; TeX output 1998.10.21:0824K7 Y̫LK`y cmr10CHAPTER4e[Nff cmbx12TheffIdealClassGroup"T"V cmbx101.De nitionsH1.1.F ractionalzjideals.Inordertokeepthealgebrasomewhatmorepleas- 6ant,sitwillbGeusefultointroGducethenotionoffractionalideals.Speci cally*,sthe6idealsoftheringofintegersofanumbGer elddonotformagroup,asthereare6no inverses.F*ractionalideals,7ontheotherhand,formagroup;d@therelationship6bGetweenfractionalidealsandidealsisquitesimilartotherelationshipbetweena6numbGerUU eldanditsringofintegers.HLeto b> cmmi10K&bGeanumbero eldwithringofintegers!", cmsy10O0ercmmi7K.DLet%n eufm10rbeanon-zerosubset6ofK@whichisanOK-moGdule;thatis,̱risclosedunderadditionandundermulti-6plicationqbyelementsofOK._{SuchanrissaidtobGea': cmti10fr}'actionalahidealqifthereexist6म ٓRcmr71|s;:::; m _2rUUsuchthata0r=f 1|s 1S+8g+8 m m _j id2OKg;6thatZis,risgeneratedoverZOK Xbythe iTL.(Therelevqantthinghereisthatris 6 nitelygeneratedover̯OK.-NoteveryOK-submoGduleofKEhasthisproperty;see6ExerciseUU4.1.)HThereTaretwoTfundamentalexamplesoffractionalideals.p"Firstofall,every6non-zero|idealaofOK O/isalsoafractionalideal:aisanOK-moGdulebyde nition6and)ithasa nitegeneratingsetsinceOK ܫisnoGetherian.DT*oavoid)confusion,we6shallUUrefertoidealsofOK asinte}'gralidealsUUfromnowon.HTheVsecondsortofexamplearefractionalidealsoftheform 8OK 7 forsome6म ;<2K^ O!cmsy7P.s(Onexcheckseasilythat 8OK .isanOK-moGdule,andithasthesingle6generatorˮ 8.)JSuchafractionalidealiscalledaprincip}'al(fractionalideal.JNotethat6theUUprincipalidealsofOK arepreciselytheintegralprincipalfractionalideals.HMoregenerally*,?letabGeanyidealofOK a]andlet 9beanyelementofK^P.6Thenm 8^1 Jaisafractionalideal.( 8ahasa nitegeneratingsetsinceif 1|s;:::; m6फgenerate+a,then 8 1|s;:::; m ƫgenerate+ a.)TdTheconverseofthisstatementisalso6true.?H - cmcsc10Lemma1.1.L}'et/rbeanOK-submoduleofK.jThenrisafractionalidealif6and\onlyifther}'eexists +2 K^ \suchthat 8risanintegralideal. (Infact,¹onecan6actuallytake "tob}'earationalinteger.)HवProof.pZW*esawabGovethatifaisanintegralidealand x2<@K^P,4then 8aisa6fractionalUUideal.qConversely*,ifrisafractionalideal,thenwecanwritear=f 1|s 1S+8:::g+8 m m _j id2OKg6फforsome 1|s;:::; m N2r.ByLemmaIGI.2.10thereexista1;:::;am N2  msbm10Zsuchthat 6मaiTL id2OK._mOneGnoweasilychecksthata1'|jamrisanintegralideal,)Jwhichproves6theUUlemmawith UP=a1'|jam.ffdffYffff175L*7 6ळ764.:THEIDEALCLASSGROUPYHफW*e;7willdenotebyIK thesetofallfractionalidealsofK.iIfr;s2IK,@pthen;7we 6de netheproGductrstobetheOK-modulegeneratedbyallproductsofpairsof6elementsofrands.ۨNotethatifrisgeneratedby 1|s;:::; m fandsisgenerated6byz1|s;:::;k됫,RthenrsisgeneratedbytheproGducts iTLj6.Inparticular,Rrsisalsoa6fractionalUUideal.HवCorollarUTyL1.2.ThesetIK m6isanab}'eliangroupundermultiplicationoffrac-6tionalide}'als.HवProof.pZW*e^sawabGovethatIK ǫisclosedundermultiplication.Thatthismul-6tiplicationiscommutativeandassoGciativeisclear.Theidentityelementiseasily6checkedtobGetheunitidealOK.FItremainsto ndinverses.SoletrbGeafractional6ideal*andchoGose k\2$K^ L*suchthat 8risanintegralideal.fFByPropGositionII.3.66thereisanintegralidealbsuchthat 8rbisprincipal,saygeneratedby ۨ2.O^GbK.6T*akeUUs=B K);fe2 ``b.qThensisafractionalideal,andwehaveSٍǕ%rs=<$K 8rbKwfe (֍)> .=OK:6फThusUUsisaninverseforrinIK.tffdffYffffHNotethatitisclearfromtheproGofofPropositionII.3.6thatifrisafractional6ideal,UUthenitsinverseUUisgivenby32rr1 =f UP2K~4j 8rOKg:HफW*e$canalsocharacterizefractionalidealsintermsofuniquefactorizationof 6ideals.HवProposition1.3.'Everyfr}'actionalidealrcanbewrittenasr=p:e Zcmr51l1 2bpe O \cmmi5r፰r6लwher}'e>OthepiaredistinctprimesofOK andtheeiareintegers.}(Notethatweallow 6theW`eitob}'enegative.)Thisexpressionisuniqueuptoreorderingofthefactors.6ThusIK Iisthefr}'eeabeliangroupontheset32fpjpaprimeofOKL9g:6लFinally,risaninte}'gralidealifandonlyifeachei3isnon-negative.HवProof.pZLet rbGeafractionalidealandchooseanon-zerorationalintegera2Z 6फsuchٗthatthatarisanintegralideal.HThenwecanwrite(uniquelyuptoreordering6andUUaddingfactorswithzeroexpGonent)𢍒aOK |˫=p;er0ncmsy501k1 2bp;er0r!rHȆUar=p;er04s01k1 ٯ4(p;er04s0r!r 41;6hereUUweallowsomee^0;Ziande^0N90;ZiqǫtobGezero.qThus,sinceIK isagroup, (6r=p;er04s01er01k1E&&p;er04s0rer0r!rG4:326फThisoshowsthatrhassuchanexpression;Lthefactthatitisuniquefollowsfrom6thefactthatthefactorizationsofaOK Uandarwereunique.QThefactthatrisan6integralidealifandonlyifeachei8ispGositiveisclearfromuniquefactorizationof6ideals.EnffdffYffffHNoticethatthisdecompGositionoffractionalidealsintermsofprimeidealsis6completelyҺanalogoustothedecompGositionofrationalnumbersҺintermsofrational6primes;UUseeSection1.1ofChapter2.M 7 ̭ 1.:DEFINITIONS!77YHभ1.2.The>idealclassgroup.LetKhbGeanumber eldwithringofintegers 6यOK.W*ehaveseenthatOK