909 0 Td[(fx=TrnkXi=1aix+zpi+1)]TJ/F17 7.97 Tf 16.477 13.636 Td[(kXi=1aixpi+1=TrnkXi=1aix+zpi+1)]TJ/F33 10.909 Tf 10.909 0 Td[(aixpi+1=TrnkXi=1aixpi+1+xpiz+xzpi+zpi+1)]TJ/F33 10.909 Tf 10.909 0 Td[(aixpi+1=TrnkXi=1aixpiz+xzpi+zpi+1=TrnkXi=1aizpi+1+kXi=1aixpiz+xzpi=Trnfz+TrnkXi=1aixpiz+aixzpi=Trnfz+TrnkXi=1aixpizp)]TJ/F18 5.978 Tf 5.756 0 Td[(i+aixzpi=Trnfz+TrnkXi=1ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iixzp)]TJ/F18 5.978 Tf 5.757 0 Td[(i+aixzpi=Trnfz+TrnxkXi=1ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iizp)]TJ/F18 5.978 Tf 5.756 0 Td[(i+aizpi=Trnfz+TrnxpkXi=1ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iizp)]TJ/F18 5.978 Tf 5.756 0 Td[(i+apizp+i=Trnfz+Trnxpfz;.2wherep)]TJ/F17 7.97 Tf 6.586 0 Td[(iisapositiveintegerandp)]TJ/F17 7.97 Tf 6.587 0 Td[(ipi1modpn)]TJ/F30 10.909 Tf 10.909 0 Td[(1,andfz=kXi=1ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iizp)]TJ/F18 5.978 Tf 5.756 0 Td[(i+apizp+i:Notethatf2Fpn[x]isap-polynomialwithnorepeatedroots.Thus,Trnfx+z)]TJ/F33 10.909 Tf 10.05 0 Td[(fxisconstantforallx2Fpnifandonlyifzisarootoff.Therefore,Lnf=fz2Fpn:fz=0gandlnf=dimFpfz2Fpn:fz=0g=logpjfz2Fpn:fz=0gj=logpdegf;xpn)]TJ/F33 10.909 Tf 10.909 0 Td[(x:.3Sincek=,thenthedegreeoffisalwaysp2.Let.4s=minfm:njm;lmf=2g:12

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Thus,FpsisthesplittingeldoffoverFpn.Itisthenobviousthatforallmultiplesmofn,.5lmf=lm;sf:Therefore,wecancomputethenullitylmfforall0>><>>>:76ifm=175;75ifm<175;25jm;72ifm<175;7jm;71ifm<175;25-m;7-m:ThesplittingeldoffoverF7isF7175.Writem=5a7bm,wherea;b0and7;m=1.Thenforeverym>0,lmf=8>>><>>>:6ifa2andb1;5ifa2andb=0;2ifa1andb1;1ifa1andb=0:Infact,wehavecomputedthevaluesoflmfwithpn=3;4,andpn=5,3.TheresultsaretabulatedinSection12.13

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6.Sf+bxFollowsfromSfLetf2Fpn[x]begivenin.1.OurmainobjectiveistocomputethesumSf;n.Letb2Fpn.ThesumSf+bx;n=Xx2Fpnenfx+bxisalsoimportant.Infact,whenviewedasafunctionofb,Sf+bx;nistheFouriertransform"ofTrnf.However,wewillseethattheseeminglymoregeneralsumSf+bx;nfollowseasilyfromSf;n.Hence,itsucestodetermineSf;n.Theorem6.1.Letb2Fpn.ThenSf;n Sf+bx;n=pn+lnfenfx0iffx=bphasasolutionx02Fnp;0otherwise:Proof.WehaveSf;n Sf+bx;n=Xx2FpnenfxXy2Fpn efy+by=Xx;y2Fpnenfx)]TJ/F33 10.909 Tf 10.909 0 Td[(fy)]TJ/F33 10.909 Tf 10.909 0 Td[(by=Xx;y2Fpnenfx+y)]TJ/F33 10.909 Tf 10.909 0 Td[(fy)]TJ/F33 10.909 Tf 10.909 0 Td[(by=Xx;y2Fpnenfx+yfxp)]TJ/F18 5.978 Tf 5.756 0 Td[()]TJ/F33 10.909 Tf 10.909 0 Td[(byby5.2=Xx2FpnenfxXy2Fpnen[yfxp)]TJ/F18 5.978 Tf 5.756 0 Td[()]TJ/F33 10.909 Tf 10.909 0 Td[(b]=pnXx2Fpnfx=bp)]TJ/F18 5.978 Tf 5.756 0 Td[(enfx:Iffx=bp)]TJ/F18 5.978 Tf 5.757 0 Td[(hasnosolutioninFpn,thenSf;n Sf+bx;n=0.Thus,assumefx=bphasasolution,x02Fpn.Thenthesolutionsetoffx=bpisx0+fx2Fpn:fx=0gsincefxisap-polynomial.Thesolutionsetcanalsobewrittenasx0+LnfwhereLnfisthenullspaceoff.SinceTrnfxisaquadraticformonFpn,itisconstanton14

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x0+Lnf.SoSf;n Sf+bx;n=pnXx2Fpnfx=bp)]TJ/F18 5.978 Tf 5.756 0 Td[(enfx=pnXx2x0+Lnfenfx=pnplnfenfx0=pn+lnfenfx0:Corollary6.2.Sf+bx;n= efx0Sf;niffx=bphasasolutionx02Fpn;0otherwise:Proof.SinceSf;n6=0,thenbyTheorem6.1,weknowthattheonlycasewehavetoconsideriswhenfx=bphasasolutionx02Fpn.ThenbyTheorem6.1,Sf+bx;n=pn+lnf Sf;n efx0=pn+lnf jSf;nj2 efx0Sf;n:SinceSf;n=tnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpplnf,wehavejSf;nj2=pn+lnf.Thisyieldsthedesiredresult.15

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7.FromSf;ntoSf;qsn,q6=p;2Inthissection,assumethatqisanoddprimesuchthatq6=pands0.Theorem7.1.IntheringZ[p],wehaveSf;qsnXx2Fpneqsnfxmodq:Equivalently,tqsnfgqsn)]TJ/F17 7.97 Tf 6.587 0 Td[(lqsnfpplqsnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnfmodq:Proof.LetTi=fx2FpqsnnFpn:Trqsnfx=ig;i2Fp.WecanthenwriteSf;qsn=Xx2FpqsnnFpneqsnfx+Xx2Fpneqsnfx=p)]TJ/F15 7.97 Tf 6.587 0 Td[(1Xi=0jTijip+Xx2Fpneqsnfx:TheGaloisgroupAutFpqsn=FpnactsonTi.Takex2Tiandtake2AutFpqsn=Fpn.Sincef2Fpn[x],wehavefx=fxforallx2Fpqsn.Thus,Trqsn)]TJ/F33 10.909 Tf 5 -8.836 Td[(fx=Trqsn)]TJ/F33 10.909 Tf 5 -8.836 Td[(fx=Trqsnfx=i:Secondly,notethatsincex=2Fpn,thenx=2Fpn.Otherwise,wewouldhavex=)]TJ/F15 7.97 Tf 6.587 0 Td[(1x2Fpn.Therefore,x2Ti:TiisaunionofAutFpqsn=Fpn-orbitsofcardinalityofqtopositivepowers.Recallthatthecardinalityoftheorbitisthesizeofthegroupdividedbythesizeofthestabilizer.Thus,sincejAutFpqsn=Fpnj=qs,thenthecardinalityofeveryorbitmustbeqtosomepositivepower.Ifthecardinalityoftheorbitis1,thentheelementofthatorbitisxedbyeveryautomorphism,andthatorbitiscontainedinthebaseeldFpn.But,theorbitisinTi,whichisacontradictionastheorbitcannotbeinFpn.Thus,thecardinalityofeveryorbitisequaltoqh,where0
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thentqsnfgqsn)]TJ/F17 7.97 Tf 6.587 0 Td[(lqsnfpplqsnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnfmodq:Lemma7.2.Wehaveplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf1modqand1 2[lqsnf)]TJ/F33 10.909 Tf 10.909 0 Td[(lnf]2Z.Proof.LetX=fx2FqsnpnFpn:fx=0.ThentheGaloisgroupAutFqsnp=FpnactsonXandXisaunionofAutFqsnp=Fpn-orbitsofcardinality>1.Then,plqsnf)]TJ/F33 10.909 Tf 10.909 0 Td[(plnf=jXj0modqplnfplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf)]TJ/F30 10.909 Tf 10.909 0 Td[(10modqsincepisaprimedierentfromqandlnfisaninteger,thenplqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf)]TJ/F30 10.909 Tf 9.235 0 Td[(10modq.Therefore,plqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf1modq:Toprovethesecondpartofthelemma,beginwiththeequationfromTheorem7.1,.1tqsnfgqsn)]TJ/F17 7.97 Tf 6.586 0 Td[(lqsnfpplqsnf)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnfmodq:Usingtheaboveequationandplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf1modq,wecansimplify.1tobe.2tqsnfgqsn)]TJ/F17 7.97 Tf 6.587 0 Td[(lqsnfp)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpmodq:Assumetothecontraryof1 2[lqsnf)]TJ/F33 10.909 Tf 11.017 0 Td[(lnf]=2Z,i.e.lqsnf)]TJ/F33 10.909 Tf 11.017 0 Td[(lnfisodd.Thenexactlyoneofqsn)]TJ/F33 10.909 Tf 10.227 0 Td[(lqsnforn)]TJ/F33 10.909 Tf 10.228 0 Td[(lnfisodd.Thisistrueasqsnandnhavethesameparity,butlqsnfandlnfhavedierentparities.Also,sincegp=p1 2orip1 2,theng2p=p.Thus,sinceqsn)]TJ/F33 10.909 Tf 10.912 0 Td[(lqsnforn)]TJ/F33 10.909 Tf 10.912 0 Td[(lnfisodd,then.2hasgptoanevenpowerononesideandgptoanoddpowerontheotherside.Thisgivesusptosomepowerononesideandptosomepowermultipliedbygpontheotherside,respectively.Thus,gppabmodqforsomea;b2Z;a0.Let2AutQgp=Qsuchthatgp=)]TJ/F33 10.909 Tf 8.485 0 Td[(gp.SinceQgppandQp=QisGalois,wecanextendto2AutQp=Q.Byapplyingtogppabmodq,wehave)]TJ/F33 10.909 Tf 8.485 0 Td[(gppabmodq:Bysubtractingthetwoequations,theresultis2gppa0modq,whichisacontradiction.Thus,1 2[lqsnf)]TJ/F33 10.909 Tf 10.909 0 Td[(lnf]2Z.LetoqpdenotethemultiplicativeorderofpinZ=qZ.Then,usingtheaboveequationplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf1modq,wendthat1 oqplqsnf)]TJ/F33 10.909 Tf 10.91 0 Td[(lnf2Z.Clearlyp1 2lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 oqplqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfmodq:Theorem7.3.Wehave.3tqsnf=tnf)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q pslnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]:17

