introduction to p-adic numbers and p-adic analysis - a. baker

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1、AnIntroductiontop-adicNumbersandp-adicAnalysisA.J.Baker[4/11/2002]DepartmentofMathematics,UniversityofGlasgow,GlasgowG128QW,Scotland.E-mailaddress:a.baker@maths.gla.ac.ukURL:http://www.maths.gla.ac.uk/»ajbContentsIntroduction1Chapter1.Congruencesandmod

2、ularequations3Chapter2.Thep-adicnormandthep-adicnumbers15Chapter3.Someelementaryp-adicanalysis29Chapter4.ThetopologyofQp33Chapter5.p-adicalgebraicnumbertheory45Bibliography51Problems53ProblemSet153ProblemSet254ProblemSet355ProblemSet456ProblemSet557Pro

3、blemSet6581IntroductionThesenoteswerewrittenforafinalyearundergraduatecoursewhichranatManchesterUniversityin1988/9andalsotaughtinlateryearsbyDrM.McCrudden.Irewrotethemin2000tomakethemavailabletointerestedgraduatestudents.Theapproachtakenisverydowntoeart

4、handmakesfewassumptionsbeyondstandardundergraduateanalysisandalgebra.Becauseofthisthecoursewasasselfcontainedaspossible,coveringbasicnumbertheoryandanalyticideaswhichwouldprobablybefamiliartomoreadvancedreaders.Theproblemsetsarebasedonthoseforproducedf

5、orthecourse.IwouldliketothankJavierDiaz-Vargasforpointingoutnumerouserrors.1CHAPTER1CongruencesandmodularequationsLetn2Z(wewillusuallyhaven>0).Wedefinethebinaryrelation´bynDefinition1.1.Ifx;y2Z,thenx´yifandonlyifnj(x¡y).Thisisoftenalsowrittennx´y(modn)o

6、rx´y(n).Noticethatwhenn=0,x´yifandonlyifx=y,sointhatcase´isreallyjustequality.n0Proposition1.2.Therelation´isanequivalencerelationonZ.nProof.Letx;y;z2Z.Clearly´isreflexivesincenj(x¡x)=0.Itissymmetricsincenifnj(x¡y)thenx¡y=knforsomek2Z,hencey¡x=(¡k)nands

7、onj(y¡x).Fortransitivity,supposethatnj(x¡y)andnj(y¡z);thensincex¡z=(x¡y)+(y¡z)wehavenj(x¡z).¤Wedenotetheequivalenceclassofx2Zby[x]norjust[x]ifnisunderstood;itisalsocommontousexforthisifthevalueofnisclearfromthecontext.Bydefinition,[x]n=fy2Z:y´xg=fy2Z:y=

8、x+knforsomek2Zg;nandthereareexactlyjnjsuchresidueclasses,namely[0]n;[1]n;:::;[n¡1]n:Ofcoursewecanreplacetheserepresentativesbyanyothersasrequired.Definition1.3.ThesetofallresidueclassesofZmodulonisZ=n=f[x]n:x=0;1;:::;n¡1g:Ifn=0weinterpr