lVmaynotbGeauniquefactorizationdomain,although6itdwillhaveduniquefactorizationofideals.W*ehavedalsoseen(seeExerciseIGI.2.10)6thatޯOK isaUFDifandonlyifitisaPID;thatis,?ifandonlyifeveryidealis6principal.GF*urthermore, evenwhenOK isnotaPIDitisoftenusefultoknowwhen6idealsUUareprincipal;see,forexample,PropGositionIII.1.7.HThese^factssuggestthatitwouldbGeusefultohavesomewaytodetermineifan6ideal_isprincipal. Althoughinpracticethisisoftenquitedicult,awecanproGceed6abstractlyfairlywell.De nePK xǫtobGethesubgroupofIKofprincipalfractional6ideals.bNote(thattheintegralidealsinPK ǫarepreciselytheprincipalidealsofOK.6W*eUUde netheide}'alclassgroupUUCK ofK qtobGethequotient̷{CK |˫=IK=PK:6यCK \naturallyKrelatestotheissuesraisedabGove.nFirstKofall,MCKisthetrivialgroup 6if andonlyifIK |˫=PK;26thatis,+/ifandonlyifeveryfractionalidealofK«isactually6principal.LoSince8theintegralidealsinPK Sarepreciselytheprincipalideals,pthisis6equivqalent toOK KbGeingaPID,whichinturnisequivqalenttoOK KbGeingaUFD.6Thattis,CK @'istrivialifandonlyifOKisaUFD.Secondly*,notethatafractional6idealrisprincipalifandonlyifitmapsto0inCK.Thus,ifonecouldobtaina6goGodܰdescriptionofCK,onewouldhaveamethoGdtodetermineifanarbitraryideal6isUUprincipal.HW*ewwillcalltheelementsofCK -gide}'alclasses;thusanidealclassAissimplya6cosetoofPK.\Byde nitionofCK,#twoofractionalidealsaandblieinthesameideal6classUUifandonlyifthereissome UP2K^Uwith4 8a=b:6फW*ekwillwritethisrelationasa5<b.8 ThekfollowingreinterpretationofLemma1.1 6showsD]thatfractionalidealsarenotreallyessentialtothede nitionoftheideal6classUUgroup.yYHवLemma1.4.L}'et$Abeanidealclass.J2Thenthereexistsanintegralidealain6thec}'osetA.HवProof.pZLet搱rbGeanyfractionalidealinA.%yThenbyLemma1.1thereexists6म m2׮K^suchcthat 8risanintegralideal.Since 8OK 2׮PK,g$wehave 8rׯ2A,g$which6provesUUthelemma.ńffdffYffff/_HवExample1.5.T*akeUUK~4=Q($pUW$fe 5v)andconsiderthetwoidealsXRjbu cmex102;18p 7fe X5UVb꭮;b?3;1+p 7fe X5UVb꭮:6फNoteUUthat \ffb2;18p 7fe X5UVbū= 8b#3;1+p 7fe X5UVb6फwhere@ƀ UP=<$33$p $fe 533wfev (֍;3+<$l1lwfe (֍3 G:ō6फThusটbu2;18p 7fe X5UVbůb\o3;1+p 7fe X5UVb꭮:HफAs5cwesawinSection4ofChapter2,mfthepresenceofnon-principalidealsis 6closely/8relatedtotheproGductionofcounterexamplestouniquefactorization.eThus6theUUidealclassgroupissomesortofmeasureofhowfarOK isfrombGeingaUFD.N#ޠ7 6ळ784.:THEIDEALCLASSGROUPYHफThevVdeterminationoftheidealclassgroupofanumbGervV eldisacentralproblem 6inAalgebraicnumbGerAtheory;H9itisalsoanextremelydicultprobleminmostcases.6W*ewillproveinthenextsectionthatitis nite,Zandoftenitisslightlyeasierto6determinetheclassnumb}'erhK =Xf#CK.wULaterwewillexplainhowtocomputeit6inathecaseofquadraticimaginary eldsandgiveanideaofthestateofknowledge6concerningUUidealclassgroupsofcyclotomic elds.荑Hभ1.3.TheGunitgroupandtheclassn9umbQerGformula.W*ewillneverac-6tually$]needtheresultsofthissection,Xbutwestatethemforcompleteness.The6second%fundamentalinvqariantofanumbGer eldKcAisthegroupofunitsO^bK.v6The6impGortanceofO^bK cstemsfromthefactthattheunitsarepreciselytheambiguityin6movingJfromfactorizationsintoprincipalidealstofactorizationsofelements.n7This6groupisessentiallyasdiculttocomputeastheidealclassgroup,andtheyare6closelyUUrelated.qW*ewilltryinthissectiontodescribGesomeofthoserelations.HT*oUUseethe rstrelation,notethatthereisanaturalsurjection|!ϮK msam10PK6फsending痮 I2ڮK^ 7totheprincipalfractionalideal 8OK.(Thekernelofthismapis 6justthesetof UP2K^fforwhich 8OK |˫=OK;+}these ɫareeasilyseentobGeprecisely6theUUunitsO^GbK.HW*ejalsohavejanaturalinjectionPK x,UX!IK.7Thecokernelofthismapisthe6idealclassgroupCK, ybyde nition.R,Inparticular,ifweconsiderthecompGositemap|[iKPK |ˮ,UX!IK;6फwegseethatithaskernelO^GbK andcokernelCK.Thuswehaveexhibitedasingle 6mapUUwhichconnectsthesetwofundamentalinvqariants.HF*romhereweomitallproGofs.xInordertostatethesecond(muchdeepGer)6connectionweneedtoknowabitmoreabGouttheunitgroup.XThefundamental6theoremZisduetoDirichlet.]W*e rstneedtoanalyzethecomplexembGeddingsa6bit.ǯW*eqwillsaythatacomplexembGeddingS:zKX,UX!Cqisr}'ealqifithasimage6in'uR;otherwiseitisimaginary.'IfNisimaginary*,[thenitscomplexconjugate6फisadi erentimaginarycomplexembGeddingofK.1W*eletrbethenumberof6realíembGeddingsofKzɫandsthenumbGerofcomplexconjugatepairsofimaginary6embGeddingsUUofK.qThusr+82s=n,UUwherenisthedegreeofK qoverQ.n:HवExample1.6.If2K=MܶQ(zPpUWzPfe4rd ɫ)isquadraticwithd>0,ithenr=2ands=0. z6SuchHaKdiscalledar}'eal/Vquadratic eld.lIfHKȫ=ଶQ(zPpUWzPfe4rd ɫ)withd<0,(thenr'ɫ=06andCs=1;,K_iscalledaimaginary[quadr}'atic eld.]lIfK~4=Q(m)withm>2,$zthen6everyCIembGeddingisimaginary(sinceRcontainsnoroGotsofunityoforder>2),Fso6मr5=0iands='(m)=2.]yNotethatinallofthesecaseswehaveoneofr_andsequal6toU0;VthisisbGecausethe eldsareGalois,andthusallembGeddingshavethesame A6image.qF*orUUanon-Galoisexample,takeK~4=Q(Zt3Pp_PfeE2 `):thenUUr5=1ands=1.)HवTheorem1.7(DirichletUUUnitTheorem).mhL}'et Kbeanumber eldwithrTreal6emb}'eddings6andsc}'omplexconjugatepairsofimaginaryembeddings.DŽLetWŲbethe6sub}'groupofO^GbK Iofr}'ootsofunity.Thenᛍ/OG፰KT͍ |˯+3 |˫= Wo8Zr7+s1:HफNotethatthistheoremimpliesthatO^GbK T_is niteifandonlyifr+iĮsAR=1;Wthis6oGccurs]ifandonlyifKyisQoranimaginaryquadratic eld.