algebraic number theory - milne

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1、ALGEBRAICNUMBERTHEORYJ.S.MILNEAbstract.ThesearethenotesforacoursetaughtattheUniversityofMichiganinF92asMath676.Theyareavailableatwww.math.lsa.umich.edu/∼jmilne/.Pleasesendcommentsandcorrectionstomeatjmilne@umich.edu.v2.01(August14,1996.)Firstversionontheweb

2、.v2.10(August31,1998.)Fixedmanyminorerrors;addedexercisesandindex.ContentsIntroduction..................................................................1Theringofintegers1;Factorization2;Units4;Applications5;Abriefhistoryofnumbers6;References.7.1.Preliminar

3、iesfromCommutativeAlgebra............................10Basicdefinitions10;Noetherianrings10;Localrings12;Ringsoffractions12;TheChineseremaindertheorem14;Reviewoftensorproducts15;Extensionofscalars17;Tensorproductsofalgebras17;Tensorproductsoffields17.2.Ringso

4、fIntegers........................................................19Symmetricpolynomials19;Integralelements20;ReviewofbasesofA-modules25;Reviewofnormsandtraces25;Reviewofbilinearforms26;Discriminants26;Ringsofintegersarefinitelygenerated28;Findingtheringofint

5、egers30;Algorithmsforfindingtheringofintegers33.3.DedekindDomains;Factorization.....................................37Discretevaluationrings37;Dedekinddomains38;Uniquefactorization39;Theidealclassgroup43;Discretevaluations46;IntegralclosuresofDedekinddomains

6、47;ModulesoverDedekinddomains(sketch).48;Fac-torizationinextensions49;Theprimesthatramify50;Findingfactoriza-tions53;Examplesoffactorizations54;Eisensteinextensions56.4.TheFinitenessoftheClassNumber..................................58Normsofideals58;Stateme

7、ntofthemaintheoremanditsconsequences59;Lattices62;Somecalculus67;Finitenessoftheclassnumber69;Binaryquadraticforms71;5.TheUnitTheorem......................................................73Statementofthetheorem73;ProofthatUKisfinitelygenerated74;Com-putation

8、oftherank75;S-units77;Findingfundamentalunitsinrealc1996,1998,J.S.Milne.Youmaymakeonecopyofthesenotesforyourownpersonaluse.i0J.S.MILNEquadraticfields77;Unitsincubicfieldswithnegativediscriminant78;Findi