Luận văn Cấu trúc các ideal trong vành đa thức trên miền nguyên Dedekind

CHVdNG 3 CA"UTRUC cAC IDEAL TRONG VANH fJA THUC TREN MIEN NGUYEN DEDEKIND Trangehu'dngnay,tasem6tadiu tfl.kcLlaidealba'tkytrangvanhc1athue trenmi~nnguyenDedekindD. Vi D[x]lavanhNoethernenmoiidealtrang D[x]a~ue6s\jphantfchnguyensdehu5nhoa,dod6d~m6taea'utrueet'1a idealba'tkytrangD[x]tan~6taea'utruecaeidealnguyensd.Khi a~e~pden P, I, J, <p(x),tahi~u,P la idealnguyentokhackh6ngtrangD va I, J lacae idealkh;k kh6ngeth D[x];<p(x)E D[x]la aathueddnkhCiithoa<p(x)+P[x] ha'tkhaquytrangD[.r]!P[x]. 3.1Call true cua caeidealnguyensdtrong D[x] Truoekhi1116taea'utnk euacaeidealnguyenSd,nh~eb;ddlng,neuI la idealnguyensdelh D[x]thi theoMfnh di 1.4.5,Rad(I) nguyentotrangD[~r], vadoa6 theoehu'dng1,Rad(I)toia:;tiho~ekh6ngtoia;;li.Saudaytase11n IltQ1116tad:;tngeuaidealnguyensdI ungvoim6iRad(l)nhu'tren. 3.1.1Dinh Ij (M6tacaeidealnguyensdvoicankh6ngtoia:;ti).IdealI Clla D[x]fanguyensO'v6idin khongtaldflineuvachinlu I = In v6in nguyen du:O'ngva J faidealnguyento/khongtaldfliala D[T]. Chllngmink. TheoDinh Ij 1.6.10,D[x]la Q-vanh,nentheoDinh /j 1.6.6 111Qiideal nguyen tokh6ngtoi a:;tieuaD[x]la idealnhan. Hdn nlfa mQiideal ngllyentokh6ngtoia:;tikhaekh6ngeuaD[x]d~lle6soehi~uIanhdnkh6ng, tLta6,dungBo?di 1.1trang[1]tae6ai~llphaieh(tngminh.\? 3.1.2Hf qua. TrangD[I])m9ilugthuaala idealnguyento'd~unguyensO', 18 De m6 t~td~lngcLb ide:llnguyenso \'6'jcan t6i c.1~titadn Ixi c.12sau 3.1.3Blf dl. IdealI ala D[x]tanguyensdValdin tal'd9ineuvachineuton t9iidealngllyento'P -#0alaD,dathucdonkhdiip(x)E D[:c]thodip(x)+P[:r] be/tkhdquy!rongD[xJ/P[x]vak, l nguyenduongsaochok(x)]1.:E I, pi C I. Hdnmia,khzd6Rad(I)=. Chungminh.(==?)GiasuI nguyensovoicantoid~tihlRad(I)=, trangd6 ip(x).P nhudaquyuoc.Do ip(x) E Rad(I), P c Rad(I) Denc6 k, l nguyendltongsaocho [;:(x)]1.:E I, pi C I. (~) Gia Slt c6 k, I nguyendltong sao cho [<p(.r]1.:E I, pi C I, thl ip(x)E Rad(I). P c Rad(I). Do d6, c RadJ). Theo Bo?di 2.4, tojd?i trangD[x]DenRad(I)=ho?cRad(I)=D[x].VI I -:JD[x]DenrheaM~nhdi 1.4.1ii), Rad(I) -:JD[x],do d6Rad(I) = toi d?i trangD[.r].TheoAl~nhdi 1.4.6,Q nguyenso rrangD[x]. \? 3.1.4Binh If (M6rac:k idealnguyensovoi cantoid~li).IdealI ala D[x]ta nguyensdv6ldin tald9ineuvachineuI c6dflng , !rongd6m,n nguyenduong,t kh6ngam,hi(x) ("1::;i ::;t) thwjcD[x],Pi(x) (1::; i ::;t) thu(JcP[x].Hon mJa,khzd6 Rad(I)=z:dtac6thi ch9nhi(x), Pi(x) thoadeg[hi(x)]::;(Tn- l)deg[ip(x)],deg[Pi(x)]::;Tndeg~;(x)]. Chungminh.(==?)GiasuI nguyensovoicantoid?ithiRad(I)=, trangd6ip(x),P nhuda quyuoc.Khi d6t<snt?i Tn,n ngllyenduongsaocho [ip(x)]mE I, pn c I. Do D[x]lavanhNoethernen I hUllh?nsinh 1=, trangd6 ii(X) (1 ::;i ::;t) thuQcD[x]. 