[Itisnotacoincidence6thatUUthesearethenumbGerUU eldsofwhichwehavethegreatestunderstanding.O607 ̭ 1.:DEFINITIONS!79YHफThe'~proGofofTheorem1.7restsuponthelo}'garithmiciembedding'~ofK^P.bThisis 6aUUmap:^jGK!Rr7+sS6फde nedYasfollows:let1|s;:::;rAbGetherealembGeddingsofKandletr7+1;:::;r7+s6फbGeXasetofimaginaryembeddingsofKUtcontainingoneofeachcomplexconjugate6pair.Z(Thus&11|s;:::;rm;r7+1;:::;r7+sWl;3r7+1;:::;3r7+sare&1thencomplexembGed-6dingsofK.)5MThelogarithmicembGeddingisde nedbysending گ2`K^ 悫tothe6(r+8s)-tuplemabeBlogtBj1|s( z)j;:::;log?jrm( )j;2logjr7+1( z)j;:::;2logjr7+sWl( z)jbW:m6फOne!shows(usingthefactthatthenormofaunitis1)thattheimageofO^GbK ԫlies6entirelyUUwithinthehypGerplanem#1x1S+8g+8xr7+s=0:6फF*urthermore,(byoExercise2.16oneseesthatthekernelofthelogarithmicembGedding 6isP8preciselythegroupofroGotsofunityWc.Theremainderoftheproofofthetheorem6involves(rshowingthattheimageofK^xrisalatticeofmaximalrankinther&7+s16dimensionalUUhypGerplanex1S+8g+8xr7+s=0.HW*eneedthelogarithmicembGeddingtode neanimportantinvqariantofK.F`Let6म"1|s;:::;"r7+s1bGe#>abasisforthefreepartofO^bK;3thuseveryelementofO^GbK canbGe6writtenUUuniquelyas":n1l1 FX"㍰nr,r+s1Ír7+s1S6फwithGX2Wandeachnid2Z.W*ede nether}'egulatorGXRK ofKttobGethedeterminant6ofUUthematrix抟b{iTL( j6)bߍWr7+s1 $Wi;jg=1)7:0k6फ(ItpinfactdoGesn'tmatterwhichembGeddingiHoneomitsfromthematrix,waseach6rowcanbGewrittenintermsoftheotherr_w+Zs1rows.)ATheDirichlet.classnumb}'er6formulaHstatesthat,SifK(=QisGaloiswithabGelianGaloisgroup(forexample,a6quadraticUU eldoracyclotomic eld),then1llBhK |˫=<$ wDjKjKwfe,@ (֍2rr7+sWl[ٟrsRK4glim-3fs!1C(s81)K(1):_6फHereG+wisthenumbGerG+ofrootsofunityinK,JK ޫisthediscriminantofOK,JrHand6मsg>arethenumbGerg>ofrealandpairsofcomplexconjugateimaginaryembeddings6respGectively*,UUandK istheDe}'dekindzetafunction,UUde nedforRe#(s)>1UUbymLK(s)=(X X&eufm7aȳanidealofOmK?^NG&_:KI=qymsbm7QW(a)s ;;g㍑6फwhichGisameromorphicfunctionwithananalyticcontinuationtotheentirecomplex6plane,UUwithasimplepGoleats=1.HAllofthesetermsturnouttobGereasonablyeasytocomputeexceptforthe6regulatorCandtheclassnumbGer.OneCsees,ݿtherefore,thatdeterminationofthe6regulator?isessentiallythesameasdeterminationoftheclassnumbGer.1Sinceto6compute5theregulatoronevirtuallyneedstoknowpreciselywhattheunitsare,6thismeansthatcomputingtheidealclassgroupandtheunitgrouparealmostthe6sameUUproblem.PJ7 6ळ804.:THEIDEALCLASSGROUPY1ح2.FinitenessToftheidealclassgroupH2.1.Norm8gbQounds.Thefactthattheidealclassgroupis niteindicates 6thatuniquefactorizationneverfailstoGospectacularlyinringsofintegersofnumbGer6 elds,andispGerhapsthemostimportantsinglefactinalgebraicnumbGertheory*.6InUUthissectionwewillgiveasurprisinglysimpleproGof.hcHवTheorem2.1.%L}'et:aK}beanumber eld.{ThereexistsanumberK,LIdepending6only onK,suchthateveryide}'alnon-zeroaofOK ccontainsanon-zeroelement 6लwith ɿjN *:KI=Q˫( z)jK `[N\:KI=Q!~(a):HवProof.pZLet~ 1|s;:::; nbGeanintegralbasisforOK 3īandlet1;:::;nbGethe6complexUUembGeddingsofK.qW*ewillshowthatonecantake[K |˫=ȁn ݱY ti=1Zf0 f@6n &X tsjg=1*BjiTL( j6)jZ1 AU:HफLetZ abGeanon-zeroidealofOK andletmbetheuniquepositiveintegersuch 6that ]mn8N G:KI=Q;(a)<(m8+1)nq~:ށ6फConsiderUUthe(m8+1)^nӫelements썍Ǻ8 < :|n VX t jg=1Ǯmj6 jįj0mjm;mj2Z9 = ;8d:6फSinceOK=ahasorderlessthan(mD+1)^nq~,twooftheseelementsmustbGecongruent6moGduloUUa.qT*akingtheirdi erencewe ndanelement_8ąٮ В=>n X tjg=1㉮m0፰j6 jį2aU6फwithUUjm^0;Zj6jm.qW*ecomputed;ïjN *:KI=Q˫( z)jm=ȁn ݱY ti=1fjiTL( z)j%Lm=ȁn ݱY ti=1f f f f f f diZ0 @n  X t jg=1fm0፰j6 jZ1 A>6 >6 >6 >6 >6 >6 (m=ȁn ݱY ti=1f f f f f f {n dX tjg=1$ծm0፰j6iTL( j)  $mȁn ݱY ti=1&5n fX tjg=1!jm0፰j6jjiTL( j)j"vmȁn ݱY ti=1&5n fX tjg=1!mjiTL( j6)jMtm=mnq~KmK `[N\:KI=Q!~(a)Q\R7 2.:FINITENESSOFTHEIDEALCLASSGROUPO81Y6फasUUclaimed.0RsffdffYfffffHवCorollarUTyL2.2.L}'etAbeanidealclassofCK.ĶThenAcontainsanintegral 6ide}'alofnormK.>HवProof.pZLetTbbGeanyintegralidealinA^1 t.WqByTheorem2.1wecan nd N42b6फwithLjN *:KI=Q˫( )j\K `[N\:KI=Q!~(b).UTheprincipalideal OK Viscontainedinb,Jso6byLemmaIGI.3.8thereisanintegralidealasuchthatabM= OK.)\Sinceܮ OK is6principalUUwehavea2A,UUandwecompute9Na::KI=Q3\(a)=dKjN *:KI=Q˫( )jK'Ήfe-? (֍#)N *:KI=QuL(b)5uկK:򍍍ffdffYffffHवCorollarUTyL2.3.Theide}'alclassgroupCK Iis nite.