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Proof.First,recallthatp1 2q)]TJ/F15 7.97 Tf 6.587 0 Td[(1p qmodqandnotethatqs)]TJ/F30 10.909 Tf 11.304 0 Td[(1sq)]TJ/F30 10.909 Tf 11.304 0 Td[(1mod4.Wecanstartwith7.2tqsnfgqsn)]TJ/F17 7.97 Tf 6.586 0 Td[(lqsnfp)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpmodqandsimplifyittobetqsnfgqsn)]TJ/F17 7.97 Tf 6.586 0 Td[(np)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfglqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpmodqtqsnfgnqs)]TJ/F15 7.97 Tf 6.587 0 Td[(1p)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfglqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpmodq:RecallthattheGausssumisgp=i1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12p1 2.Then,g2p=)]TJ/F33 10.909 Tf 5 -8.837 Td[(i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12p1 22=i1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(12p=)]TJ/F30 10.909 Tf 8.485 0 Td[(1p)]TJ/F16 5.978 Tf 5.756 0 Td[(1 2p:Usingthisfact,thengnqs)]TJ/F15 7.97 Tf 6.586 0 Td[(1p=[)]TJ/F30 10.909 Tf 8.485 0 Td[(1p)]TJ/F16 5.978 Tf 5.756 0 Td[(1 2p]1 2nqs)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4np)]TJ/F15 7.97 Tf 6.586 0 Td[(1qs)]TJ/F15 7.97 Tf 6.586 0 Td[(1p1 2nqs)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4np)]TJ/F15 7.97 Tf 6.586 0 Td[(1sq)]TJ/F15 7.97 Tf 6.586 0 Td[(1p1 2nsq)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4nsp)]TJ/F15 7.97 Tf 6.586 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1p1 2q)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4nsp)]TJ/F15 7.97 Tf 6.586 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 6.195 -1.457 Td[(p qsnmodq:Also,simplifyglqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfp=[)]TJ/F30 10.909 Tf 8.485 0 Td[(1p)]TJ/F16 5.978 Tf 5.756 0 Td[(1 2p]1 2[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]p1 2[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf])]TJ/F30 10.909 Tf 8.485 0 Td[(11 oqplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfmodq=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]+1 oqplqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfmodqThen,theabovevaluescanbesubstitutedintotqsnfgnqs)]TJ/F15 7.97 Tf 6.586 0 Td[(1p)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfglqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpmodqastqsnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4nsp)]TJ/F15 7.97 Tf 6.587 0 Td[(1q)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F33 10.909 Tf 6.196 -1.456 Td[(p qsn)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]modq18

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tqsnftnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf)]TJ/F33 10.909 Tf 6.195 -1.456 Td[(p q)]TJ/F17 7.97 Tf 6.587 0 Td[(sn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf])]TJ/F16 5.978 Tf 7.782 3.258 Td[(1 4nsp)]TJ/F15 7.97 Tf 6.587 0 Td[(1q)]TJ/F15 7.97 Tf 6.587 0 Td[(1modq=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf)]TJ/F33 10.909 Tf 6.195 -1.456 Td[(p qsn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf])]TJ/F16 5.978 Tf 7.782 3.258 Td[(1 4nsp)]TJ/F15 7.97 Tf 6.586 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q psn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf)]TJ/F33 10.909 Tf 6.195 -1.456 Td[(p qsn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.837 Td[([lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf])]TJ/F17 7.97 Tf 6.587 0 Td[(snq)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q p)]TJ/F17 7.97 Tf 6.586 0 Td[(slnf)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q psn)]TJ/F33 10.909 Tf 6.195 -1.457 Td[(p qsn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.836 Td[([lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf])]TJ/F17 7.97 Tf 6.586 0 Td[(snq)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q pslnf)]TJ/F33 10.909 Tf 6.31 -1.456 Td[(q p)]TJ/F33 10.909 Tf 11.196 -1.456 Td[(p qsn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.837 Td[([lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf])]TJ/F17 7.97 Tf 6.587 0 Td[(snq)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]:Intheaboveequation,bothsidesare1.Thus,thetwosidesareequal.Wecanusethelawofquadraticreciprocitywhichstates)]TJ/F33 10.909 Tf 6.195 -1.457 Td[(p q)]TJ/F33 10.909 Tf 11.31 -1.457 Td[(q p=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1:Thisallowsustofurthersimplifyourequation.tqsnf=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q pslnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q p)]TJ/F33 10.909 Tf 11.196 -1.456 Td[(p qsn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.837 Td[([lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf])]TJ/F17 7.97 Tf 6.587 0 Td[(snq)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.457 Td[(q pslnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1sn)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.836 Td[([lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf])]TJ/F17 7.97 Tf 6.587 0 Td[(snq)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q pslnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4snp)]TJ/F15 7.97 Tf 6.587 0 Td[(1q)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F15 7.97 Tf 5 -8.836 Td[([lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf])]TJ/F17 7.97 Tf 6.586 0 Td[(snq)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q pslnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 5 -8.837 Td[(snq)]TJ/F15 7.97 Tf 6.586 0 Td[(1+[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf])]TJ/F17 7.97 Tf 6.586 0 Td[(snq)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnf)]TJ/F33 10.909 Tf 6.309 -1.456 Td[(q pslnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1[lqsnf)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]+1 oqp[lqsnf)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]:Example7.4.Letfx=2x31+1+2x32+1+x33+12F3[x]asinExample5.1.Thenwefoundthenullityoffxforeverymtobelmf=6if26jm;0otherwise:SinceSf;1=Xx2F3e1fx=Xx2F3e1)]TJ/F33 10.909 Tf 8.485 0 Td[(x2=)]TJ/F30 10.909 Tf 8.485 0 Td[(1Xx2F3e1x2=)]TJ/F33 10.909 Tf 8.485 0 Td[(g3;wehavet1f=)]TJ/F30 10.909 Tf 8.485 0 Td[(1.Now,letmbeoddandnotdivisibleby3.Sincemisnotdivisibleby26,thenlmf=0anditfollowsfrom.3thattmf=)]TJ/F30 10.909 Tf 8.485 0 Td[(1:19

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Example7.5.Letfx=3x71+1+3x72+1+x73+12F7[x]asinExample5.2.Thenwefoundthenullityoffxforeverym=5a7bm,m;57=1,tobelmf=8>>><>>>:6ifa2andb1;5ifa2andb=0;2if0a1andb1;1if0a1andb=0:SinceSf;1=Xx2F7e1fx=Xx2F71=7;wehavet1f=1.Now,letmbeoddandnotdivisibleby7.Writem=5amwherem;257=1.Notethato5=4and5 7=)]TJ/F30 10.909 Tf 8.485 0 Td[(1.Itfollowsfrom.3thattmf=)]TJ/F30 10.909 Tf 8.485 0 Td[(1a)]TJ/F17 7.97 Tf 6.196 -4.541 Td[(m 7ifa1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1a+1)]TJ/F17 7.97 Tf 6.195 -4.542 Td[(m 7ifa2:Inthenextsection,wewillrevisittheseexamplesanddeterminetmfforallevenmnotdivisibleby7.20