19 Do I chClatrongRud(I) = nen Ill') E, Suyfa Ji(x) =:p(x)hi(:r)+p;(l'), trangdo h;(x)E D[x],Pi(X)E P[x], V~y 1= , ({=)TruoctientachungminhIi- D[x].Giasl'iI =D[x]thl1E I vadodo 1 = [,,(:r)!mu(x)+llV(X)+t[\O(X)hi(X)+P,(X)]g:.r) (pE Po) t t = [cp(x)]rnu(x)+;;(x)L h;(.r)gi(X)+pu(:r)+L Pi(:r)g;(x) k=l i=1 t t = 'P(x)(['P(x)]m-Iu(x)+{; hi(x)9i(X)) + (PU(X)+~/i(X )9i(X)). D~t t r(x) = [:p(x)]m-1u(x)+L h;(X)g,(l'), k=l t s(x) = pv(x)+LPi(X)gi(X) E Pix:, ;=1 thl 1=cp(x)r(x)+s(x), Suyfa 1+P[x]= :p(x)r(x)+P[x], hay 1+P[:r]= (y{r)+P[x])(r(x)+P[.r} Tli ding thuc\'1:iathu dliQ'cta suyfa :p(x)+ P[x] kh?lnghichtrangD[x]jP[x], mall thu~n.V~yI i- D[x]. Do [:p(x)]mE I, pn c 1 nen theoBo"dl 3.1.3,taco I nguyensavoi cantoi cl~i va Rad(1)=. 20 Hqnlilia,do;(.r)Gonkhdilien['P(x)]m-I,[<p(£)]'"Gonkhdi.Bangeachchia hi(x) cho [:;(£)]"'-1vaPi(X) cho[ip(x)]m,ta nh~ndliQ'Ccaephandu tliong.ling Ia1li(X) Vat';(x). V~y 1 =, trang do u;(x), L'i(x)(l ::; i ::; t) thuQcD[x] thaa degui(x)::; deg[<p(x)]m-l, degv;(x) ::; deg[.;(x)]m.D(it hi(x) +- Ui(X) va Pi(X) +- Ci(X), ta codi~uphai changminh.\? 3.1.5Bif di. ChoD ldmiennguyenDedekind,P ldidealnguyent6'khdckh6ng cuaD. Khi do, 1)pn[x]= (P[x]t,Valm9in nguyendu:ang. 2) C6 PoE P saDchop[x]+pn[x]= . Chlcngminh. Kha don gi~lnlien dliQ'Cba qua. Tv fJtnh Ii 3.1.4,tac6 hai m~nhd~sau 3.1.6M~nhdl. ChoI ldidealcuaD[x]chua;p(x)vagidszlRad(I)=. Khid6I nguyensava1=Valn nguyendl1ang. Chungminh. Do Rad(I)=la idealtoid~i(fJtnh/f 2.6)lienI nguyenso. Do P C Rad(I) lien co n nguyenduongthoapn C I va gia Slr n la s6 nhanh?lthoadi~uki~nnay. D~th?lyc I. NgvQ'cl;,li, tase changminhI C, di~unay tliong QUangvoi chang minh J c trangD[x]=D[x]jpn[x]. Ta co Rad(J)=. VI D la mi~nnguyenDedekindliencoPoE p thoaP[x]=. Voi f(x) E J, tasechangminhbangqui n;,lptheoi (1::;i ::;n), f(x) bi~udi~n duQ'cdvoi d;,lng f(x) =<P(X)gi(X)+pbhi(x), trangdogi(:r).hi(x)E D[x]. 21 Do f(x) E I nen Tr0E RllIl(I) =, suy fa c6 91(X),III(x) E D[x] sao cho f(x) = ';(1')gl(:r)+pl,hI(J;), V~yphatbieudungvaii =1.Giasuphatbieu aCingvai i =k (1::;k ::;11- 1),nghialacog.,(x),hd:r)E D[x]saGcho M =f(X)gk(X) +p~hdx), Suyfap~hk(x)= f(x) - f(X)gk(X)E7.Neup~E 7thlpk[:r]C 7,d{lndenpkc I, mall thu{'in\'6i k < n. \)y p~~7. Do I nguyenso nen hk(x)E Rad(l) =< .;(x),Po >, suy fa c6 u(x). dx) E D[x]saGcho hdx) =.;{x)u(x)+pov(x).V~y f(x) = rp(x)gdx)+p~hk(:r) = <p(X)gk(X)+p~(cp(X)u(X)+podX)) = f(X)(~ +p~U[;))+p~+IV(X) V~yf(x) =';(X)gk+l(x) +p~+lhk+l(x) vai gk+l(x) =gdx) +p~u(x)va hk+l(x) = v(x). V~yphatbieu dungvaii =k +1. Theo nguyenly quy nq.p,vai mQi i (1::;i ::;n), Hi) =CP(~)gi(X)+pbhi(x),trongd6gi(X),hi(x)E D[x].Vai i =n, taco f(x) = .;(x)gn(x)+Pohn(x)=cp(x)gn(x)E. Do d6 7chuatrong hay I C. V~yI =. c::; Hoanroantl1angtv m~nhd~tren,taco 3.1.7M~nhdi. ChoI zaidealcuaD[x]chuap vagiGszlRad(I)=. Khi doI nguyens6va1=v{jimnguyenduong. Nhl1v~ytadam6taxongCalltruccuaidealnguyensotrongD[x].Tuy nhiendq.ngcuaidealnguyensotrongBinh ii 3.1.4chl1agQn,hannuarhea Binh ii 1.5.6mQiidealnguyensotrongD[x]la giaoclh hUllhq.ncacideal kh6ngthe rutgQnc6 Clingcan,nentier rheata se016radq.ngclla ideal kh6ngtherutgQn. 