>HवProof.pZByQCorollary2.2everyidealclasscontainsanidealofnormatmost6मK.Byd^Exercise4.2thereareonly nitelymanyidealsofnormϮK, sothis6means1 thateveryidealclasscontainsoneofa nitesetofideals.eInparticular,8MCK6फmustUUbGe nite.!RqffdffYffffHThebGoundgivenaboveisnotterriblyusefulinactuallycomputingtheideal6classfgroup,k&bGothbecauseitisdiculttocomputeandbecauseitgetslargefairly6fast.nARmuchbGetterboundcanbeobtainedusingMinkowski'stheoreminthe6geometryßofnumbGers;1weßstateithereandwilluseitinthenextsectiontocompute6idealUUclassgroupsofimaginaryquadratic elds.>HवTheorem2.4(MinkowskiUUbGound).tL}'etJK۲beanumber eldofdegreen.3Then6everyide}'alclassofOK Icontainsanidealasatisfying'N:KI=Q(a)K |˫=<$O|n!Kwfe r (֍nrnJ>^<$at4wfe (֍"6^)x۰s/i5p9i5feD ˍjKjOѮ:|6लHer}'esisthenumberofconjugatepairsofimaginaryembeddingsofK.Hभ2.2.Computations"ofidealclassgroupsofcyclotomic elds.There6areD someimmediateapplicationsoftheMinkowskiD bGound.l F*orexample,GtakeK~4=6शQ(5|s).This\ eldhasdiscriminantK f+=x5^3]ϫands=2,^sotheMinkowski\bGound6showsUUthateveryidealclasscontainsanidealofnormatmostqIK |˫=<$4!Kwfe |t (֍4r4T^<$kЫ4Awfe (֍ &^'۳2-!=Vp5x=Vfeª125GƓ1:6992079064:|6फThus@meveryidealclasscontainsanidealofnorm1.jButtheonlyidealofnorm1is6यOK,b/so_everyidealclasscontainsOK;dthusthereisonlyoneidealclass,b/andCK Pis6trivial.qItUUfollowsimmediatelythatZ[5|s]isaUFD.HF*orfaslightlymoreinvolvedexample,ktakeK#=Q(7|s).Thistimewecompute6thatHtheMinkowskiHbGoundisK |˯4:12952833191:Thuseveryidealclasscontains6anFXidealofnormatmost4.DLetabGesuchanideal,andassumethataXů6=OK.6Since&_theonlypGossibleprimefactorsofN `:KI=Qx(a)are2and3,/everyprimefactorof6ऱaUUmustlieover2or3.HLetusnowdeterminetheseprimes.TDSince2hasorder3in(Z=7Z)^,theprimes6lying~-over2willhaveinertialdegree3.PInparticular,ctheywillhavenorm2^3= +8;6thustheycannotappGearasprimefactorsofa.USimilarly*,since3hasorder6in6(Z=7Z)^,ditactuallyremainsprimeinOK andhasnorm3^6C=729.OItcannotoGccurRjp7 6ळ824.:THEIDEALCLASSGROUPY6फasafactorofaeither;1thusamustbGeOK._ItfollowsthatCK vistrivialandZ[7|s]is 6aUUUFD.HEvenKtheMinkowskibGoundbecomessomewhatdiculttousepastthispoint;6thisUUagainillustrateshowdicultitcanbGetocomputeidealclassgroups.HW*eswillconcludethissectionwithafewcommentsonthestudyofidealclass6groupsofcyclotomic elds;thisremainsanimpGortantandactiveareaofnumbGer6theory*.XwThe edeterminationofallcyclotomic eldsofclassnumbGer e1wascompleted6inr1971byMasley*,lusingworkofSiegel,lMontgomeryandUchida.Z{Recallthatifm6फis9oGddthenQ(m)=Q(2m ), so9wecanrestrictourattentiontothosemwhichare6notUUcongruentto2moGdulo4.pwHवTheorem2.5(Masley).ӲL}'et: mbeanintegerwhichisnotcongruentto2mod-6ulo,4.NeThenQ(m)hastrivialide}'alclassgroup(andthusZ[m]isaUFD)ifand6onlyifbCm=u1;3;4;5;7;8;9;11;12;13;15;16;17;19;20;21;24;25;27;28;uث32;33;35;36;40;44;45;48;60;84: HवProof.pZTheUUproGofisquiteintricate;see[20 ,Chapter11].O`ffdffYffff*HTheԨ rstcyclotomic eldwithnon-trivialidealclassgroupisQ(23x),dwhichhas6classUUnumbGer3.HInthegeneralcasethe rststepistobreaktheidealclassgroupintosmaller6pieces.R!Letbuswritehm fortheclassnumbGerbofQ(m)andh^+፰mfortheclassnumbGer6oftherealsub eldQ(m)^+.Onecanshowthath^+፰m I=divideshm(theproGofofthis6isaneasyapplicationofclass eldtheory*,7but,bGeinganapplicationofclass eld6theory*,isK0.(AswesaidabGove,6weucannothaveImgw(1|s=2)=0,}sinceuthenwouldnotcontainanR-basisforC.6Thus,if$Im(1|s=2)<0,we$canswitchtheorderofthei/ptogettheimaginarypart6pGositive.)qLetUUH=fz72CjImcz>0gUUbGetheupperhalf-planeandde ne(j(1|s;2)=<$K1Kwfe Qʟ (֍2F`2H:6फNoteUUthatforany В2C^,Zፒ˱j( z1|s; 2)=j(1;2);6फwhichbsuggeststhatjisadecentplacetostartintheclassi cationoflatticesupto 6homotopy*.HUnfortunately*,j(1|s;2)zJdepGendsnotonlyonbutalsoonthechoiceofbasis6म1|s;2.mInZordertousejtoclassifylatticesuptohomothetywemustremovethis6basisUUdepGendence.HW*edothisbydeterminingtheotherpGossiblebasesforandseeinghowj6फdepGends>uponthechoice.Bystandardlinearalgebra,18thebasesforareofthe6form'$0፳1C=a1S+8b2|s;0፳2=c1S+8d2AH6फwhereAҟ^dȞFa\Zbcvd^?t2GL2\p(Z);6फtheYintegermatricesofdeterminant1.However,asabGovewewanttorestrictto6onlythebases^0l1|s;^0l2 oforderedsothatImv(^0l1=^0l2)I>0.\Onecheckseasilythat6thematriceswhichpreservethisconditionarepreciselythoseinSL:2H(Z);RTthatis,6thoseUUofdeterminant1.qW*ecomputeforthesebasesuޝj(a1S+8b2|s;c1+d2|s)=<$Ka1+b2Kwfe*pL (֍Ǯc1+d22d=<$Kaj(1|s;2)+bKwfe9x? (֍cj(1|s;2)+d>:TȠ7 6ळ844.:THEIDEALCLASSGROUPYHफTheseD"computationssuggestthefollowingapproach.>/W*ede neanactionof 6मSL2|s(Z)UUonHby1)^dƍaKbHcd ?