二次互反律的傅里叶分析证明（英文）

¡¡¡¡Asidefromitsunquestionablenovelty£¬leadingtoitsinclusioninmostifnotallintroductorycoursesinnumbertheory£¬thelawofquadraticreciprocitystandsoutasoneofthedeepestfactsofthetheoryofalgebraicnumberfields.ThiswascertainlyalreadyunderstoodbyGauss£¬whoinhislifetimegavesixproofsofthisbeautifultheoremfirstconjecturedbyEuler£¬ThereareanumberofgoodsourcesavailabletreatingthiscentralthemeofGauss\'arithmeticalwork£¬amongwhichwerecommendVariationenubereinZahlentheoretischesThemavonCarlFriedrichGauss[Pi78]£¬andtheindicatedsectionofScharlau-Opolka[SO84].¡¡¡¡Gauss\'worklaidbaredeepconnectionsbetweenatfirstglanceratherdisparateaspectsofthebehaviorofringsofintegersofalgebraicnumberfields.PresentlyitbecameclearthatthesplittingofprimesinquadraticextensionsiscompletelygovernedbythefinestructureoftheLegendresymbol£¬thatis£¬byquadraticreciprocity£¬andthissetthestageforGauss\'workonthegeneraofquadraticforms.¡¡¡¡IfthereisatoolparexcellenceinGauss\'armoryforthesearithmeticalinvestigationsitissurelythemethodofGausssums.TheirrelationtotheLegendresymbolisfundamental£»itisaneasyexercisetoshowthatGausssumstransformver£©rnicelyundertheLegendresymbol\'snaturalaction.Itisaquickstepfromtheretotheformulationofquadraticreciprocityasanidentitybetweenso-calledreciprocalGausssums.Butwherearethequadraticforms?