22 3.2Cliu trueeuacaeidealkhongthi rzltgQIltrollg D[l:] Trangnwc nay,tase m6tac1cidealkh6ngth~rutgQncua D[:!;].Do trang r , vanhNoetherI11Qiidealkh6ngth~rutgQndellnguyensaHenl11Qiidealkh6ng nguyensodellcoth~rutgQn.VI the,trangnwcnaytachIc:lnxettrangt?P dc idealnguyenso thayVIt~ptatcacacidealcuaD[x]. Calltruccuacacidealkh6ngth~rutgQnvaicantoi ti~uduQ'cm6taqua dinhly sau 3.2.1DinhIi. ChoI taidealczlaD[x].Khi do,neuI nguyensavaidin kh6ng t6ldC;ZithiI kh6ngthe'nitg9n. Chungmink.Gia Sll I nguyensavai cankh6ngtoi d?i co Rad(I)=J vaI coth~rutgQn.Do I nguyensonS'icankh6ngtoid?i HenrheaDinhIi 3.1.1 1= Y. Do I coth~rutgQnHenrheaM4nhdi 1.4.8,taco Ii, 12nguyensa, I] =II =I12thoa I = I] n h vaRad(I1)=Rad(I2)=J. TheoDinhIi 3.1.1,co 1.:],k2nguyendliangsaocho Ii =Jkl va 12=Jk2.Kh6ngmattinht6ngquat, taco th~giasuk1~1.:2,khi do I] ~ 12,Sur ra I = h, mallthuan.Do do I kh6ngth~rutgQn.Q Nhuv~y,rheaDinh Ii 3.2.1,mQiidealnguyensavai cankh6ngtoi d?i d~ukh6ngth~flItgQn.Tuynhien,mQtidealnguyensovo1cantoid?ikh6ng nhatthietla idealkh6ngth~flit gQn;ch~ngh?n,xetD =Z va trangZ[x], ideal=n nguyensovai cantoi d?i va co th~ rutgQn.f)~m6taCalltrucdb cacidealkh6ngth~rutgQnvai cantoi d?i, ta c:1nmQtso be)d~sau 3.2.2Bfl d€. Cho I ta idealczlaD[x], I = vai T,s nguyen dl1o'ng.Khi doT,s taeaes6~nguyendl1angnhonh{[{thoG[<p(xWE I, ps c I. 23 Chung minh. Giii Sl(T kh6nglaso nguyendl(cfngnhonhatthoa['P(:r)j"E I, nghla la co k nguyendl(ang,k < T sao cho ['P(xW E J. Do dot6n t~t'if(x), g(:r) E D[:r:],Ps E ps saocho [<p(X)]k= [<p(x)tf(x) +Psg(x), sur fa [<p(X)]k+P[x]= ['P(x)tf(x) +P[x]. t((cla ([<p(x)]+ p[X])k=«p(x)+P[x]Y(J(x) +P[x]), nen 1+P[x]= «p(x)+p[x])r-k(J(x)+P[.rJ, hay 1+P[x]= «p(x)+P[x])[«p(x)+p[X]y-k-l(J(l') +P[x])]. Sur fa <p(x)+P[x]khii nghichtrongD[x]/P[x],mauthu~n.V?y T la so nguyen duangnho nhatsaocho [<P(xWE I. Gia su s kh6ngla so nguyenduangnhonhatthO3ps C I, t((cla co I nguyenduong,I < s saocho pi C I. Xet I trangD[x]= D[xJlPS[x].Ta co I =. TheoEo?dl 3.1.5,coPoE P thoa pi[x]=. Vi pi C I nen Pb=[<p(x)]rh(x).M~tkh,k do pI[x]nguyensava [<p(x)]r"tf-PrxJ= Rad(pI[x]))nenh(x)E pI[x],sur fah(x)=Pbk(x).DodoPb=P&[<p(x)]r"k(x)hay Pb(1- [<p(X)]rk(X))= O. Ma I < s nen Pbi=0, sur fa I - [<p(x)]rk(x)E P (do 0 nguyensa,oaRad(O)=F). Do do 1+P[x]= (['P(xW+P[xJ) (k(X)+P[x]) hay[<p(xW+P[:r]khanghichtrongD[xJlP[x],mallthu~n.V?ys la songuyen dt(angnhonhatsaochops c I. Q 3.2.3Eo?dl. ChoI ld idealnguyenSrJc6din tOldr;1icuaD[x]vaiRad(I)=< . Gr;;im,'n ld ale 56nguyendZ1rJngnhdnh6.tsaGcho[<p(x)]mE I, 24 P" c I. Khz'd6c6 k, I nguyenduong,k ::;II vaI ::;rn,(hoa 1: = . I:pn-l[x] = . Chung minh. Ta