^[z7=<$Kazw+8bKwfe֟ (֍czw+8d T;[t6weu#leaveittothereadertocheckthatthisreallyisagroupaction.2Letusdenote6byίYKthequotientspaceofHbythisaction.TRecallthatthismeansthatYKconsists6ofUUtheorbitsoftheSL#2(Z)actiononH:qforanyz72H,itsorbitissimplytheset>f 8z7j UP2SLS2ƫ(Z)g:HफThis%actionofSLƟ2p9(Z)isde nedinsuchawaythatif1|s;2and^0l1;^0l2aretwo 6correctlydorderedbasesofalattice,thenj(1|s;2)dandj(^0l1|s;^0l2)dwilllieinthesame6SLBߟ2G+R(Z)UUorbitofH;thatis,theywillbGeequalinY}.HThisUUtellsusthatifwecompGoseourmap>txj:orderedUUbasesoflatticesn!H6फwithUUthequotientmapH!Y},UUweobtainamapj:lattices$ޯ!Y};6the1Jj:homothetyUUclassesoflattices̯!Y}:6फByUUthiswemeanthatifand^0#arehomothetic,thenj()=j(^09). HThise,mapjiseasilyseentobGesurjectiveanditcanalsobeshowntobeinjective.6Thus{jestablishesasetbijectionbGetweenhomothetyclassesoflatticesandY}.I)This6means7^c8z72C;ImGz>0;<$ۯ1۟wfe  (֍2E1^Cȯ[\>7^c8z72C;jzpj=1;0ReQ(zp)<<$K1Kwfe (֍2 -^뛡[8\( Gz72C;Re x(zp)=<$K1Kwfe (֍2 -;Im q(z)<$KPp OPfeE3Kwfe UX (֍*2֟\):H6लThen/:Yhc}'ontainsexactlyoneelementofeachSLu2y(Z)orbitofH;|thatis,VYhisin6natur}'albijectionwithY}.HवProof.pZSeeUU[17 ,Chapter7,Section1.2]or[18,PropGosition1.5].0X$ffdffYffff퍑HThe lastthingweneedisagoGod waytodeterminewhichelementofYan6elementUUofHcorrespGondsto.HवProposition3.2.'SetESSZ=^d#0#1#1"0/꬟^8Ȯ;T*=^d#1#1#1#0(#^1*:F6लThenS'tandTvgener}'ateSLb"2ޕ(Z).HवProof.pZSeeUU[17 ,Chapter7,Section1.2]or[18,PropGosition1.5].0X$ffdffYffffU7 n+3.:IDEALCLASSGROUPSOFIMAGINARZYQUADRATICFIELDS,^85Y(45PSfile="fd.eps" llx=0 lly=0 urx=203 ury=175 rwi=2030 VjFigure1.TheUUfundamentaldomainY9fortheSL2|s(Z)actiononHHफNoteUUthatS(zp)= B³133&fez LandUUTc(z)=zw+81. HThese#resultsgiveusthefollowingalgorithmfordeterminingthehomothety6class-}ofalatticewithbasis1|s;2.dFirst,5ucompute-}j=j()=1|s=2.W*e-}wantto6moGdify)(jbySandTtogetitintoY8.cIfImřj<0,1replace)(jby1=j;7thiscorrespGonds6toKXswappingthetwobasiselements.nsNow,MWifjisinY8,thenwearedone.nsIfjisnot6inUUY8,then rstaddanintegermtojsothatZ<$^1^wfe  (֍2<ReQ(j8+m)<$K1Kwfe (֍2 -:6फIf+ jg+m+i2Y8,`thenwearedone.)Ifnot,replacejg+mby 133&fe:j+mVandstartover. !6PropGosition?V3.2(ormorehonestlyitsproof)guaranteesthatthiswilleventually6yieldUUanelementofY8.ԬHवExample3.3.LetUU=5Z8+(1+i)Z.qW*ecomputeՍaj()=<$ ϴ5Kwfeӟ (֍18+ii=<$K5Kwfe (֍2 f_<$l5lwfe (֍2 Gi;W6फsoUUwereplaceitby'j<$1ȟwfe (֍j()=<$K1Kwfe (֍5 f_+<$l1lwfe (֍5 Gi2Hgi6फThisDdoGesnotyetlieinY8,{asithasabsolutevqalue<1.=Sinceitsrealpartisalready6bGetweenUU 33133&fes2 .and 1&fes2,wereplaceitbyitsnegativereciproGcal,whichis۠]<$33533wfe (֍2 G+<$l5lwfe (֍2i:6फAddingUU3tothisweobtaintheelement<$1wfe (֍2D+<$l5lwfe (֍2 Gi6फofUUY8. HSuppGosethatweusedthebasis23 8+3i=4(5) 8+3(1+i);17+2i=3(5) 8+2(1+i)6ofUUinstead.qW*ecompute7^j()=<$K238+3iKwfe՟ (֍178+2i&k=<$K397Kwfe (֍293fa+<$l5lwfe (֍293Ii:V7 6ळ864.:THEIDEALCLASSGROUPYHफSubtractingS1yields 104&fe Y293+ 5&fe Y293>i,whichhasabsolutevqalue<nū1.lhItsnegative 6reciproGcalUUisn<$104wfe! (֍c37D+<$5lwfe  (֍37Hi;?m6addingUU3yields<$y7txwfe  (֍37+<$5lwfe  (֍37Hi;6फwhichUUstillhasabsolutevqalue<1.qItsnegativereciproGcalis]<$33733wfe (֍2 G+<$l5lwfe (֍2i;Y6addingUU4yields<$)1)ҟwfe (֍2ڕ+<$l5lwfe (֍2 Gi2Y9;6फasUUbGefore.&Hभ3.2.Idealgeneratorsandlatticegenerators.In4ordertotakeadvqantage 6ofourlatticeclassi cationoftheprevioussectionweneedamethoGdtogofromideal6generatorswtolatticegenerators.B(Thatis, givenanideala=(a1|s;a2)wwewantto nd6a1Z-basisfora.VfThegeneralalgorithmislittlemorethanGaussianelimination:Hwe6know-thata1|s;a2PformasetofZ[ z]-generatorsfora,soa1;a1 z;a2;a2 ݧform-asetof6शZ-generatorsfora.8W*riteallfouroutintermsofthebasis1; )ofOK.NowpGerform6your:[favoriteGaussianeliminationalgorithmonthesefourvectorstoobtainatwo6vectorbasis;&bonemustremembGerthatsinceweareworkingonlywithZ-moGdules6andUUnotwithvectorspaces,theonlyscalarsallowedareintegers.HवExample3.4.T*akeEK~4=Q($pUW$fe 5v)anda=(10; ;+5),where В=$p o$fe 5.ZThen610;10 z;5+ ;(5+ ) В=5+5 C.are9Z-generatorsfora;BthuswewishtopGerform6GaussianUUeliminationonthematrix =}^d010D304555D10210415T^}:6फAddingUU5timesthethirdcolumntothelastcolumnyieldsࠟ^d'10ާ0'5'20ʧ0'10'10'ş^:6फSubtractingAtwicethe rstcolumnfromthelastcolumnnoweliminatesthelast6column.qSubtractingUU10timesthethirdcolumnfromthesecondcolumnyields^dD010D250 R5100 R1 S^:6फFinally*,>=adding8w5timesthe rstcolumntothesecondcolumnshowsthattheideal6generatorsUU10; BZ+85arealsoalatticebasisfora.&HInfact,itveryoften(butpGossiblynotalways;SI[haven'tyetfoundacoun-6terexample,Qbut@itseemsthattherecouldbGeone)happensthatthe\natural"6idealMgeneratorsarealsoalatticebasis.̮F*orexample,PExercise4.3showsthatif6ऱp=(p; BZ+8m)UUisaprimeidealofOK,thenpand +8marealatticebasisforp. HNoteUUalsothatifa=b\oabꬫisUUaprincipalideal,thenahaslatticebasisa;a z.W7 n+3.:IDEALCLASSGROUPSOFIMAGINARZYQUADRATICFIELDS,^87YHभ3.3.Computing@idealclassgroups.W*enowhaveallofthetoGolswewill DA6needvtocomputeidealclassgroupsofimaginaryquadratic elds.LetK=Q(zPpUWzPfe4rd ɫ) 6andde ne 0andf(x)asbGefore.The rststepistodeterminegeneratorsfor6the`idealclassgroup.T*odothis,=computetheMinkowski`bGound:forimaginary6quadraticUU elds,itworksoutasCK |˫=\(. 