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8.FromSf;ntoSf;2sn;s>0TondSf;2sn,ifweusethecongruenceweusedintheprevioussection,itwouldyieldananswerfort2snfwithcongruencemodulo2.Since1and-1arecongruentmod2,thatdoesnothelpustodeterminethesign.Inthiscase,insteadoflookingatt2snfmodulo2,wewilllookatt2snfmodulo4.LetTi=fx2Fp2snnFp2n:Tr2snfx=ig.WecanwriteSf;2sn=Xx2Fp2snnFp2ne2snfx+Xx2Fp2ne2snfx=p)]TJ/F15 7.97 Tf 6.587 0 Td[(1Xi=0jTijip+Xx2Fp2ne2snfx:TheGaloisgroupAutFp2sn=FpnactsonTi,andTiisaunionofAutFp2sn=Fpn-orbitsofcardinalityof2tosomepowergreaterthanorequalto2.RecallthatAutFp2sn=Fpniscyclicwithanorderof2s.Ifx2Fp2snnFp2n,thenthestabilizerofxinAutFp2sn=FpndoesnotcontainAutFp2sn=Fp2nsothestabilizerofxmustbeproperlycontainedinAutFp2sn=Fp2n,i.e.containedinAutFp2sn=Fp4n.Notethatallsubgroupsofacyclicgroupoforder2sformachain.Thus,theAutFp2sn=Fpn-orbitshavecardinalitythatisdivisibleby4.SojTij0mod4forall0ip)]TJ/F30 10.909 Tf 10.909 0 Td[(1.Therefore,Xx2Fp2snnFp2ne2snfx0mod4:Thisgivesus.1Sf;2sn=Xx2Fp2ne2snfxmod4:ConsidertheelementsofFp2n.Foreveryx2Fp2n,xp2n)]TJ/F15 7.97 Tf 6.587 0 Td[(1=1ifandonlyifx2Fpn.So,wecanpartitionFp2nintoFp2n=Fpn[A[B;whereA=fx2Fp2n:xpn)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F30 10.909 Tf 8.484 0 Td[(1gB=fx2Fp2n:xpn)]TJ/F15 7.97 Tf 6.587 0 Td[(16=1g:ThesetBcanbepartitionedevenfurtherintofour-elementsubsetsoftheformfx;xpng.Sincef)]TJ/F33 10.909 Tf 8.485 0 Td[(x=fxandfxpn=fxpn,thene2snfxisconstantonfx;xpng.Wecanthenwrite.2Xx2Be2snfx0mod4:Now,considerA.Choose2Fpnsuchthatisanonsquare.Also,letx20=wherex02Fp2n.Wethenndthatxpn)]TJ/F15 7.97 Tf 6.586 0 Td[(10=)]TJ/F30 10.909 Tf 8.485 0 Td[(1.Also,foreveryy2A,ycanbewrittenas21

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y=x0xwherex2Fpn,soAcanberewrittenasA=x0Fpn.Recallthatfx=kXi=1aixpi+1:So,Xx2Ae2snfx=Xx2Ae2snkXi=1aixpi+1=Xx2Fpne2snkXi=1aixpi+10xpi+1=Xx2Fpne2snkXi=1ai1 2pi+1xpi+1=Xx2Fpne2sn)]TJ/F30 10.909 Tf 7.349 -5.958 Td[(~fx)]TJ/F30 10.909 Tf 10.909 0 Td[(1;.3where~fx=kXi=1ai1 2pi+1xpi+12Fpn[x]:Noticethat~~fx=fxwhenisxed.Combine8.1through.3.WehaveSf;2snXx2Fp2ne2snfxmod4=Xx2Fpn+Xx2A+Xx2Be2snfx=Xx2Fpne2snfx+Xx2Fpne2sn~fx)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4=2 psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfSf;n+2 psn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fS~f;n)]TJ/F30 10.909 Tf 10.909 0 Td[(1:SinceSf;n=tnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpplnf,thenS~f;n=tn~fgn)]TJ/F17 7.97 Tf 6.586 0 Td[(ln~fppln~fandSf;2sn=t2snfg2sn)]TJ/F17 7.97 Tf 6.587 0 Td[(l2snfppl2snf.Bysubstitutingthesethreevaluesintotheaboveequation,wehavet2snfg2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfppl2snf2 psn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfpplnf+2 psn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~ftn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fppln~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4:Since2 p=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1andp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1mod4,wecanfurthersimplifytheabovetot2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1l2snfg2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.586 0 Td[(ln~f+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fp)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4:22

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Theorem8.1.Letfx=kXi=1aixpi+1and~fx=kXi=1ai1 2pi+1xpi+1andlets>0.Thenlnf+ln~f+l2snfiseven.Moreover,.4t2snf=tnftn~fiflnfln~fmod2;tnftn~f)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1siflnf6ln~fmod2:Proof.Webegintheproofbyshowingthatlnf+ln~f+l2snfiseven.Supposetothecontrarythatlnf+ln~f+l2snfisodd.Then,eitheroneorallthreetermsinlnf+ln~f+l2snfmustbeodd.Thismeansthatexactlyoneorthreeofn)]TJ/F33 10.909 Tf 11.433 0 Td[(lnf,n)]TJ/F33 10.909 Tf 10.127 0 Td[(ln~f,or2sn)]TJ/F33 10.909 Tf 10.127 0 Td[(l2snfmustbeodd.Recallthatg2p=i1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(12pandthatp)]TJ/F30 10.909 Tf 8.484 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1mod4,sog2pi1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(12)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1mod4=i1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(12ip)]TJ/F15 7.97 Tf 6.586 0 Td[(1=i1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(12+p)]TJ/F15 7.97 Tf 6.586 0 Td[(1=i1 2p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1=1:Weusetheabovefacts,intheequationt2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1l2snfg2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.586 0 Td[(ln~f+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fp)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4:.5Theaboveequationbecomesgpumod4forsomeu2Zifonlyoneofthethreenumbersn)]TJ/F33 10.909 Tf 10.064 0 Td[(lnf,n)]TJ/F33 10.909 Tf 10.064 0 Td[(ln~f,or2sn)]TJ/F33 10.909 Tf 10.064 0 Td[(l2snfisodd;theaboveequationbecomes)]TJ/F30 10.909 Tf 8.484 0 Td[(1vgpmod4forsomev2Zifallthreenumbersn)]TJ/F33 10.909 Tf 11.303 0 Td[(lnf,n)]TJ/F33 10.909 Tf 11.303 0 Td[(ln~f,and2sn)]TJ/F33 10.909 Tf 11.303 0 Td[(l2snfareodd.Let2AutQp=Qsuchthatgp=)]TJ/F33 10.909 Tf 8.485 0 Td[(gp.Wecanapplytobothoftheabovecases.Intherstcase,wegetgpumod4;)]TJ/F33 10.909 Tf 8.484 0 Td[(gpumod4:Thisimplies2gp0mod4,whichisacontradiction.Forthesecondcase,wehave)]TJ/F30 10.909 Tf 8.485 0 Td[(1vgpmod4;)]TJ/F30 10.909 Tf 8.485 0 Td[(1)]TJ/F33 10.909 Tf 20 0 Td[(vgpmod4:Thisleadsto)]TJ/F30 10.909 Tf 8.484 0 Td[(20mod4whichisacontradictionaswell.Therefore,lnf+ln~f+l2snfiseven.23

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Toprovethesecondpartofthetheorem,webeginbyassumingthatlnfln~fmod2:Sincelnfandln~fhavethesameparity,thenl2snfiseven.Considert2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1l2snfg2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.586 0 Td[(ln~f+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fp)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4:Thus,sinceg2p1mod4,theng2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfp1mod4.Wecanusethisandtheassumptionthatlnfln~fmod2tocontinuewiththeabove.t2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1l2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnftnfgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnftn~fgn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4tnf+tn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1lnf)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4tnf+tn~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4where2f1;gpg.Since1 2gp)]TJ/F30 10.909 Tf 11.467 0 Td[(1isintegraloverQ,thengp1mod2.Wecanrewritethistobe2gp2mod4andgeneralizeitfurtherto22mod4.Returningtot2snftnf+tn~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4;recallthatastnfisthetypeoff,itiseither)]TJ/F30 10.909 Tf 8.484 0 Td[(1or1.Thentnf+tn~fcanbeeither0;2;or)]TJ/F30 10.909 Tf 8.484 0 Td[(2.Iftnf+tn~f=0,thent2snf)]TJ/F30 10.909 Tf 21.001 0 Td[(1mod4.Iftnf+tn~f=2,thent2snf2)]TJ/F30 10.909 Tf 10.756 0 Td[(11mod4.Iftnf+tn~f=)]TJ/F30 10.909 Tf 8.485 0 Td[(2,thent2snf)]TJ/F30 10.909 Tf 20 0 Td[(2)]TJ/F30 10.909 Tf 10.757 0 Td[(11mod4.Thus,t2snf=)]TJ/F30 10.909 Tf 8.485 0 Td[(1iftnf+tn~f=01iftnf+tn~f=2=tnftn~f:Nowassumethatlnf6ln~fmod2.Sincetheyhavedierentparitiesbutthesumiseven,thenl2snfisodd.Withoutlossofgenerality,assumethatn)]TJ/F33 10.909 Tf 11.174 0 Td[(lnfisoddandn)]TJ/F33 10.909 Tf 10.909 0 Td[(ln~fiseven.Then,wecansimplify.5.t2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1l2snfg2sn)]TJ/F17 7.97 Tf 6.586 0 Td[(l2snfp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1lnftnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfp+)]TJ/F30 10.909 Tf 8.484 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1sn)]TJ/F17 7.97 Tf 6.586 0 Td[(ln~f+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~fgn)]TJ/F17 7.97 Tf 6.587 0 Td[(ln~fp)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4t2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1gp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1lnftnfgp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1ln~ftn~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~f+1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1tnfgp+)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1t2snf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1gp)]TJ/F30 10.909 Tf 10.909 0 Td[()]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~f+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1tnfgp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod4gp)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1t2snf)]TJ/F33 10.909 Tf 10.909 0 Td[(tnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~ftn~f)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~f)]TJ/F30 10.909 Tf 10.909 0 Td[(1mod424