chungminhd~ngthCrcthu nhat. Neu Tn= 1thl 1: = I: R = I. Mi;itkhacdo [y(xWE I nenrheaM~nhdi 3.1.6,taco I =v6'ir ngllyendLio'ng.Ho'nmia,rheaBo?di 3.2.2,T 1a 56 nguyendlro'ngnho nhat thoap,.c I nenr =n. \)y b6dedlfQ'chungminhv6'iTn=1. Neu Tn> 1, ta co [ip(x)]rn= [. Do [ nguyenSo'co Rad(I :. TheoH~ qua3.1.6,taco 1:=, trangdo k nguyenduo'ng. TheoBo?di 3.2.2,k 1asonguyendLio'ngnho nhatthoapk c. Mi;it khac pn C I nen pn C I :=. V~yk ::;n va d~ngthuc thC£nhatdlfQ'Cchungminhxong. Hoan roantLio'ngt\j, taco d~ngthC£cthC£hai, nghia1aco I nguyenduo'ng, I <Tn saccho I: pn-l[x]=.v 3.2.4Btfdi. ChoI lil idealczlaD[x]{hoa[<p(x)YE I, ps c I, (rangd6T,s lilale 56/nguyenduang;hannaG,1:= ho?fcI: ps-l[X]=< [ip(x)]",P >. Khi do 1= . Chzlngminh. Do [ip(xWE I, ps c I nen ~ I. Ta se chC£ng minh I ~. 25 Xet tnianghQ'p[ :=. Trang D[x]= D[x]!PS[x],ta se chCtngminh [ =. Voi mQif(x) E [, ta c6 T(X) E [ : cp(X)"-1=< ~ >. Suy fa t6n t~i~ E ~ thoa f(x) =CP(X)Ul(X).Gia Slt f(x) = cpk(X)Uk(X),k < f, khi d6 .pk(x)udx)E [, do d6 udx) E [ : cp(x)k.Do k <l' nen [: <.p(X)kC [: p(X)r-l. Suy ra c6 Uk+l(X)E D[x] thoa ~ =<.p(X)Uk+1(X), V?y f(x) =<.pk+l(X)Uk+l(X),Theo nguyenIy qui n<;lp,f(x) = <.pi(X)Ui(X),voi mQi - - 1 ::;i ::;T. Voi i =T tac6 f(x) =cpr(x)ur(x)E. V?y tada chungminh du'Q'cI chay[ c. Xet tru'ang hQ'p [ : ps-l [x] =. Trang D[x] = D[x]1PS[x],ta sechungminh[ =. Voi mQif(x) E I, ta c6 f(x) E [ : PS-l[X]=< - - ~ , [:p(xW,P[x]>=,trongd6 P[x]= "oi PoE P (doBo de 3.1.5).Suyra f(x) =[cp(xWhdx)+pok1(x) Bang qui n<;lptheo i(l ::; i ::;B), ta chung minh du'Q'c f(x) =[;;(xWhi(x)+pbki(x) Voi i =s ta du'Q'c f(x) = [<p(xWhs(x)+Poks(x) = [<p(xWhs(x) V?yI chayI c. T6mIvab6d~du'Q'Cch{jngminhxong.\7 3.2.5Bo?dl. V6'im9in nguyenduCJng,ideal[=kh6ngthl rutg9n trangD[x]. Chungminh.GiaSlt[ =coth~rutgQntrangD[x],nghialac6 Jr, [2Ia c:k idealcuaD[x]saocho [ = Jrnh (1) 26 v6'i It 1= I 1= 12. Ta c6 ';7(.£)E I, pn c I nen f(x) Ell, 12,pn ell, h. TheaBtf di 3.1.3,tac6 11.12ngllyenSdva Rad(Id==Rad(I2)'TheaM4nhdi 3.1.5,t6n t~linl,n2ngllyendltdngsaacha II = , 12 = . (2) Ttt(1)va(2)tac6 = n . Da vai fro cua nl va n2nhv nhallnen ta xem nl 2:n2.Khi d6 chuatrang nen = n . V~y =, hay II =I, mall thu~nv6'idi~ugia SUoT6m l la idealkh6ngth~rut gQnclla D[x].Q 3.2.6B6?di. Valm91m,n nguyendl1(jng,idealI =kh6ngthl rUtg9ntrangD[x]. Chlingminh.Ne'um=1,rheaB6di 3.2.5,tac6di~llphaichungminh. XettrLtanghQ'pm > 1. TheaB6?di 3.2.2,m,n la cacso ngllyendvdng nho nhatthoa[.p(x)]mE I, pn C I. TheaB6?di 3.2.3,tac6 1:= (3) 27 VCJih::::;n. Tier theo,U se cht'tngminh h:= n, nghia I~chang rninh7 :< [-;(X)]IIL-I>=trangD[x]= D[x]/P"[x]. Dt?thayc 7 :< I [~(x)]m-l>. :'\gu'Q'cl~ti,v6'imQif(x) E 7 :, tac6 f(x)[<p(x)]m-lE 7 =. Suy fa c6 g(x) E D[x] sao cho M[c;(x)]m-l = [<p(x)]mg(x) hay ['P(.r)]m-l(f(X) - 'P(X19(X))= O. M;;U khac [<p(x)]m-lf/.