4 &feo)(p~fe d1zd2;3 (moGd4)fc 2 &feo)(p~fe d1zd1 (moGd4):6फNext,T,for!4everypGositiverationalprimepK,T,determine!4thefactorizationofp6फinto;zprimesofOK -asinChapter3,@Section1.1.i)Ifpisinert,thentheidealpOK -is6principal,BCso=itisirrelevqantincomputingtheidealclassgroup.iThusweneedonly6considerethosepwhichsplitorramify*.MLetP0eثbGethesetofprimesofOK lyingover6theseUUp.HW*eclaimthatP0scontainsgeneratorsfortheidealclassgroupCK.ZToseethis,6letAbGeanyidealclass.uW*eknowthatthereissomea`ͯ2AwithN 1:KI=Q(a)`ͯK.6By{uniquefactorizationofideals,Fafactorsintoprimeideals,andeachsuchprime6mustJ havenorm^K.OThusacanbGewrittenasaproductofprimesofnorm6य~,K;this.showsthattheidealclassAisgeneratedbyidealclassesofprimesin6मP0|s,UUandthusthatP0ȫgeneratesCK.HThe}nextstepistodeterminewhichofthesegeneratorsareequalintheideal6class)group.CFirstonecomputesj(OK)y2Y and)j(p)2Yfor)eachpy2P0|s.CIf)for6anyp;q2P0 onehasj(p)=j(q),2thenweknowthatpandqarehomotheticas6complexlattices.6Thatis,thereisan =24C^/suchthatp= zq.6Oneshowseasily6thatY bmustactuallylieinK^(seeExercise4.4)sopeq.}ThusYpandqareequal6in0CK,7andonemustonlyincludeoneofpandqasageneratorofCK.eSimilarly*,7if6ऱj(p)4=j(OK),thenpistrivialinCK,andthusirrelevqanttothecomputation.7Let6मP1.abGeasetcontainingoneelementofP0.aforeachj-vqalueobtained;fP1.astillgenerates6यCK andUUitselementsaredistinctinCK.HF*romLhereoneneedstocomputethefullgroupCK Vsimultaneouslywithamulti-6plicationtable.ZNote rstofallthatwealreadyknowtheinversesofeveryelement6ofkFP1|s,psinceforeachp몯2P1繫therekFisap^02몮P0suchthatpp^0=(p)isprincipal.Ifp6फandԱqaretwoprimesofP15Gwhicharenotinverses,!wecompute rstidealgenerators6ofTpq,U andfromthesewecomputealatticebasis.qW*ethendeterminej(pq)2Y8.If6thisequalsj(a)forsomeidealwehavealreadycomputed,cthenwehavepq5կaǫin6यCK.u{OtherwiseVweobtainanewelementofCK Dwhichweaddtothemultiplication6table.F*romhereonecontinuesuntileverypGossibleproducthasbeendetermined;6oftenonecanusepreviouslydeterminedrelationstodetermineothersandthus6simplifyjthecomputations.Theendresultisamultiplicationtablefortheideal6classUUgroupCK,togetherwiththej-invqariantsUUofeachidealclass.HNoteZthatasaspGecialcaseofthisalgorithmwegetasimplemethodtodeter-6mine)zifanidealisprincipal:[simplycomputealatticebasis,2?fromthatcomputeits6ऱj-invqariant,andcompareittoj(OK);+theywillbGeequalifandonlyiftheidealis6principal.TMoregenerallyonecandeterminewhichelementoftheidealclassgroup6aUUgivenidealisequivqalenttointhesamemanner.Hभ3.4.Examplec:rQ($pUW$fe 14w).T*akeȰK=2Q($pUW$fe 14).Inthissectionwewill6compute$CK.aW*ecomputeK |˯4:764026148,.Dsotheonlyprimesweneedconsider6areUU2and3.q2OK factorsasGEn2OK |˫=b\o2;p fe X14b \v߳2Xe7 6ळ884.:THEIDEALCLASSGROUPY6फandUU3OK factorsasЍ3OK |˫=b\o3;p fe X14+81bWb *3;p fe X14+2bW:6फSetUUa1C=OK,a2=b\o2;$p $fe 14b \v,a3=b\o3;$p $fe 14+81bW,a^0l3=b\o3;$p $fe 14+82bW. HW*eUUnowcomputejofeachoftheseideals.qW*ehave j(a1|s)==Vp o=Vfe ª14qi:6फBycExercise4.3weknowthat2and$p A$fe 14=arealatticebasisfora2|s,asowe ndthat[퍒̱j(a2|s)=<$KPp OPfe E14KwfeUY (֍2׮i:i6फSimilarUUcomputationsfora3ȫanda^0l3yield呍 j(a3|s)=<$K1Kwfe (֍3 f_+<$lPp jPfe E14lwfeUY (֍3i;͍}j(a0፳3|s)=<$33133wfe (֍3 G+<$lPp jPfe E14lwfeUY (֍3i:H6फThusUUallthreegeneratorsaredistinctinCK.HW*eUUnowcomputeproGducts.qWealreadyhaveUUthemultiplicationtable(8ʍ1q ffҤ4a1!Աa2ta3 a^0l3&ffat fda1 I@ ff}a1+@a2?xa3Ra^0l3a2 I@ ff}a2+@a1a3 I@ ff}a3Ra1a^0l3 I@ ff}a^0l3?xa1(W6फW*eUUcompute# a2|sa3c=b\o2; zbџb 4(3; BZ+81buc=b\o6;2 BZ+82;3 z; 2ͫ+8 bc=b\o6;2 BZ+82;3 z; 814bc=b\o6; BZ+84bW:36फCall`thisideala.gOneeasilychecks`that6and J +@4arealatticebasisofa,csowe 6compute(a^6l3/la^3l4a2|s,>a^8l3a^4l4a8|s,6ऱa^10l3 sݯa^5l4wja1|s.%T*ottcomputetheoGddpowersttofa3wesimplyneedtomultiplyeach6ofUUthesebya3|s.HW*eUU ndthata^qfa33}a3|sa4C=b\o3; zbџb 4(4; BZ+82b=b12;3 BZ+86;4 z; 2ͫ+82 b}=b\o12;3 BZ+86;4 z;3 830b\o=b12; +86bW:l6फThisUUhaslatticebasis12; BZ+86,andj-invqariant<$I1̟wfe  (֍105+<$lPp jPfeE119lwfeUZ (֍10i;[6फsoұa^3l3_Sincea3|Ehasorder10anda䍴15=a^0l5|s,*qthisalsotellsusthata^7l3_a^0l5|s.6Sincewealsohavea^9l3(=ua^0l3|s,KeverygeneratorisapGowerofa3|s;|thusCK sͫiscyclicof6orderUU10withgeneratora3|s.HTheUUonlyremainingpGowerUUtoexplicitlycomputeisa^5l3|s.qW*e ndthatvfa53a3|sa0፳2C=b\o3; zbџb 4(2; BZ+81b=b6;2 z;3 BZ+83; 2ͫ+8 b=b\o6;2 z;3 BZ+83;2 830b\o=b6; +83bW:l6फCallUUthisideala6|s.qIthaslatticebasis6; BZ+83,and j(a6|s)=<$zL5Kwfe  (֍12f`+<$lPp jPfeE119lwfeUZ (֍12i:/.6फThisUUcompletesthecalculationofCK.