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So,t2snf=tnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.586 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~fiftn~f)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~f=1;)]TJ/F33 10.909 Tf 8.485 0 Td[(tnf)]TJ/F30 10.909 Tf 8.485 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1s+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~fiftn~f)]TJ/F30 10.909 Tf 8.485 0 Td[(11 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1ln~f=)]TJ/F30 10.909 Tf 8.485 0 Td[(1=tnftn~f)]TJ/F30 10.909 Tf 8.484 0 Td[(11 8p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1s:Thus,result.Example8.2.Letfx=2x31+1+2x32+1+x33+12F3[x]asinExample7.4wheret1f=)]TJ/F30 10.909 Tf 8.484 0 Td[(1andl1f=0.Choose=2asthenonsquareinF3.Then~fx=2x31+1+x32+1+2x33+1:SinceS~f;1=Xx2F3e1~fx=Xx2F3e1)]TJ/F33 10.909 Tf 8.485 0 Td[(x2=)]TJ/F30 10.909 Tf 8.485 0 Td[(1Xx2F3=)]TJ/F33 10.909 Tf 8.485 0 Td[(g3thent1~f=)]TJ/F30 10.909 Tf 8.485 0 Td[(1.Also,l1~f=0.By.4,t2sf=1;fors>0:Letm=2s13bmwherem;2313=1.Notethato13=3.Then,by.3,tmf=)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifb1;1otherwise:Example8.3.Letfx=3x71+1+3x72+1+x73+12F7[x]asinExample7.5wheret1f=1andl1f=1.Choose=3asthenonsquareinF7.Then~fx=4x71+1+3x72+1+4x73+1:SinceS~f;1=Xx2F7e1~fx=Xx2F7e1x2=g7thent1~f=1.Also,l1~f=0.By.4,t2sf=1;fors>0:Letm=2s5amwheres>0andm;257=1.Then,by.3,tmf=)]TJ/F30 10.909 Tf 8.485 0 Td[(1a)]TJ/F17 7.97 Tf 6.195 -4.541 Td[(m 7ifa1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1a+1)]TJ/F17 7.97 Tf 6.195 -4.541 Td[(m 7ifa2:.25

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9.FromSf;ntoSf;pnNotethatbothSf;nandSf;pncontainapowerofp.Ifweusedthecongruencemethodfromthetwoprevioussections,wewouldonlyget00modpwhichisuseless.Therefore,tondarelativeformulaforSf;pnintermsofSf;n,wehavetouseadierentmethod.Letb2FpnsuchthatTrnb6=0.UsingtheArtin-SchreierTheorem[10,Ch.5,Prop.7.8],[12,Ch.VI,Thm6.4],wehaveFpm=Fpnwherep=+b,andtherootsoftheirreduciblepolynomialxp)]TJ/F33 10.909 Tf 10.909 0 Td[(x)]TJ/F33 10.909 Tf 10.909 0 Td[(bare+j;j2Fp.Then,foreveryintegert0,wehaveTrpn=nt=Xj2Fp+jt=Xj2FptXs=0tsjst)]TJ/F17 7.97 Tf 6.587 0 Td[(s=tXs=0tst)]TJ/F17 7.97 Tf 6.587 0 Td[(sXj2FpjsAccordingto[13,Lemma7.3],Xj2Fpjs=)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifs>0ands0modp)]TJ/F30 10.909 Tf 10.909 0 Td[(1;0otherwise:Sowehave.1Trpn=nt=)]TJ/F34 10.909 Tf 10.303 10.364 Td[(Xi>0tip)]TJ/F30 10.909 Tf 10.909 0 Td[(1t)]TJ/F17 7.97 Tf 6.587 0 Td[(ip)]TJ/F15 7.97 Tf 6.586 0 Td[(1:Lemma9.1.Letu;vbeintegerssuchthat0u;vp)]TJ/F30 10.909 Tf 10.909 0 Td[(1.ThenTrpn=nu+v=0ifu+v6=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifu+v=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1:Ifisapositiveinteger,thenTrpn=nu+vp=)]TJ/F34 10.909 Tf 10.909 15.382 Td[(vu+v)]TJ/F30 10.909 Tf 10.909 0 Td[(p)]TJ/F30 10.909 Tf 10.909 0 Td[(1bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.586 0 Td[(p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+0ifu+v6=2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifu+v=2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1:26

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Proof.Writeu+v=u0+v0p,where0u0;v0p)]TJ/F30 10.909 Tf 10.909 0 Td[(1.WenowhaveTrpn=nu+v=)]TJ/F34 10.909 Tf 10.303 10.363 Td[(Xi>0u+vip)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F17 7.97 Tf 6.587 0 Td[(ip)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F34 10.909 Tf 8.485 15.382 Td[(u+vp)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.587 0 Td[(p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F34 10.909 Tf 10.909 15.382 Td[(u+v2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.586 0 Td[(2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F34 10.909 Tf 8.485 15.382 Td[(u0+v0pp)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.587 0 Td[(p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F34 10.909 Tf 10.909 15.382 Td[(u0+v0p2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.587 0 Td[(2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F34 10.909 Tf 8.485 15.382 Td[(u0+v0pp)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.586 0 Td[(p)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F34 10.909 Tf 10.909 15.382 Td[(u0+v0pp)]TJ/F30 10.909 Tf 10.909 0 Td[(2+pu+v)]TJ/F15 7.97 Tf 6.586 0 Td[(2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1=)]TJ/F34 10.909 Tf 8.485 15.382 Td[(u0p)]TJ/F30 10.909 Tf 10.909 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.587 0 Td[(p)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F34 10.909 Tf 10.909 15.382 Td[(u0p)]TJ/F30 10.909 Tf 10.909 0 Td[(2v01u+v)]TJ/F15 7.97 Tf 6.587 0 Td[(2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifu0;v0=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;0orp)]TJ/F30 10.909 Tf 10.909 0 Td[(2;1,i.e.u+v=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1or2p)]TJ/F30 10.909 Tf 10.909 0 Td[(10otherwise:Using.1againasweprovethesecondpartofthetheorem,Trpn=nu+vp=)]TJ/F34 10.909 Tf 10.303 10.363 Td[(Xi>0u+vpip)]TJ/F30 10.909 Tf 10.91 0 Td[(1u+vp)]TJ/F17 7.97 Tf 6.586 0 Td[(ip)]TJ/F15 7.97 Tf 6.587 0 Td[(1:Writeip)]TJ/F33 10.909 Tf 10.513 0 Td[(q=s+s1p++s)]TJ/F15 7.97 Tf 6.586 0 Td[(1p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+tpwhere0s;s1;:::;s)]TJ/F15 7.97 Tf 6.586 0 Td[(1;pp)]TJ/F30 10.909 Tf 10.513 0 Td[(1.BytheLucasTheorem[14],u+vpip)]TJ/F30 10.909 Tf 10.909 0 Td[(1)]TJ/F17 7.97 Tf 5 -3.995 Td[(us)]TJ/F17 7.97 Tf 10 -3.995 Td[(vtmodpifs1==s)]TJ/F15 7.97 Tf 6.587 0 Td[(1=0;0modpotherwise:So,Trpn=nu+vp=)]TJ/F34 10.909 Tf 52.7 10.363 Td[(X0s;tp)]TJ/F15 7.97 Tf 6.587 0 Td[(10s+tu+vmodp)]TJ/F15 7.97 Tf 6.586 0 Td[(1usvts+tps7!u)]TJ/F33 10.909 Tf 10.909 0 Td[(s;t7!v)]TJ/F33 10.909 Tf 10.909 0 Td[(t:27

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Sincep=+b,then,byinduction,wehavep=+bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1.Thus,Trpn=nu+vp=)]TJ/F34 10.909 Tf 60.185 10.364 Td[(X0su0tvu+v>s+tu+vmodp)]TJ/F15 7.97 Tf 6.587 0 Td[(1usvts+bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1t=)]TJ/F34 10.909 Tf 60.185 10.363 Td[(X0su0tvu+v>s+tu+vmodp)]TJ/F15 7.97 Tf 6.587 0 Td[(1usvttX=0t+bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1t)]TJ/F17 7.97 Tf 6.586 0 Td[(s+Intheabovesum,s+s+tu+v)]TJ/F30 10.909 Tf 10.438 0 Td[(p)]TJ/F30 10.909 Tf 10.438 0 Td[(1p)]TJ/F30 10.909 Tf 10.437 0 Td[(1.SinceTrpn=nu+vp2Fpn,thenweonlyhavetosumthetermswheres+=0whichiswhens==0.So,Trpn=nu+vp=)]TJ/F34 10.909 Tf 54.328 10.364 Td[(X0tvu+v>tu+vmodp)]TJ/F15 7.97 Tf 6.587 0 Td[(1vt+bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1t=)]TJ/F34 10.909 Tf 8.485 15.382 Td[(vu+v)]TJ/F30 10.909 Tf 10.909 0 Td[(p)]TJ/F30 10.909 Tf 10.909 0 Td[(1bp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1u+v)]TJ/F15 7.97 Tf 6.586 0 Td[(p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+0ifu+v6=2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifu+v=2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1:Thus,result.Theorem9.2.Letfbeaspreviouslystated.Assumepnv,wehave.2bp0++bpi)]TJ/F16 5.978 Tf 5.756 0 Td[(1=Trib=0:Notethatifi=0thenbp0++bp)]TJ/F16 5.978 Tf 5.756 0 Td[(1isanemptysum.RecallthatTrpn=nu+vpi=0ifu+v6=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1)]TJ/F30 10.909 Tf 8.485 0 Td[(1ifu+v=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1;2p)]TJ/F30 10.909 Tf 10.909 0 Td[(1:28