~ = Rad(O)nen f(x) - . V~y 7 :=. V~yk =n vatu (3)tac6 1:= . (4) Bay gi6',ta cht'tngminh I kh6ngth~rutgQn. Gia SlTngLtQ'cI~ti,tltcla c6 cae ideal II. h db D[x]saocho 1=it n12 v6'i it =1= I =1= h. Suy fa I : = (II n12): = (it:< [cp(X)]m-l» n (I2 :< [;P(X)]m-l». (5) Tu (4)va (5)tac6 (II :. Theo Bli di 3.2.5, kh6ng th~rut gQnnen (it : ) = ho?c (h :. Hdn nua [<p(x)]mE it,I2, pn c it. h nenrheaB6di 3.2.4,tac6it =ho?c12=< [. tlTcla 11= I ho?c12= I, mau thu:1n.V~yI = kh6ngth~rutgQntrangD[x].Q Tu B6?di 3.2.6,li~uc6 phaimQiidealkh6ngth~rutgQnc6 canla < a~uc6dhaykh6ng? Cautral6'ila kh£ngainh vatacanmQtsob6a~sau 28 3.2.7Rti dl. Xel vanhdiGphllorzgRAIlvaiR =D[x]va/vl=.Khi d6 « [cp(X)Jk,pI »M =« [Ct/(x)]k',pt' >hI nlu vachinlu k =k' val =L'. Chung minh. (~) Hi~nnhien. (==::;,)Gia si't« [cp(x)]k.pl»M =« [cp(xW',p/>h/' Do \'ai tfOclb k va k' nhu nhaunen taco th~gia si'tk ~ k'. Ta chungminh, k = k'. Th?t \'?y,gia si'tk > k'. Ta co [cp(~)]k E « [cp(X)]k',pt' >hl = « [cp(x)]\pi >hI nen co f(x), g(x)E D[x],PI E pi va m(x),n(x) E D[x]\ M sao cho [;(r)]k' [;(X}]kf(x) PI g(x)- = +-- I 1 m(x) 1n(x)' Suyfa m(x)n(x)[;(x)t'= [f(x)]kf(x)n(x)+PIg(x)m(x), nen m(x)n(r):;(X)]k'+P[x]= [cp(X)]kf(x)n(x) +P[x]. Gian lu'Q'cn(x)+P[x].[;(x)]k'+P[x]( do n(x)+P[x],[;(x)]k'+P[x]=I-0+P[x]) haiv~clladangthuc\-uathudu'Q'c,taco , m(x)+P[x]= [cp(x)]k-kf(x) + P[x], suyfa m(x)E=AI, mallthu1n.V?y k = k', Do vai tfOclh l va [' nhltnhaunen taco th~gia si'tl ~ z'. Ta chungmini) , l = ['. Gia si'tl > ['. D~tI =, J' = [, Khi do, trong D[x]= D[x]/pl[rJ, ta co Y= l' =, trongdo =P[x]vaiPoE P (doRd dl 3.1.5), , - - -- VI I = I nenI =1'. Dod6 I M =l'M 7 7 [( )]k- Ta co Po E.\1nen Po = <px ~ voi f(x) E D[x]M'm(x) E 1 1 1 m(x) 29 D[x]\ A{.Dodo co s(x)E D[.r]\ 1\1saocho P~m(x)s(x) = [cp(X)]kf(x)s(x).( *) - - I - I SHYfa [cp(x)]kf(x)s(x)E. M?t khac [cp(X)]krt p = Rad(< P~ » nen , ,- - - f(x)s(x) E, SHY fa f(x)s(x) = p~h(x),h(x) E D[.r]. Ket hQ'pv&i (*) ta dLtQ'C , ,- p~m(x)s(x)=[cp(X)]kp~h(x). SHY fa ~(m(x)s(x) - [cp(x)]kh(X)) =O. M~itkhacP~i=0 nen m(:r)s(x)- [;(x)]kh(x)E Rad(O)=P. SHYfa m(x)s(x)E< [cp(X)]k.P >c -'I. Mauthuc1ndom(x)s(x)rt!vI.V~Yl =I'.r;) 3.2.8Blf die Cho1 la idealala D[x],1=v6it,snguyendurJng. Khi do 1: P[x] = . 1: = . HrJn mIa,voiAI la h? nhdnczlaD[x],fa co 1M : (P[xDA! = « [cp(x)t,ps-l >)M, hI: « cp(x) »M = « [cp(X)t-l,p$ »AI. Chung minh. Trangvanh D[x]=D[x]jPS[x],taco 1= . Bautien [a ch:"tngminh 1: P[x] = hay I :=, tfOng d6 P[x]= v6'iPoE P (doRlf di 3.1.5).Do PflJ50=Po =0nenp~-lE I :. Hdnmla [cp(xWE I c I : nen c I :. NguQ'cl~ineuf(x) E I :,b~ngquin~ptheoi (1:::;i :::;s- 1),tasechung minh f(x) = [<p(x)]rh(x)+phki(x),(*) 30 --- trangdoh(x), J.;i(:r)E D[x]. VI f(1:) E I : nenpof(x) E I, dodopof(:r)=[:p(x)]rg(x).M~Hkhk [cp(xWttnen~ E(donguyento), dodo g(x)= Puh(x). Suyfa pof(.c)=[.p(xWpoh(x)hay Po(7(X)- [~(:~Wh(X))=0(**). Ma Po=1= 0 nenf(x) - [-;(x)]rh(x)E Rad(O)=. Suyfa f(x) = [cp(x)]rh(x)+ P6g1(X),v~y (*) dung yO'ii =1. Gia su (*) dung v6'ii = k < s - 1, nghiala f(x) =[cp(x)]rh(x)+P§9k\X).Ta co f(.r) - [cp(x)]rh(x)=p~9dx),ket hQ'pv6'i (**) ta dltQ'cP~+19k(X)=O.