\XHभ3.6.Imaginarynquadratic eldsofclassn9umbQern1.W*ehavealreadyseen6a٨fewimaginaryquadratic eldswithclassnumbGer٨1:3Q(i),dQ($pUW$fe 2v),andQ($pUW$fe 3v).6It}rturnsoutthatthereareexactly6moreimaginaryquadratic eldsofclassnumbGer61.TheyPareQ($pUW$fe 7v),OQ($pUW$fe 11w),Q($pUW$fe 19),Q($pUW$fe 43),Q($pUW$fe 67)PandQ($pUW$fe!163x).6ItisquiteeasyusingourtechniquestoshowthattheseallhaveclassnumbGer1.6W*eUUwilldothecaseofK~4=Q($pUW$fe!163x),whichisthemostinteresting.HIn\jthiscasewe ndthatK 8:12781715683,^0sowemustchecktheprimes2,63,5طand7.H=RecallthatanoGddrationalprimepisinertinOK jifandonlyifwehave,Bȟ^<$ O8163owfe8 (֍ qp>^b=1:[A7 @4.:APPLICAZTIONSTOQUADRATICFORMSUm91Y6फW*eUUcomputeS%^<${8163B̟wfe8 (֍ p3خ^l=^<$ V2 Vwfe (֍3^=1;%^<${8163B̟wfe8 (֍ p5خ^l=^<$ V2 Vwfe (֍5^=1;%^<${8163B̟wfe8 (֍ p7خ^l=^<$ V5 Vwfe (֍7^=1:T6फThusnoneoftheseprimessplitinOK.{EF*orp?=2,weneedtodeterminethe 6factorizationofx^2>x+41inF2|s[x];titisirreducible, Gso2doGesn'tspliteither.SWThus6ourUUsetofgeneratorsofCK istrivial,soCKitselfmustbGetrivial.q̍HContinuing@theLegendresymbGolcalculationsabove,zone@ ndsthat` W163 W);fe5Z p6`'w=q͍6य1L[forallpb˯37.VThisL[hasanamusingconsequence.ConsiderthepGolynomial6मf(x)=x^2 Yx+41.ItjhasbGeenobservedthatthispolynomialyieldsprimeswith6remarkqablefrequency;infact,ityieldsaprimeforeachofx=1;:::;40.NUsingthe6LegendreUUsymbGolcalculationswecangiveaquickproGofofthis.HLetUUx0ȫbGeanintegerandsupposethatsomeprimepdividesf(x0|s).qThen^>zx20S8x0+410 (moGdp):6फThus YSoX(2x0S81)2C163 (moGdp);r6फsoBv` n163 n);fe5Z p`&=0Bvor1.k}ButwehaveshownthatthisdoGesnothappenforanyp37.6ThusUUnop37UUdividesf(x0|s)foranyx0|s.HNext,notesthatf(x)ispGositiveandincreasingforx>1=2sandf(40)=1601<6फ41^2|s;thusjf(x)j<41^2E^forall1x40.̊Itfollowsthatiff(x)isnotprimefor6suchx,thenf(x)isdivisiblebysomeprime37.NSinceweshowedabGovethatthis6doGesAnothappen,Eeveryvqaluef(x)with1x40AmustbGeprime.NMoregenerally*,6thefactthatvqaluesf(x)arenotdivisiblebyanysmallprimessuggeststhatthey6shouldUUbGeprimeunusuallyoften.HItmismuchmhardertoshowthattheabGovearetheonlyimaginaryquadratic6 eldsUUwithclassnumbGerUU1;thiswasprovedonlyin1967byStark.HThecaseofrealquadratic eldsisquitedi erent;{infact, itisconjecturedthat6mostUUrealquadratic eldshaveUUclassnumbGerUU1.A c4.ApplicationsTtoquadraticformsH4.1.Example$:Q($pUW$fe 5v).Ourexplicitcalculationsofidealclassgroupsof6imaginary.quadratic eldscanbGeusedtoyieldsomeinterestingre nementsofour6earlierO\resultsonquadraticforms.uW*ebGeginwiththecaseK~4=Q($pUW$fe 5v)toillustrate6theHbasicidea.mRecallthatwerelatedthis eldtothequadraticformx^2+5y[ٟ^2L;Lwe6showed#wthatan(unrami ed)pGositiverationalprimepcouldberepresentedbythis6quadraticformifandonlyifitsplitintoprincipalprimesinOK.BUnfortunately*,6weɊhadnogoGodɊcharacterizationofwhichprimesthesewere;that` 5 );fe ^<$s1swfe (֍4y4y[ٟ2,+8xy+x2S+<$l5lwfe (֍4 Gy[ٟ2Tƫ=p^<$ :Oay :OwfeSF (֍4+<$lbylwfeT (֍*`2,+<$laylwfe ( (֍4dN+<$laxlwfe L (֍&2^}x2S+8xy+<$l3lwfe (֍2 Gy[ٟ2Tƫ=p^<$ :Oax8by :Owfe ̟ (֍ f2,4N^}x2S+8xy+<$l3lwfe (֍2 Gy[ٟ2Tƫ=<$KpKwfe (֍2 5~:si6फThusrg2x2S+82xy+3y[ٟ2d=p:6फInUUparticular,pcanbGerepresentedbythequadraticform2x^2S+82xy+3y[ٟ^2L.HLetussummarizeourresultstothispGoint.AW*ebeginwithanypositiverationalq̍6primepsuchthat` 45 4);fe 7 6ळ944.:THEIDEALCLASSGROUPY6फThus,GifDzallweknowisthat` p5 p);fe 5> );fe :A6फSinceUU51q(moGd4),quadraticreciproGcitytellsusthat` ܳ5 l);feRp`=` /p /);feRp5E`>;thusڍ^<$z85Kwfe8ߟ (֍ppη^8=^<$ 81 Vwfe8ߟ (֍ppџ^&^<$/Yp/Ywfe (֍55ǟ^>:h6फTheseUULegendresymbGolsevqaluateashO^<$j81ߊwfe8ߟ (֍ppK^o(=\(. S1!sp1 (moGd4);fc S1!sp3 (moGd4);h6andCo^<$?p?wfe (֍5zI^ի=\(. S1!sp1;4 (moGd5);fc S1!sp2;3 (moGd5):O6फCombiningUUthesetwocomputationswe ndthat7^<$m854ޟwfe8ߟ (֍pp^|=\(. S1!sp1;3;7;9 (moGd20);fc S1!sp11;13;17;19 (moGd20):HफPutUUtogether,ourabGoveUUcomputationsyieldthefollowingtheorem._X7 @4.:APPLICAZTIONSTOQUADRATICFORMSUm95YHवTheorem4.1.%L}'etpǯ6=2;5b}'eapositiverationalprime.Thenpcanberepre- 6sente}'dbyatleastoneofthequadraticformsox2S+85y[ٟ2L;2x2+2xy+3y[ٟ26लifandonlyif Ӎ>p1;3;7;9 (moGd20):HफIn6fact, itturnsoutthatthe rstformrepresentsthosepsuchthatp֍1;96(moGd20)yandthesecondthosesuchthatp֯3;7q(moGd20),butythebGestproofof6thisUUrequiresclass eldtheory*.'Hभ4.2.Thegeneralcase.Theargumentsoftheprevioussectiongeneralize DA6easily*.LetǮKF=*Q(zPpUWzPfe4rd ɫ)bGeanimaginaryquadratic eld;twebeginwiththecase6मd^2;3q(moGd4).NoSuppGoseIthata1|s;:::;ah xareidealrepresentativesIforitsideal6classgroup. LetpbGeanypositiverationalprimesuchthat` ܰd ܟ);fe)0pd`u=Q1andletҀ6ऱp=b\op;zPp zPfe4rdmQ+8mbꬫbGeUUoneoftheprimesofOK lyingoverUUp.HBy(thede nitionoftheidealclassgroupwehavep."aitfor(auniquei.+?Note6thatUUitisclearfromourde nitionofjthatj(aiTL)2K;UUthuswecanwritey!̱j(aiTL)=r+8sVpUWVfe4r d6फforUUsomer;s2Q.qSinceUUpaiTL,thede nitionofjtellsusthatthereissome8Ӎʵ^d')a=bˡcpYd˟^W2SLS2ƫ(Z)8ԍ6suchUUthatڍY^dab:yNxöj^̽`Ҷjr+8sVpUWVfe4r d ɟ`!=<$KmKwfeǷ (֍p.