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forall1ikandall0u;vp)]TJ/F30 10.909 Tf 11.345 0 Td[(1.Letx=x00++xp)]TJ/F15 7.97 Tf 6.587 0 Td[(1p)]TJ/F15 7.97 Tf 6.586 0 Td[(12Fppn,wherexu2Fpn,0up)]TJ/F30 10.909 Tf 10.909 0 Td[(1.ThenTrpn)]TJ/F33 10.909 Tf 5 -8.837 Td[(fx=TrpnkXi=1aix00++xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1p)]TJ/F15 7.97 Tf 6.587 0 Td[(11+pi=TrpnkXi=1aiX0u;vp)]TJ/F15 7.97 Tf 6.587 0 Td[(1xuxpivu+vpi=TrnhkXi=1aiX0u;vp)]TJ/F15 7.97 Tf 6.587 0 Td[(1xuxpivTrpn=nu+vpii=)]TJ/F30 10.909 Tf 10.909 0 Td[(TrnhkXi=1aiX0u;vp)]TJ/F15 7.97 Tf 6.587 0 Td[(1u+v=p)]TJ/F15 7.97 Tf 6.586 0 Td[(1;2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1xuxpivi=)]TJ/F30 10.909 Tf 10.909 0 Td[(TrnhkXi=1aix1+pip)]TJ/F15 7.97 Tf 6.587 0 Td[(1+x0xpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1xpi0+x1+pi1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1xuxpip)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(u+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(uxpiui=)]TJ/F30 10.909 Tf 10.909 0 Td[(TrnhkXi=1aix1+pip)]TJ/F15 7.97 Tf 6.587 0 Td[(1+x0xpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1xpi0+x1+pi0)]TJ/F33 10.909 Tf 10.909 0 Td[(x1+pi0+x1+pi1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1xuxpip)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(u+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(uxpiui=)]TJ/F30 10.909 Tf 10.909 0 Td[(TrnhkXi=1aix0+xp)]TJ/F15 7.97 Tf 6.587 0 Td[(11+pi)]TJ/F33 10.909 Tf 10.909 0 Td[(x1+pi0+x1+pi1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1kXi=1aixuxpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(u+xp)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(uxpiui=)]TJ/F30 10.909 Tf 10.909 0 Td[(Trnhfx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 10.909 0 Td[(fx0+fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1kXi=1aixuxpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(u+aixp)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(uxpiui=)]TJ/F30 10.909 Tf 10.909 0 Td[(Trnhfx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 10.909 0 Td[(fx0+fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1kXi=1aixuxpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(u+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iixp)]TJ/F18 5.978 Tf 5.756 0 Td[(ip)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(uxui=)]TJ/F30 10.909 Tf 10.909 0 Td[(Trnhfx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 10.909 0 Td[(fx0+fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1kXi=1xuaixpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(u+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iixp)]TJ/F18 5.978 Tf 5.756 0 Td[(ip)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(ui=)]TJ/F30 10.909 Tf 10.909 0 Td[(Trnhfx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 10.909 0 Td[(fx0+fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1xukXi=1aixpip)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(u+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iixp)]TJ/F18 5.978 Tf 5.756 0 Td[(ip)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.586 0 Td[(ui=)]TJ/F30 10.909 Tf 10.909 0 Td[(Trnhfx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F33 10.909 Tf 10.909 0 Td[(fx0+fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1+1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Xu=1xufxp)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(up)]TJ/F18 5.978 Tf 5.756 0 Td[(i:29

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Then,Sf;pn=Xx0;:::;xp)]TJ/F16 5.978 Tf 5.756 0 Td[(12Fppnenh)]TJ/F33 10.909 Tf 8.485 0 Td[(fx0+xp)]TJ/F15 7.97 Tf 6.586 0 Td[(1+fx0)]TJ/F33 10.909 Tf 10.909 0 Td[(fx1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F16 5.978 Tf 12.105 19.05 Td[(1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3Xu=1xufxp)]TJ/F15 7.97 Tf 6.586 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(up)]TJ/F18 5.978 Tf 5.756 0 Td[(i= Sf;n2Sf;n1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3Yn=1Xxu;xp)]TJ/F16 5.978 Tf 5.756 0 Td[(1)]TJ/F18 5.978 Tf 5.756 0 Td[(u2Fpnen)]TJ/F31 10.909 Tf 5 -8.836 Td[()]TJ/F33 10.909 Tf 8.485 0 Td[(xufxp)]TJ/F15 7.97 Tf 6.587 0 Td[(1)]TJ/F17 7.97 Tf 6.587 0 Td[(up)]TJ/F18 5.978 Tf 5.756 0 Td[(=jSf;nj2 Sf;n1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3Yn=1plnf+n=p1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3n+lnfjSf;nj2 Sf;n:SinceSf;pn=tpngpn)]TJ/F17 7.97 Tf 6.587 0 Td[(lpnfpplpnf,thentpnfgpn)]TJ/F17 7.97 Tf 6.586 0 Td[(lpnfpplpnf=p1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3n+lnfjSf;nj2 Sf;nConsiderthetwosidesseparately.Theleft-handsideequalstpnfi1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12p1 2pn)]TJ/F17 7.97 Tf 6.586 0 Td[(lpnfplpnf=tpnfi1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12pn)]TJ/F17 7.97 Tf 6.587 0 Td[(lpnfp1 2pn)]TJ/F17 7.97 Tf 6.587 0 Td[(lpnf+lpnf=tpnfi1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12pn)]TJ/F17 7.97 Tf 6.587 0 Td[(lpnfp1 2pn+lpnf:Whenwelookattheright-handside,wegetp1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3n+lnfjSf;nj2 Sf;n=p1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3n+lnfjtnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnfj2 tnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnf=p1 2p)]TJ/F15 7.97 Tf 6.587 0 Td[(3n+lnfpn+lnftnfi)]TJ/F16 5.978 Tf 5.757 0 Td[(1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12p1 2n)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfplnf=tnfp1 2p)]TJ/F15 7.97 Tf 6.586 0 Td[(3n+lnfpn+lnfi)]TJ/F16 5.978 Tf 5.756 0 Td[(1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp1 2n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf+lnf=tnfp1 2pn+lnf)]TJ/F16 5.978 Tf 7.782 3.258 Td[(3 2n+lnfpn+lnfi)]TJ/F16 5.978 Tf 7.782 3.258 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp1 2n+lnf=tnfp1 2pn+lnfi)]TJ/F16 5.978 Tf 7.782 3.259 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf:Thisyieldstpnfi1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12pn)]TJ/F17 7.97 Tf 6.586 0 Td[(lpnfp1 2pn+lpnf=tnfp1 2pn+lnfi)]TJ/F16 5.978 Tf 7.782 3.259 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf:Thus,lpnf=plnf.Wecanthensimplifyfurthertpnf=tnfi)]TJ/F16 5.978 Tf 7.782 3.258 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12[n)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+pn)]TJ/F17 7.97 Tf 6.587 0 Td[(lpnf]=tnfi)]TJ/F16 5.978 Tf 7.782 3.259 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12[n)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf+pn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf]=tnfi)]TJ/F16 5.978 Tf 7.782 3.259 Td[(1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12[p+1n)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf]=tnf:Thus,result.30

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10.When21=22==2kLemma10.1.Let1;:::;k0beintegers.Thengcdp1+1;:::;pk+1>2,21==2k<1:When21==2k<1,gcdp1+1;:::;pk+1=pgcd1;:::;k+1:Proof.Itissucienttoprovethelemmawithk=2.Sincei 1;2,i=1;2,areodd,p1;2+1jpi+1fori=1;2.Thus,p1;2+1jp1+1;p2+1.Also,since1 2p1+1;p2+1j1 21 2p21)]TJ/F30 10.909 Tf 10.909 0 Td[(1;1 2p22)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 4p21;2)]TJ/F30 10.909 Tf 10.909 0 Td[(1=p1;2+1 2p1;2+1 2andsincep1+1 2;p1;2)]TJ/F15 7.97 Tf 6.586 0 Td[(1 2=1,wehave1 2p1+1;p2+1j1 2p1;2+1whichisp1+1;p2+1jp1;2+1.Thus,p1+1;p2+1=p1;2+1.Clearly,i>0forevery1ik.Assumetothecontrarythat1>2.Supposethat1=2i01and2=2j02wherei>jand01and02areodd.Then,p1+1;p2+1jp2i0102+1;p22)]TJ/F30 10.909 Tf 10.909 0 Td[(1jp2i0102+1;p2i12)]TJ/F30 10.909 Tf 10.909 0 Td[(1=2whichisacontradiction.Thus,wehaveproventhelemma.Lemma10.2.Let;0beintegers.Thenp+1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1=p;+1if2>2;2if22:Proof.Since;m=m,thenifanyoris0,theconclusionisobvious.Thus,assume;>0.31

