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1、CLASSFIELDTHEORYJ.S.MilnePreface.These12arethenotesforMath776,UniversityofMichigan,Winter1997,slightlyrevisedfromthosehandedoutduringthecourse.Theyhavebeensubstantiallyrevisedandexpandedfromanearlierversion,basedonmynotesfrom1993(v2.01).Myapproachtocla
2、ssfieldtheoryinthesenotesiseclectic.Althoughitispossibletoprovethemaintheoremsinclassfieldtheoryusingneitheranalysisnorcohomology,therearemajortheoremsthatcannotevenbestatedwithoutusingoneortheother,forexample,theoremsondensitiesofprimes,ortheoremsaboutt
3、hecohomologygroupsassociatedwithnumberfields.Whenitshedsadditionallight,Ihavenothesitatedtoincludemorethanoneproofofaresult.Theheartofthecourseistheoddnumberedchapters.ChapterII,whichisonthecohomologyofgroups,isbasicfortherestofthecourse,butChaptersIV,V
4、I,andVIIIarenotessentialforreadingChaptersIII,V,andVII.Exceptforitsfirstsection,ChapterIcanbeskippedbythosenotinterestedinexplicitlocalclassfieldtheory.ReferencesoftheformMathxxxaretocoursenotesavailableathttp://www.math.lsa.umich.edu/∼jmilne.Pleasesendc
5、ommentsandcorrectionstomeatjmilne@umich.edu.BooksincludingclassfieldtheoryArtin,E.,AlgebraicNumbersandAlgebraicFunctions,NYU,1951.(ReprintedbyGordonandBreach,1967).Artin,E.,andTate,J.,ClassFieldTheory.NotesofaSeminaratPrinceton,1951/52.(HarvardUniversit
6、y,MathematicsDepartment,1961;Benjamin,1968;AddisonWesley,1991).Cassels,J.W.S.,andFr¨ohlich,A.,(Eds),AlgebraicNumberTheory,AcademicPress,1967.Fesenko,I.B.,andVostokov,S.V.,LocalFieldsandtheirExtensions,AMS,1993.Goldstein,L.J.,AnalyticNumberTheory,Prenti
7、ceHall,1971.Iwasawa,K.,LocalClassFieldTheory,Oxford,1986.Iyanaga,S.,(ed.),TheTheoryofNumbers,North-Holland,1975.Janusz,G.J.,AlgebraicNumberFields,AcademicPress,1973;SecondEdition,AMS,1996.Koch,H.,NumberTheoryII,AlgebraicNumberTheory,Springer,1992.(Ency
8、-clopaediaofMathematicalSciences,Vol.62,Parshin,A.N.,andShafarevich,I.R.,Eds).Lang,S.,AlgebraicNumberTheory,Addison-Wesley,1970.Neukirch,J.,ClassFieldTheory,Springer,1985.Neukirch,J.,AlgebraischeZahlentheorie,Springer,1992.Serre,J-P.,Co
