Number Theory II_Kimball Martin

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1、NumberTheoryIISpring2010NotesKimballMartinFebruary16,2010Exercise0.1.Readtheintroduction.It’saroadmapforthecourse.Infact,youmaywanttorereaditseveraltimesthroughoutthecoursetorememberwherewe’vebeenandwherewe’regoing.IntroductionLastsemester,wesawsomeofthepowerofAlgebraicNumberTheo

2、ry.Thebasicideawasthefollowing.Ifforexample,wewantedtodetermineWhichnumbersareoftheformx2+ny2?(1)Brahmagupta’scompositionlawtellsusthattheproductoftwonumbersofthisformisagainofthisform,andthereforeitmakesensetofirstaskWhichprimespareoftheformx2+ny2=p?(2)√TheideaofAlgebraicNumberTh

3、eoryistoworkwiththeringZ[−n]soanypsuchthat√√√p=x2+ny2=(x+y−n)(x−y−n)factorsoverZ[−n].AtthispointonewouldliketousethePrimeDivisorProperty(orequivalently,UniqueFactorization)tosaythatthismeanspisnot√√primeinZ[−n].UnfortunatelythisdoesnotalwaysholdinZ[−n],andthereweretwothingswedidt

4、oovercomethisobstacle.ThefirstwastoworkwithO√whichissometimeslarger√√Q(−n)thanZ[−n],andmayhaveuniquefactorizationwhenZ[−n]doesnot(wesawthisforthecasen=3—ithappensforothervaluesofnalso,butstillonlyfinitelymanytimeswhenn>0).Otherwise,weshoulduseDedekind’sidealtheory.Themainideahereis

5、wehavethePrimeDivisorPropertyandUniqueFactorizationatthelevelofideas.Henceifp=x2+ny2,theideal(p)=pOQ(√−n)inOQ(√−n)isnotaprimeidealandfactorsintotwoprincipalprimeideals(not√√necessarilydistinct)(p)=p1p2,eachofnormp.Further,p1=(x+y−n)andp2=(x−y−n).Infact,withsomeslightmodifications,

6、theconverseisalsotrue.Tounderstandthis,wefirstneedtounderstandthemorebasicquestionWhenispOQ(√−n)aprimeideal,andwhendoesitfactor?(3)Onceweknowforwhichprimesp∈N,(p)isnotprimeinOQ(√−n)(inwhichcasewesaypsplits√inQ(−n)),weneedtoknow√WhatistheclassgroupofQ(−n)?(4)1todeterminewhen(p)isap

7、roductoftwoprincipalidealsinOQ(√−n).Thefirstpartofthesemesterwillbemotivatedbythesequestions,thoughweshallspendalotofourtimepursuingrelatedquestionsandtopicsalongtheway.Inotherwords,ourgoalisnotsomuchtoseekadefinitiveanswertothequestion(2)(see[Cox]),butrathertouseitasaguidetounders

8、tandandpursuesomeimportanttopicsinnumber