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Firstassume2>2.Recallthat2isthe2-adicorderfunction.Since ;isodd,thenp;+1jp+1.Since ;iseven,thenp;+1jp)]TJ/F30 10.909 Tf 11.913 0 Td[(1.Thus,p;+1jp+1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1.Notethat1 2p+1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1j1 2p2)]TJ/F30 10.909 Tf 10.91 0 Td[(1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 2p;)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 2p2;)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 2p;)]TJ/F30 10.909 Tf 10.909 0 Td[(1p;+1:Since1 2p+1;1 2p;)]TJ/F30 10.909 Tf 11.613 0 Td[(1=1thisimpliesthat1 2p+1;p)]TJ/F30 10.909 Tf 11.613 0 Td[(1jp;+1.Foreachx2Zandanoddintegerk>0,wehave2+xk=2+x.Bythisweget2p+1=2p;+1and2p)]TJ/F30 10.909 Tf 8.898 0 Td[(12p+1.Then2p+1;p)]TJ/F30 10.909 Tf 8.898 0 Td[(1=2p;+1.Then,using1 2p+1;p)]TJ/F30 10.909 Tf 11.937 0 Td[(1jp;+1,wegetp+1;p)]TJ/F30 10.909 Tf 11.938 0 Td[(1jp;+1.Thus,p+1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1jp;+1.Forthesecondpart,assumethat22.Then,p+1 2p)]TJ/F30 10.909 Tf 10.91 0 Td[(1 2;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1 2j1 2p2)]TJ/F15 7.97 Tf 6.587 0 Td[(1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 2p;)]TJ/F30 10.909 Tf 10.909 0 Td[(1=1 2p;)]TJ/F30 10.909 Tf 10.909 0 Td[(1=p)]TJ/F30 10.909 Tf 10.909 0 Td[(1 2;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1 2:Sincep+1 2;p)]TJ/F15 7.97 Tf 6.586 0 Td[(1 2=1,thenp+1 2;p)]TJ/F15 7.97 Tf 6.586 0 Td[(1 2=1whichgivesusp+1;p)]TJ/F30 10.909 Tf 10.909 0 Td[(1=2:Thiscompletestheproof.Theorem10.3.Letfbeaspreviouslystated.Assumethat21==2k=andthat2n>.Then2+1jlnfandtnf=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(122+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1=8<:)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1if=0;)]TJ/F30 10.909 Tf 8.485 0 Td[(1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1if>0:Proof.Usingthetwolemmasabove,wehavegcdp1+1;:::;pk+1;pn)]TJ/F30 10.909 Tf 10.909 0 Td[(1=pgcd1;:::;k+1;pn)]TJ/F30 10.909 Tf 10.909 0 Td[(1=pgcd1;:::;k;n+10modp2+1:.1Letq=p2+1.Then2+1isthemultiplicativeorderofpmodq,namely,oqp=2+1.Since2+1jn,thenqjpn)]TJ/F30 10.909 Tf 10.909 0 Td[(1.Firstweneedtoshowthat2+1jlnf.Todothat,weshowthatfx2Fpn:fx=0gisavectorspaceoverFp2+1.Letx2Fpnsuchthatfx=0.Also,lety2Fp2+1.Then,32

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wewanttoshowthatfxy=0.Usingthedenitionoff,wehavefyxp)]TJ/F18 5.978 Tf 5.757 0 Td[(=kXi=1aiypixpi+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iiyp)]TJ/F18 5.978 Tf 5.756 0 Td[(ixp)]TJ/F18 5.978 Tf 5.756 0 Td[(i:Weclaimthatypi=ypforall1ik.Byequation.1,pi)]TJ/F30 10.909 Tf 22.299 0 Td[(1modqwhichimpliesthatpi)]TJ/F30 10.909 Tf 21.449 0 Td[(1modq.Then,p)]TJ/F30 10.909 Tf 21.449 0 Td[(1pimodq.Thisleadstopi1modqforall1ik.Sinceoqp=2+1,theni0mod2+1.Thus,ypi=ywhichimpliesthatyp=yp.Thenfyxp)]TJ/F18 5.978 Tf 5.756 0 Td[(becomesfyxp)]TJ/F18 5.978 Tf 5.756 0 Td[(=kXi=1aiypxpi+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iiypxp)]TJ/F18 5.978 Tf 5.756 0 Td[(i=yp)]TJ/F17 7.97 Tf 10.568 4.8 Td[(kXi=1aixpi+ap)]TJ/F18 5.978 Tf 5.756 0 Td[(iixp)]TJ/F18 5.978 Tf 5.756 0 Td[(i=ypfxp)]TJ/F18 5.978 Tf 5.756 0 Td[(=0:Thus,fyx=0.So,wehaveprovedthat2+1jlnf.Now,choosez2Fpnsuchthatoz=q.Sincepi+10modqforall1ik,wehavefyx=fxforally2hzi.So,tnfgn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnf=Sf;n=1+Xx2Fpnenfx=1+Xx2Fpn=hziXy2hzienfxy=1+Xx2Fpn=hzienfxq=1+qXx2Fpn=hzienfx1modq:Intheabove,plnf1modqsinceoqp=2+1jlnf.Also,g2+1p=hi1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12p1 2i2+1=i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(1222p1 22=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(122p2)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(122+1modq;hence,gn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnfp=g2+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1p)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(122+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1modq:Thisleadstotnf=)]TJ/F30 10.909 Tf 8.484 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(122+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 2+1:Corollary10.4.Assumep)]TJ/F30 10.909 Tf 21.672 0 Td[(1mod4,1;:::;karealloddandniseven.Thentnf=1.Proof.ThisfollowsimmediatelyfromTheorem10.3.33

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Example10.5.Letfx=2x31+1+x33+12F3[x].Thenfx=x30+2x32+2x34+x36.ThesplittingeldoffoverF3isF312andl2a3bmf=8>>><>>>:6ifa2,b14ifa=1,b1ora2,b=0;2ifa=1,b=0ora=0,b1;1ifa;b=0:wherem;23=1.ByCorollary10.4,forevenn,tnf=1:34

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11.TheFormulaforSax1+pTheorem10.3helpstoprovideaprooffortheevaluationofthesumofSaxp+1;n,aspecialcaseofthesuminTheorem10.3Corollary11.1.Leta2Fpnandlet0.iIf2n2,Saxp+1;n=a)]TJ/F30 10.909 Tf 8.485 0 Td[(1n)]TJ/F15 7.97 Tf 6.587 0 Td[(1i1 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12np1 2n:iiIf2n=2+1,Saxp+1;n=8<:p1 2[n+2;n]ifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1;)]TJ/F33 10.909 Tf 8.485 0 Td[(p1 2notherwise:iiiIf2n>2+1,Saxp+1;n=8<:)]TJ/F33 10.909 Tf 8.485 0 Td[(p1 2[n+2;n]ifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=1;p1 2notherwise:Proof.Werstneedtodeterminelnf.Weclaimthat.1lnf=8<:;nifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.757 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(10otherwise:Notethatfx=apxp2+ax=apxxp2)]TJ/F15 7.97 Tf 6.586 0 Td[(1+a1)]TJ/F17 7.97 Tf 6.586 0 Td[(p.So,iffx=0hasasolutioninFpn,thenumberofsolutionswillbep2)]TJ/F30 10.909 Tf 10.909 0 Td[(1;pn)]TJ/F30 10.909 Tf 10.909 0 Td[(1=p;n)]TJ/F30 10.909 Tf 10.909 0 Td[(1.So,lnf=;niffx=0hasasolutioninFpn;0otherwise:Noticethatfx=0hasasolutioninFpnifandonlyif)]TJ/F33 10.909 Tf 8.485 0 Td[(ap)]TJ/F15 7.97 Tf 6.587 0 Td[(1=xp2)]TJ/F15 7.97 Tf 6.587 0 Td[(1forsomex2Fpn.Thishappensifandonlyif)]TJ/F33 10.909 Tf 8.485 0 Td[(ap)]TJ/F15 7.97 Tf 6.587 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p2)]TJ/F16 5.978 Tf 5.756 0 Td[(1;pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1=1;)]TJ/F33 10.909 Tf 8.485 0 Td[(ap)]TJ/F15 7.97 Tf 6.587 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.757 0 Td[(1=1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1pn)]TJ/F16 5.978 Tf 5.757 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1ap)]TJ/F15 7.97 Tf 6.587 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p2;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=1;)]TJ/F30 10.909 Tf 8.485 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1ap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=1;ap)]TJ/F16 5.978 Tf 5.757 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.757 0 Td[(1:Thisyields.1.35