\1 k+ 1, - - hay9dx) =PO9k+l(X),9k-l(X)E D[x].V~y f(.r) = [:p(x)]rh(x)+P~+19k+l(x) Theo giil thietqui n:;1p f(x) =[:p(x)]rh(x)+pb9i(X), v6'imQi 1 :::;i :::;s - 1. V6'i i = s - 1 ta duQ'c f(x) = [.;(x)]rh(x)+pg-19s-1(X)E V~ytada chungminhdltQ'cI: PIT] =. Tiep theota se changminh I :=, hay chungminh 1:=. D~tha'yc I :. NguQ'c l<,liv6'i mQi 7[X) E I :< cp(x)>, ta co f(x)y?(x)E I =, sur fa f(x)cp(x)= [y?(x)]rh(x), h(1') E D]Xj, hay :p(X) (1(1) -1<P(X)]'-lh(X)) =0. Ma ",(x)rtRad(O)= Plx[nen f(x) - [Y?(x)]r-lh(x)=0 vadodo f(x) E. Sur fa I :c< [Y?(x)]r-l>.V~ytaetachltngminhdltQ'c1:=. 31 TheoM~nhdi 1.4.9),(! : P[:r;])M = hf : (P[:r:]hf va (I :< cp(x)»M = hI : « y(:r) >Lu va do do ta thu dltQ'Chai d5ng tht'tccon l~li.Q Hai b6 d~sauduQ'ctrichtu[9](Dinh if 34vaDinh if 35 ctlachlldng 4). 3.2.9Bil di. ChoR ldl)(lnhdiaphuongNoethergiaohoancodonvivaf ld idealnguyensrJcuaR co Rad(1)t6ldc;zi.Khi d6neuf kh6ngthe'rUtg9nthi f: (f : J) =J vaim9idealJ chuaf. 3.2.10Bil di. ChoR ldvanhdiaphuongNoethergiaohoancodonvi va f ld idealkh6ngthe'rUtg9n CllaR, J ld idealcuaR chuaf. Khi d6 J kh6ngthe'rUt g9nneuvachineuf : J chinhmodulof. 136d~saul;)chi~ungltQ'ccuaBo?di 3.2.6 3.2.11Bo?di. Chof ld idealkh6ngthe'rUtg9ntrongD[x]vai Rad(I) =< -p(:r),P >=JI. Khi d6,fontflim,n nguyendurJngsaGcho f =. Chung minh. D?t S = { Q I Q kh6ngth~nit gQntrangD[x], Rad(Q) =, Q i-< [cp(x)]a,pb > voi mQia,bnguyen dltdng } Ta se chang minh S = 0 b~ng phan chung. Gia su S i- 0. VI. D[x] la v~mh Noether nen S c6 ph~n ttt toi d?i la fa. VI. Rad(Io) = nen c6 m,n nguyen dvdng saDcho [,p(x)]mE fa, pn C fa va ta c6 th~xem m,n la cae so nguyen dltO'ngnho nh{{thoa tinh chat nay. Ne'um = 1,rheaM~nhdi 3.1.6,fa=, mall thuKn VI fa thuQc S. 32 l\i2\1f/, = 1,theoM41llzdf 3.1.7,fo= [;(J,,)]TII,P >, mf1LlthLl~n. Nell Tn> 1 \':1n > 1, rhea Blf df 3.2.3, ta co 10: = , fo : = , (6) (7) trongc161::;k ::;n va 1::;l ::;117. Neu k = n ha?c l =m, rheaBo?df 3.2.4,10=, mall thuan. Giastfk <n va l <m. D?t S' = { hI 1 hI kh6ng th~n.'1tgQntrang (D[x])Ju,Rad(I1\1)= « ip(:r),P > );\1./." =1=« [ipCrW. pb >~,'J voi mQia.b nguyen cllfo'ng}. Ta co (Io)Julaph:lntv toid~ici.'taS'. Th~tv~y,ghiSlfco (It)M thuQcS' saGcha (Iohl chuatrang(IdA/. TnJoc lien, tachUngminh II thuQCS. VI (IdM kh6ng th~flit gQnnenIt kh6ngth~rutgQnva II ~ M. Sur fa RadIt ~ AI; m;)tkhac, Rad((It)M)= (Rad(It)).u= « ~(x),p »M vaRad(Id nguyentonenrheaBinh /f 1.4.10,tacoRad(I1J=. Ho'nnuaII =1= vCJimQi a,bnguyendVa'ng,VIneucoao,bonguyendvo'ngthoaIt =thl (It)M = « [r.p(x)]ao,pbo>L\1,mall thuan. V~y It thuQcS. Tiep thea,ta chungmint 10~ II. Gia sv 10:l It; ta co u E 10\ It. Sur fa '!.:E (IO)M;va da do '!.:E (IdA! nen co 9 E It,h rt !v!saGcha '!.:= -hg. Sur fa1 1 - 1 uh = 9 E II. VI It kh6ngth~rut gQn,da do nguyenso',va h rt AI = Rad(1d nen u E It, mallthuan.V~y10~ It. VI 10la ph:ln tlf t6i d~icuaS nen10= It. Sur fa (IO)M= (Idlvl. V~y(10)1>1la ph:ln tlf t6i d;;1icuaS'. Tiep thea,tase chungminh co T,s,U,v la c:k 56nguyendvo'ngsaGcha (f0: P[X])lvI = « [~(xW,pS »A/, (Io :hI. (8) (9) TrVCJclien, dam,n la ck 56nguyendvo'ngnho nhatthoa[~(x)]mE 10,pn c 10 33 V~l 11/.I/. >1nen (10)M ~ (10:P[x]),\!. (/O)M =I (/0 : P[.T])AJ, (10) (11)(1ohJ ~ (fo:< :p(x)»M, (/O)M=I(/0:< ;p(:£)»M. Do (fa)M kh6ngth2 rutg()nnen rheaDinh Ii 3.2.9 (/0);\1: [(/oh! : (10+P[X])M] =(/0+P[x])A! hay (foh! : : fo).\! : (fa -+-P[X])M] =(fO)M +(P[x]h! M<;itkhclcdo P;.r]la idealnhannen (P[X])iV!chinh,tudayrheaDinh Ii 3.2.10, (fO).\1: (fo +P[l'])M kh6ng th~ [(it gQn . M<;itkhac (fOi.\! : (fo:- P[l'DAf = (10)M:[(/oh!+(P[X])Al] = [(10)M:(1ohrJn [(fohl : (P[X])M] = (10)1\11:(P[X])M nen (fohI : (P[.r])Mkh6ngth~nIt gQn. DoRnd(foh!=« :p(x). P »M nenRad((10)AI: (P[X]hI)=« y(x). P >)M. V?y (fo: P[:T])Afkh6ngth~rLItgQntrong(D[X])M,Rad(1o:P[x])J',1= « :p(x),P >hI. Ho'nnCta,(10).\[,ph~ntlTtoi d?i ct'Ia5', chuatrongva khac (fa : P[X])AInen t6nt?i T,S nguyendtTongsaGcho (10: P[xDA[=« [rp(x)t, ps >)Af. V?y taco (8). Hoanroantuongt\T,tathuduQ'c(9). B~ngquy n?p rheai(l :::;i :::;n - 1),ta sechungminh (10:pi[X])M=« [tp(x)t,ps-i+l »Af. 34 Voi 'i = 1,tu(8),ta co di~l1phai chungminh. Gi!l Sli ding thlic Cling voi i =j(1 :::; j :::; 11- 2),nghiala (10: pJ [J;])M =« [:p(x)]'", ps-j+L »Al. Suy fa (ps-j+1[X])M(Pj[X])M = (ps+L[X])MC (10)M do do pHI C 10.Soy fa 8+ 1;::n. Do 1 :::;j :::;n - 2 nen (8+ 1) - j =8 - j +1;::2. Ta co (10:ph-I[X])Al = [(10:pj[x]) :P[x]Lu = (10:pJ[.r]Lu : (P[x])Al = « [<;(~.W,ps-j+l »M : (P[X])M = « [:p(xW,ps-j »M CtheoB(f di 3.2.8). V?y d:1ngthuc dung ,"oi i = j + 1. TheonguyenIy quy n?p, d:1ngthuc (10: Pi[xDA! = ~< [<p(x)(ps-i+l)Al Cling voi ffiQi i, trong do 1 :::;i :::;n - 1. Voi i =n - 1,taco (10: pn-l [xDA! = « [tp(xW,ps-n+2 >)M. (12) Tu'o'ngtv, tlt \9)va bangquyn?p taco (10:).\f = « [tp(x)t-m+2, PV >)M. (13) Tu (6)va (7)faco (10:pn-l[X])M = «[hI, (10:< [tp(x)]m-l»M = « <p(x),pk)M. (14) (15) 35 Tli (12),(14)vaBlf di 3.2.7taco T =1..'3- n +2= 1, hay T=l,s=n-l. Tu'o'ngtIj, tv (13)va(15),taco u- m+2=1,v=k, hay u=m- 1,v=k. Tv (8),(9),(16).(17)taco (fo : P[X])M = « [tp(X)]l, pn-l >)M, (10:)M. Tv (18),(19)vaBII di 3.2.8,taco (10:P[X])M: « tp(x)»M = « [tp(x)]l-l,pn-l »M, (fo :hI : (P[X])M = « [tp(X)]m-l, pk-l >)M. Ho'nnU'a (10: P[X]);\I : « hI = (10:P[X]<tp(X)»M = (10: P[X])M = (10:<tp(X)»M : (P[X])Mo Tti (20),(21)va (22)taco « [tp(x)]l-l.pn-l ».