+<$p1lwfe (֍p FVpVfe4r d1:tHफExpandingUUouttheSL#2(Z)actionyields][<$^Wm^WwfeǷ (֍pj!+<$p1lwfe (֍p FVpVfe4r df=<$ha(r+8szPpUWzPfe4rd ɫ)+bKwfeB6˟ y[٫(r+8szPpUWzPfe4rd ɫ)+x2f=<$L(ar+8b)+aszPpUWzPfe4rdKwfeGyß (y[r+8x)+yszPpUWzPfe4rdqf=Lb ?(ar+8b)+aszPpUWzPfe4rd ɟb bw(y[r+x)y[szPpUWzPfe4rd ɟbKfeH bW(y[r+8x)+yszPpUWzPfe4rd ɟb bw(yr+x)yszPpUWzPfe4rd ɟbf=<$(㟯KwfeLş (֍(y[r+8x)r2Sdyr2Lsr2T#+<$l(ar+8b)(y[s)+as(yr+x)lwfeuP (֍(y[r+8x)r2Sdyr2Lsr2ztVpVfe4r dՍ6फwhereUUissomerealnumbGer.qEquatingUUimaginarypartsyields7pū=<$(y[r+8x)^2Sdy^2Ls^2KwfeuP (֍(ar+8b)(y[s)+as(yr+x)⍍ep(7(ar+8b)(y[s)+as(yr+x))=(y[r+8x)2Sdy2Ls2ukqp(arGsy8bsy+arGsy+asx)=rG2Ðy[ٟ2,+82rGxy+x2Sds2|sy[ٟ2îps(ax8by[٫)=rG2Ðy[ٟ2,+82rGxy+x2Sds2|sy[ٟ2Q7pū=<$K1Kwfe (֍(s -x2S+<$l2rlwfe  (֍ si[xy+<$lrG^2p8ds^2lwfe#J (֍%s(y[ٟ2L;`i7 6ळ964.:THEIDEALCLASSGROUPY6फusingvthefactthataxnKby}=L1.b+NotevthatthequadraticformdepGendsonlyonr 6फand)s;7thatis,1onlyonj(aiTL).cW*ehave)thereforeshownthatifpai,1then)pcanbGe6representedUUbythequadraticformu<$1wfe (֍(sEOx2S+<$l2rlwfe  (֍ si[xy+<$lrG^2p8ds^2lwfe#J (֍%s(y[ٟ2L: 6फSinceGeveryprimeplyingoverarationalprimepwith` @^d @^);fe)0pC`|=U1isequivqalenttoq͍6somex˱aj6,weobtainthefollowingtheorem.(W*ewillsaythataprimepisr}'elatively6prime!toarationalnumbGer!q}ifpdoesnotdividethenumeratorordenominatorof6मq.(inUUlowestterms).;HवTheorem4.2.%L}'et7߮d2;3q(moGd4)b}'eanegativeintegerandleta1|s;:::;ahffbe z6r}'epresentativesfortheide}'alclassesinQ(zPpUWzPfe4rd ɫ).Write농čj(aiTL)=ri,+8siVp Vfe4r d:6लTheneveryp}'ositiverationalprimepsuchthat` K'd K');fe)0p `d =01canberepresentedbyat6le}'astoneofthehquadraticformsu<$1 ^wfeL (֍si@ݮx2S+<$l2rilwfe D (֍i|sivxy+<$lr^G2;Zip8ds^2;Zilwfe#J (֍ si(y[ٟ2L:ፑ6लF;urthermor}'e,letpbeaprimewhichisrelativelyprimetoallofthecoecientsof DA6allP&ofthesequadr}'aticformsandwhichisnotrami edinQ(zPpUWzPfe4rd ɫ).Ifforsuchapwe6have` d );fe)0p`߫=1,thenpc}'annotberepresentedbyanyofthesequadraticforms.HवProof.pZThe onlynewinformationisthelaststatement.XSoletpbGeapositive6rationalprimewhichisrelatielyprimetoallofthecoGecients.SZSupposethatpcan6bGeUUrepresentedasB\<$G1Gwfe (֍(sz6x2S+<$l2rlwfe  (֍ si[xy+<$lrG^2p8ds^2lwfe#J (֍%s(y[ٟ2d=pއ6फforUUsomex;y"2Z,with(r;s)=(riTL;si)UUforsomei.qW*emustshowthat` ld l);fe)0pQ`M=1.HNote_thatunderthehypGothesisthatpisrelativelyprimetothecoGecients6wemusthavebGothxandylrelativelyprimetop;2ifonewerenot,cthentheother6wouldgalsobGedivisiblebypandtheentireleft-handsideoftheexpressionwould6bGeywdivisiblebyp^2|s.-Inparticular,wemusthavethatyPisinvertiblemoGdulop.-The6representationUUabGoveyieldsasolutiontothecongruenceupH07a<$K1Kwfe (֍(s -x2S+<$l2rlwfe  (֍ si[xy+<$lrG^2p8ds^2lwfe#J (֍%s(y[ٟ2<(moGdp)fpH07a^<$ Vx Vwfe (֍:yA^۳2 Rի+82rş^<$ lx lwfe (֍:yk^+rG2pds2|s:捑6फ(W*eMcancancelthe 1&fess ]sincebyhypGothesispisrelativelyprimetoallofthecoGecients 嗍6ofLallofthequadraticformsandthecoGecientofx^2Zis 1&fess.)Bythequadraticformula,6theUUroGotsofthisare췍 ̍q2r8s0p 8s0feK&ϟЍ4rGr2p84(rGr2dsr2|s)qffer (֍6p2==r<$l@p j@fe`4dsr2lwfe= (֍ [2#=r8sVpUWVfe4r d :i⍑6फInUUparticular,ifpcanbGerepresentedbythequadraticform,then <$Ӟ1Ӟwfe (֍(s|~^<$ %x %wfe (֍:y/T+8rG^a}67 @4.:APPLICAZTIONSTOQUADRATICFORMSUm97Y6फwillUUbGeasquarerootofdmodulop.qThus,` ld l);fe)0pQ`M=1.{>ÄffdffYffffuKHTheTanalysisinthed1q(moGd4)Tcaseisentirelysimilar,exceptthatwebGegin 6withUUtheideal)p=^ #p;m8+<$l1lwfe (֍2 '+<$l1lwfe (֍2 GVpVfe4r d)^Z4w:~6फTheUUonlye ectthishasisremovinganadditionalfactorof2..荍HवTheorem4.3.%L}'etҮd41q(moGd4)b}'eanegativeintegerandleta1|s;:::;ah @Ybe z6r}'epresentativesfortheide}'alclassesinQ(zPpUWzPfe4rd ɫ).WriteԼčj(aiTL)=ri,+8siVp Vfe4r d:M6लTheneveryp}'ositiverationalprimepsuchthat` K'd K');fe)0p `d =01canberepresentedbyatq͍6le}'astoneofthehquadraticforms^č<$1ٟwfe M (֍2si*Yx2S+<$rilwfeL (֍si xy+<$lr^G2;Zip8ds^2;Zilwfe#J (֍ m2si(y[ٟ2L:6लF;urthermor}'e,letpbeaprimewhichisrelativelyprimetoallofthecoecientsof DA6allP&ofthesequadr}'aticformsandwhichisnotrami edinQ(zPpUWzPfe4rd ɫ).Ifforsuchapwe6have` d );fe)0p`߫=1,thenpc}'annotberepresentedbyanyofthesequadraticforms.HवExample4.4.T*akevd=14.3x2S+82xy+5y[ٟ26फand RM3x2S82xy+5y[ٟ2L;k6फso8weonlyneedoneofthosequadraticformstorepresentallsuchp. q(Notethat6thisUUisobviousonreplacingxbyx,aswell.)HOne+caneasilyusequadraticreciproGcitytocharacterizethosepsuchthatq̍6ट`> 14> );fe8 ˰pMx՟`V8ѫ=1;UUone ndsthatthisoGccursifandonlyifN|"p1;3;5;9;13;15;19;23;25;27;39;45 (moGd56):6फW*eUUconcludethatforp6=2;3;5;7,UUpcanbGerepresentedbyatleastoneofx2S+814y[ٟ2L;x2+7y[ٟ2L;3x2+2xy+5y[ٟ2b7 6ळ984.:THEIDEALCLASSGROUPY6फifUUandonlyif32|"p1;3;5;9;13;15;19;23;25;27;39;45 (moGd56):V;7 u cmex10 Zcmr5ٓRcmr7K`y cmr10 O \cmmi50ercmmi7 b> cmmi100ncmsy5 O!cmsy7!", cmsy10Nff cmbx12"V cmbx10': cmti10 msam10  msbm10qymsbm7 - cmcsc10%n eufm10X&eufm7T