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iSince2n2,wecanuseLemma10.2whichsaysp+1;pn)]TJ/F30 10.909 Tf 11.026 0 Td[(1=2.Then,x7!xp+1isa2-to-1mapfromFpntoFpn2.Thus,Saxp+1;n=1+Xx2Fpnenaxp+1=1+2Xx2Fpn2enax=1+Xx2Fpnenax2=Xx2Fpnenax2=aXx2Fpnenx2=a)]TJ/F30 10.909 Tf 8.485 0 Td[(1n)]TJ/F15 7.97 Tf 6.586 0 Td[(1gnpbytheDavenport-Hassetheorem[5],[13,x5.2]=a)]TJ/F30 10.909 Tf 8.485 0 Td[(1n)]TJ/F15 7.97 Tf 6.586 0 Td[(1i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12np1 2n:iiSince2n=2;n,thenpn)]TJ/F15 7.97 Tf 6.586 0 Td[(1 p;n)]TJ/F15 7.97 Tf 6.586 0 Td[(1isodd.By.1,lnf=8<:;nifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1;0otherwise:WecanthenuseTheorem10.3,tnf=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 22+1:Iflnf=;n,thenn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnf 22+1isevensince2n)]TJ/F30 10.909 Tf 11.662 0 Td[(;n>2n=2+1.So,tnf=1;henceSaxp+1;n=gn)]TJ/F17 7.97 Tf 6.587 0 Td[(lnfpplnf=i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12n)]TJ/F15 7.97 Tf 6.587 0 Td[(;np1 2[n+2;n]=p1 2[n+2;n]since2n+;n2.Iflnf=0,thenn)]TJ/F17 7.97 Tf 6.586 0 Td[(lnf 22+1isoddandtnf=)]TJ/F30 10.909 Tf 8.485 0 Td[()]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12.So,Saxp+1;n=)]TJ/F30 10.909 Tf 8.485 0 Td[()]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12gnp=)]TJ/F30 10.909 Tf 8.485 0 Td[()]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12i1 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12np1 2n=)]TJ/F30 10.909 Tf 8.485 0 Td[()]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.586 0 Td[(12n 2p1 2n=)]TJ/F33 10.909 Tf 8.485 0 Td[(p1 2nsince2+n 2>20.Tosummarize,wehaveSaxp+1;n=8<:p1 2[n+2;n]ifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.756 0 Td[(1=)]TJ/F30 10.909 Tf 8.485 0 Td[(1;)]TJ/F33 10.909 Tf 8.485 0 Td[(p1 2notherwise:iiiInthiscase,pn)]TJ/F15 7.97 Tf 6.587 0 Td[(1 p2;n)]TJ/F15 7.97 Tf 6.586 0 Td[(1iseven.By.1,lnf=8<:;nifap)]TJ/F16 5.978 Tf 5.756 0 Td[(1pn)]TJ/F16 5.978 Tf 5.756 0 Td[(1 p;n)]TJ/F16 5.978 Tf 5.757 0 Td[(1=1;0otherwise:ByTheorem10.3,tnf=)]TJ/F30 10.909 Tf 8.485 0 Td[(11 4p)]TJ/F15 7.97 Tf 6.587 0 Td[(12+1n)]TJ/F18 5.978 Tf 5.756 0 Td[(lnf 22+1:Theconclusionfollowsthesamewayasinii.36

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12.TablesofNumericalResultsLetfx=kXi=1aixpi+12Fpn[x];01<
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Table1.Valuesoflmfwithpn=3,4 a0;:::;ak s m;lmf;mjs 1 1 ,001 4 ,0,0,211 6 ,0,1,0,221 3 ,1,2001 8 ,0,0,0,4101 12 ,0,0,0,2,012,4201 6 ,1,2,2,4011 18 ,0,1,0,3,018,4111 12 ,1,1,2,3,212,4211 5 ,0,4021 9 ,1,3,4121 12 ,0,1,0,3,212,4221 10 ,0,0,0,40001 12 ,0,0,0,2,012,61001 18 ,0,1,0,3,018,62001 9 ,1,3,60101 8 ,0,0,2,61101 30 ,1,1,2,1,210,5,2,62101 30 ,0,1,0,4,210,5,4,60201 12 ,1,2,2,4,412,61201 13 ,0,62201 26 ,0,0,0,60011 30 ,0,1,0,0,210,5,0,61011 12 ,1,1,2,3,212,62011 28 ,0,0,0,0,028,60111 24 ,1,1,2,1,28,5,2,61111 36 ,0,1,0,3,39,0,5,4,62111 7 ,0,60211 13 ,0,61211 20 ,0,0,2,4,420,62211 18 ,1,2,3,5,418,60021 15 ,1,2,5,61021 28 ,0,0,0,0,028,62021 12 ,0,1,0,3,212,60121 24 ,0,1,0,1,28,5,2,61121 14 ,0,0,0,62121 36 ,1,1,3,3,39,4,5,4,60221 26 ,0,0,0,61221 18 ,1,2,2,5,218,62221 20 ,0,0,2,0,420,6 38

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Table1.Continued a0;:::;ak s m;lmf;mjs 00001 16 ,0,0,0,0,810001 24 ,0,0,0,0,08,4,0,820001 12 ,1,2,2,4,412,801001 90 ,0,1,0,0,39,0,5,0,4,7,0,811001 84 ,1,1,2,1,27,1,2,1,2,7,2,821001 52 ,0,0,2,0,652,802001 45 ,1,3,5,4,745,812001 84 ,0,1,0,1,27,0,2,1,0,7,2,822001 52 ,0,0,2,6,652,800101 36 ,0,0,0,2,09,0,6,0,810101 24 ,1,2,2,2,48,6,4,820101 10 ,0,0,4,801101 36 ,1,1,3,3,39,6,5,6,811101 41 ,0,821101 42 ,0,1,0,2,614,7,6,802101 36 ,0,1,0,3,39,0,5,6,812101 82 ,0,0,0,822101 42 ,1,1,2,2,114,7,2,800201 18 ,1,2,3,6,418,810201 24 ,0,0,0,2,08,6,4,820201 20 ,0,0,0,0,020,801201 41 ,0,811201 60 ,0,1,0,3,46,2,5,4,4,7,6,821201 78 ,1,1,2,2,126,7,2,802201 82 ,0,0,0,812201 60 ,1,1,2,3,16,2,5,4,2,7,6,822201 78 ,0,1,0,2,626,7,6,800011 42 ,0,1,0,2,014,7,0,810011 78 ,1,1,2,2,126,7,2,820011 60 ,0,0,0,2,46,0,4,4,4,6,4,801011 39 ,1,2,7,811011 72 ,0,1,0,1,38,5,0,3,4,7,4,821011 52 ,0,0,2,6,652,802011 80 ,0,0,0,0,010,0,0,0,0,812011 30 ,0,0,0,0,010,4,0,822011 36 ,1,2,3,4,59,4,7,6,800111 12 ,1,1,2,3,212,810111 41 ,0,820111 78 ,0,1,0,2,026,7,0,801111 60 ,0,1,0,3,06,2,5,4,0,7,6,811111 40 ,0,0,0,4,410,4,4,8 39

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Table1.Continued a0;:::;ak s m;lmf;mjs 21111 9 ,1,3,802111 28 ,0,0,2,0,028,812111 18 ,1,2,2,5,218,822111 80 ,0,0,0,0,010,0,0,0,0,800211 41 ,0,810211 36 ,0,1,0,3,39,0,7,4,820211 90 ,1,1,3,1,39,4,5,3,4,7,4,801211 82 ,0,0,0,811211 24 ,1,1,2,3,28,7,4,821211 84 ,0,1,0,1,27,0,2,1,0,7,2,802211 30 ,1,2,2,5,410,6,6,812211 28 ,0,0,2,6,628,822211 40 ,0,0,0,0,010,0,0,800021 21 ,1,2,7,810021 78 ,0,1,0,2,626,7,6,820021 60 ,0,0,0,2,06,0,4,4,0,6,4,801021 80 ,0,0,0,0,010,0,0,0,0,811021 15 ,0,0,4,821021 36 ,1,2,2,4,59,2,7,6,802021 78 ,0,1,0,2,026,7,0,812021 72 ,1,1,3,1,38,5,4,3,4,7,4,822021 52 ,0,0,2,0,652,800121 12 ,0,1,0,3,212,810121 82 ,0,0,0,820121 39 ,1,2,7,801121 28 ,0,0,2,0,028,811121 18 ,1,2,3,5,618,821121 80 ,0,0,0,0,010,0,0,0,0,802121 60 ,1,1,2,3,56,2,5,4,6,7,6,812121 40 ,0,0,0,0,410,4,4,822121 18 ,0,1,0,3,018,800221 82 ,0,0,0,810221 36 ,1,1,3,3,39,4,7,4,820221 90 ,0,1,0,4,39,0,5,4,4,7,4,801221 30 ,1,2,2,1,410,6,2,811221 28 ,0,0,2,0,628,821221 40 ,0,0,0,0,010,0,0,802221 41 ,0,812221 24 ,0,1,0,3,28,7,4,822221 84 ,1,1,2,1,27,1,2,1,2,7,2,8 40