\1 =« [<p(x)]m-l,pk-l »,'v/o 36 (16) (17) (18) (19) (20) (21) (22) (23) Tv (23)va Bd dl 3.2.7,ta co I - 1 = m - 1,n - 1 = Ie- 1 tlie la I = m, n = k, mall thufin voi tru'O'nghQ'pdang xet. V;;ly, trang mQi tnlO'nghQ'p,ta dell co mall thufin. Do do S = 0, nghia la, neLl I la ideal kh6ng th~[(It gQn clla D[x]voi Rad(I) = th1,tcSnt<;li m,n nguyen dvo'ngsao cho I = , C/ Tv Bif dl 3.2.6,Bo?dl 3.2.11"J Bo?dl 3.2.2,taco 3.2.12Dink Ii (M6 racacidealkh6ngth~rutgQnvoi cantoi d<;li).IdealI Clla D[:r]fakhangthinitg9nvaidintaldr;zinlu vachinlu tontr;ziidealnguyenta/ PolO CllaD, da thucdonkhrJi-p(.r)E"D[x]thod<p(x)+P[x] batkhdquytrang D[xJlP[x]vak, 1nguyenduringsaocho 1=, trongd6 m,n nguyenduring, Honnaa,khzd6Rad(I)=vam,nfacacsa/nguyenduringnhonhdt thoa[<p(x)]mE I, pn c I, Nhu'V?y,tada m6tadIu trueclla caeidealkh6ngth~[(itgQntrangvanh da th{TCD[x],trangdo D la mi~nnguyen.Dedekind. Tv s1jm6 ta nay, ideal nguyenso'vCiican toi ti~uchinh li idealkh6ngth~flit gQnvai can toi ti~u. Tuy nhien, idealnguyenso'voi cantoi d<;likh6ng nha'tthiet la idealkh6ng th~rutgQnvo-icantoid<;li.DungDink Ii 1.5.6vaDink Ii 3.2.12ta co mQt s1jm6ra khacvoiDinh Ii 3.1.4v~d<;lngcih idealnguyensovoi cantoid<;li nhlTsaD 3.2.13H~qua. IdealI cuaD[x]fanguyensrivaicantaldr;zinlu va chi nlu 37 J = n.trongdo i chC;ZYhau hc;zn,Tni,fli ld cacs6'nguyen duo'ng. I Ho'nnaa,khidoRad(I)=. 3.3 Ci{lltruc cac idealtrollg vanhda th,lCtrill miln ngllyen Dedekind Trongnwcnay,tasekhaasatcalltrucclb m(>tidealbfitkycllaD[x].Thea M~nhdi 1.5.1.mQiideald~ulagiaoheluh?n cllacacidealkh6ngth~rutgQn. Ttl day,kethQ'pvai dc dinhly m6tac:1utflk clb cacidealkh6ngth~nit gQn trongD[x],calltruccuaI1l(>tidealbfltkytrongD[x]dllQ'Cm6taquadinhlysau 3.3.1Dinh if. iVf9idealI cdaD[.r]dell du(jcbie'udienduoidc;zng r S 1=(nJ;"') n(n< [<pj(x)]mJ,P?», i=1 j=1 trang doT, s, ki (1 ::;i :::;T), mj' nj (1 ::;j ::;s) ld cacs6'nguyendl1fJng,Ji ld idealnguyent6'kh6ngt61dC;ZicdaD[x],Pj faidealnguyent6'khackh6ngtrong D, Zpj(x)E D[x]thodyj(x)+Pj[x]bit khdquytrangD[.rJ/Pj[x], Bi~udi~ncuaideal1 trongBinh if 3.3.1co th~kh6ngduynhat. Tuy nhien,do D[x]la Q-\'anhnenthea [3],idealbatky cuaD[x]co duynh:1tS\f phantichtichnguyensochu:1nhaa.Lu'uydingneu1=h..,Invai Ii la ideal Pi-nguyenso thlS\lphantichthanhtichnayclla1 OllQ'CgQilas~(phanlith lith nguyensd chua?nhod neu P; =I-Pj vai mQi i =I-j ya I =I-I1...Ji-lli+l...In vai I11Qi(1 ::;i ::;n). Dinh l~'sau ouQ'ctrich d~n tu [3] 3.3.2Dinh if. M9i idealI cda D[x]dell co duy nhrltSljphantfchtfchnguyen s(jchudnhoa. 38 Vi d~lXet [([:1'.U]v6'i K IJ tntong.VI K[.1:]Ij PID \'~tdo do la mi~nnguyen Dedekind nen [([x, V] l~tq-vanh. Do do mQi ideal du'Q'cphan tlch duy nhat thanhtlchdic idealnguyenso.Tacoideal=n ,cE [(, conhi~uhanmOtphantlchnguyensochu5nhoa,nhltngcoduynh{lt mOtphan tlchtlch nguyensochu:lnhoa=.