PAGE 45

Table2.Valuesoflmfwithpn=5,3 a0;:::;ak s m;lmf;mjs 1 1 ,001 4 ,0,0,211 10 ,0,1,0,221 6 ,0,0,0,231 3 ,0,241 5 ,1,2001 8 ,0,0,0,4101 20 ,0,0,2,0,020,4201 12 ,0,0,0,0,012,4301 6 ,0,0,2,4401 10 ,1,2,2,4011 30 ,0,1,0,0,310,2,0,4111 12 ,0,0,2,2,212,4211 13 ,0,4311 5 ,1,4411 24 ,0,0,0,0,08,0,0,4021 26 ,0,0,0,4121 20 ,0,1,3,0,220,4221 30 ,1,1,1,2,310,2,2,4321 13 ,0,4421 15 ,0,2,0,4031 13 ,0,4131 20 ,1,1,3,2,220,4231 30 ,0,1,2,0,310,2,2,4331 26 ,0,0,0,4431 30 ,0,0,0,0,210,0,0,4041 15 ,1,3,2,4141 12 ,0,0,0,2,212,4241 26 ,0,0,0,4341 10 ,0,1,0,4441 24 ,0,0,0,0,08,0,0,40001 12 ,0,0,0,2,012,61001 30 ,0,1,0,0,310,2,0,62001 18 ,0,0,0,0,018,63001 9 ,0,0,64001 15 ,1,3,2,60101 8 ,0,0,2,61101 126 ,0,0,0,0,09,0,0,0,0,0,0,62101 30 ,0,1,2,0,310,4,2,63101 30 ,1,1,1,4,310,4,4,64101 63 ,0,0,0,0,063,60201 12 ,0,0,2,2,412,6 41

PAGE 46

Table2.Continued a0;:::;ak s m;lmf;mjs 1201 21 ,0,0,0,62201 130 ,1,1,2,2,126,5,2,63201 130 ,0,1,0,2,426,5,4,64201 42 ,0,0,0,0,014,0,0,60301 20 ,0,0,2,0,020,61301 65 ,1,2,5,62301 24 ,0,0,0,0,28,0,2,63301 24 ,0,0,2,0,28,0,2,64301 130 ,0,1,0,2,026,5,0,60401 20 ,1,2,4,2,420,61401 78 ,0,0,2,2,026,4,2,62401 62 ,0,0,0,63401 31 ,0,64401 78 ,0,0,0,2,426,4,4,60011 50 ,0,1,0,5,050,61011 52 ,0,0,2,4,452,62011 12 ,0,0,2,0,212,63011 65 ,1,2,5,64011 78 ,0,0,0,2,426,4,4,60111 24 ,0,0,2,0,28,4,2,61111 60 ,0,1,0,3,06,3,2,5,0,4,4,62111 130 ,1,1,2,2,126,5,2,63111 7 ,0,64111 62 ,0,0,0,60211 63 ,0,0,0,0,063,61211 20 ,1,1,3,4,420,62211 120 ,0,1,0,1,06,1,1,2,1,0,2,5 ,2,2,2,63211 30 ,0,0,2,0,410,0,4,64211 31 ,0,60311 30 ,1,1,1,2,310,2,2,61311 60 ,0,0,2,2,06,2,0,4,2,4,2,62311 42 ,0,0,0,0,014,0,0,63311 130 ,0,1,0,2,026,5,0,64311 124 ,0,0,0,0,0124,60411 124 ,0,0,0,0,0124,61411 24 ,0,0,0,2,08,2,2,62411 78 ,0,0,0,2,026,4,0,63411 124 ,0,0,0,0,0124,64411 30 ,1,2,3,2,410,4,4,60021 24 ,0,0,2,0,28,0,2,6 42

PAGE 47

Table2.Continued a0;:::;ak s m;lmf;mjs 1021 63 ,0,0,0,0,063,62021 60 ,1,1,1,3,26,3,2,5,2,4,4,63021 124 ,0,0,0,0,0124,64021 60 ,0,1,0,1,06,1,2,5,0,2,2,60121 40 ,0,1,1,0,510,2,2,61121 120 ,1,1,1,1,26,1,1,2,1,2,2,5 ,2,2,2,62121 52 ,0,0,2,0,452,63121 39 ,0,2,4,64121 30 ,0,0,0,0,210,0,0,60221 130 ,1,1,2,2,126,5,2,61221 30 ,0,1,2,0,510,2,2,62221 52 ,0,0,2,4,452,63221 62 ,0,0,0,64221 63 ,0,0,0,0,063,60321 63 ,0,0,0,0,063,61321 126 ,0,0,0,0,09,0,0,0,0,0,0,62321 20 ,0,1,3,0,220,63321 78 ,0,0,0,2,426,4,4,64321 15 ,1,3,4,60421 78 ,0,0,0,2,026,4,0,61421 62 ,0,0,0,62421 60 ,0,0,2,2,06,2,0,4,4,2,4,63421 10 ,1,2,2,64421 124 ,0,0,0,0,0124,60031 24 ,0,0,0,0,28,0,2,61031 60 ,1,1,1,1,26,1,2,5,2,2,2,62031 124 ,0,0,0,0,0124,63031 60 ,0,1,2,3,06,3,2,5,2,4,4,64031 126 ,0,0,0,0,09,0,0,0,0,0,0,60131 40 ,1,1,1,2,510,2,2,61131 15 ,0,2,0,62131 78 ,0,0,0,2,026,4,0,63131 52 ,0,0,2,4,452,64131 120 ,0,1,0,1,06,1,1,2,1,0,2,5 ,2,2,2,60231 130 ,0,1,0,2,426,5,4,61231 126 ,0,0,0,0,09,0,0,0,0,0,0,62231 31 ,0,63231 52 ,0,0,2,0,452,64231 30 ,1,1,3,2,510,2,4,60331 126 ,0,0,0,0,09,0,0,0,0,0,0,6 43

PAGE 48

Table2.Continued a0;:::;ak s m;lmf;mjs 1331 30 ,0,1,0,0,310,4,0,62331 78 ,0,0,2,2,026,4,2,63331 20 ,1,1,3,2,220,64331 63 ,0,0,0,0,063,60431 39 ,0,2,4,61431 124 ,0,0,0,0,0124,62431 10 ,1,2,4,63431 60 ,0,0,0,2,06,2,0,4,0,2,4,64431 31 ,0,60041 25 ,1,5,61041 78 ,0,0,2,2,026,4,2,62041 130 ,0,1,0,2,026,5,0,63041 12 ,0,0,0,0,212,64041 52 ,0,0,2,0,452,60141 24 ,0,0,0,0,28,4,2,61141 31 ,0,62141 14 ,0,0,0,63141 130 ,0,1,0,2,426,5,4,64141 60 ,1,1,3,3,26,3,2,5,4,4,4,60241 126 ,0,0,0,0,09,0,0,0,0,0,0,61241 62 ,0,0,0,62241 30 ,0,0,2,0,410,0,2,63241 120 ,1,1,1,1,26,1,1,2,1,2,2,5 ,2,2,2,64241 20 ,0,1,3,0,420,60341 30 ,0,1,2,0,310,2,4,61341 124 ,0,0,0,0,0124,62341 65 ,1,2,5,63341 21 ,0,0,0,64341 60 ,0,0,0,2,06,2,0,4,0,4,2,60441 124 ,0,0,0,0,0124,61441 30 ,1,2,1,2,410,4,2,62441 124 ,0,0,0,0,0124,63441 39 ,0,2,4,64441 24 ,0,0,0,2,08,2,2,6 44

PAGE 49

References[1]C.Arf,UntersuchungenuberquadratischeFormeninKorpernderCharakteristik2,J.ReineAngew.Math.183,148{167.[2]L.BaumertandR.McEliece,Weightsofirreduciblecycliccodes,Inform.Control20,158{175.[3]L.Carlitz,Explicitevaluationofcertainexponentialsums,Math.Scand.44,5{16.[4]L.Carlitz,Evaluationofsomeexponentialsumsoveraniteeld,Math.Nachr.96,319{339.[5]H.DavenportandH.Hasse,DieNullstellenderKongruenzzetafunktioneningewissenzyklischenFallen,J.ReineAngew.Math.172,151{182.[6]L.E.Dickson,LinearGroups,Dover,NewYork,1958.[7]T.Helleseth,Someresultsaboutthecross-correlationfunctionbetweentwomaximallinearsequences,DiscreteMath.16,209{232.[8]X.Hou,Explicitevaluationofcertainexponentialsumsofbinaryquadraticfunctions,preprint.[9]X.Hou,LecturesonFiniteFields,preprint.[10]T.Hungerford,Algebra,Springer-Verlag,NewYork,1974.[11]K.IrelandandM.Rosen,AClassicalIntroductiontoModernNumberTheory,Springer,NewYork,1982.[12]S.Lang,Algebra,Addison-Wesley,Reading,MA,1993.[13]R.LidlandH.Niederreiter,Finiteelds,CambridgeUniversityPress,Cambridge,1997.[14]E.Lucas,Surlescongruencesdesnombreseuleriensetdescoecientsdierentielsdesfonctionstrigonometriques,suivantunmodulepremier,Bull.Soc.Math.France6{1878,49{54.[15]WolframResearch,Inc.,Mathematica,Version5.1,Champaign,IL,2004.45


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Draper, Sandra D.
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Evalutaion of certain exponential sums of quadratic functions over a finite fields of odd characteristic
h [electronic resource] /
by Sandra D. Draper.
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[Tampa, Fla] :
b University of South Florida,
2006.
3 520
ABSTRACT: Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)[x] where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)).^^ We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.
502
Thesis (M.A.)--University of South Florida, 2006.
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Includes bibliographical references.
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Text (Electronic thesis) in PDF format.
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System requirements: World Wide Web browser and PDF reader.
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Adviser: Xiang-Dong Hou, Ph.D.
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Artin-Schreier Theorem.
Gauss sum.
Law of quadratic reciprocity.
Legendre symbol.
Quadratic form.
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Dissertations, Academic
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Masters.
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t USF Electronic Theses and Dissertations.
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