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tài liệu tham khảo cấu trúc các ideal trong vành đa thức trên miền nguyên dedekind
CHVdNG CA"U TRUC cAC IDEAL TRONG VANH fJA THUC TREN MIEN NGUYEN DEDEKIND Trang ehu'dng nay, ta se m6 ta diu tfl.k cLlaideal ba't ky trang vanh c1athue tren mi~n nguyen Dedekind D Vi D[x] la vanh Noether nen moi ideal trang D[x]a~u e6 s\j phan tfch nguyen sd ehu5n hoa, d6 d~ m6 ta ea'u true et'1a ideal ba't ky trang D[x] ta n~6ta ea'u true cae ideal nguyen sd Khi a~ e~p den P, I, J, toj d?i trang D[x]Den Rad(I) =< ip(x),P > ho?c Rad(I) = D[x].VI I -:JD[x] Den rhea M~nh di 1.4.1 ii), Rad(I) -:JD[x], d6 Rad(I) =< ip(x), P > toi d?i trang D[.r] Theo Al~nh di 1.4.6, Q nguyen so rrang D[x] \? 3.1.4 Binh If (M6 c:k ideal nguyen so voi can toi d~li).Ideal I ala D[x]ta nguyen sd v6l din tal d9i neu va chi neu I c6 dflng < [;p(x)]m,pn, ip(x)h1(x)+ Pl(X), , ip(x)ht(x) -l-Pt(x) >, !rong d6 m, n nguyen duong, t kh6ng am, hi (x) ("1::; i ::; t) thwjc D[x], Pi(x) (1 ::; i ::; t) thu(Jc P[x] Hon mJa, khz d6 Rad(I) =< ip(x), P > z:dta c6 thi ch9n hi(x), Pi(x) thoa deg[hi(x)] ::; (Tn- l)deg[ip(x)], deg[Pi(x)] ::; Tndeg~;(x)] Chung minh (==?)Gia su I nguyenso voi can toi d?i thi Rad(I) =< ip(x), P >, trang d6 ip(x), P nhuda quy uoc Khi d6 t, trang d6 ii(X) (1 ::; i ::; t) thuQc D[x] 19 Do I chCla Rud(I) =< :p(J'),P > nen Ill') E< ;(,r), P >, Suy fa Ji(x) = :p(x)hi(:r)+ p;(l'), trang h;(x) E D[x], Pi(X) E P[x], V~y 1=< [p(x)]m, pn, ;;(x)h1(:r) + P1(X),, , cp(x)ht(r) + Pt(x) > , ({=) Truoc tien ta chung minh Ii- D[x] Gia sl'i I = D[x]thl E I va do = [,,(:r)!mu(x)+ llV(X)+ t = [cp(x)]rnu(x)+ ;;(x) t [\O(X)hi(X) + P,(X)]g:.r) L h;(.r)gi(X)+ pu(:r) + L Pi(:r)g;(x) k=l i=1 t = (p E Po) t t 'P(x) (['P(x) ]m-Iu(x) + {; hi(x )9i(X)) + (PU(X)+ ~/i(X )9i(X)) D~t t r(x) L h;(X)g,(l'), = [:p(x)]m-1u(x) + k=l t s(x) = pv(x) + LPi(X)gi(X) ;=1 E Pix:, thl = cp(x)r(x) + s(x), Suy fa + P[x] = :p(x)r(x) + P[x], hay + P[:r]= (y{r) + P[x])(r(x) + P[.r} Tli ding thuc \'1:iathu dliQ'c ta suy fa :p(x) + P[x] kh?l nghich trang D[x]jP[x], mall thu~n V~y I i- D[x] Do [:p(x)]mE I, pn c nen theo Bo"dl 3.1.3, ta co I nguyen sa voi can toi cl~i va Rad(1) =< cp(x),P > 20 Hqn lilia, ;(.r) Gon khdi lien ['P(x)]m-I,[ la ideal toi d~i (fJtnh /f 2.6) lien I nguyen so Do P C Rad(I) lien co n nguyen duong thoa pn C I va gia Slr n la s6 nha nh?lt thoa di~u ki~n D~ th?ly < c I NgvQ'c l;,li, ta se chang minh I C, di~u tliong QUang voi chang minh J c< trang D[x] = D[x]jpn[x] Ta co Rad(J) =< y(x), p[x] > VI D la mi~n nguyen Dedekind lien co Po E p tho a P[x] =< Po > Voi f(x) E J, ta se chang minh bang qui n;,lp theo i (1 ::;i ::;n), f(x) bi~u di~n duQ'c dvoi d;,lng f(x) = , suy fa c6 u(x) dx) E D[x] saG cho hdx) f(x) V~y f(x) = ';(X)gk+l (x) = ;{x)u(x) + pov(x) = rp(x)gdx) + p~hk(:r) = c::; Hoan roan tl1ang tv m~nh d~ tren, ta co 3.1.7 M~nh di Cho I za ideal cua D[x]chua p va giGszl Rad(I) =< Khi I nguyen s6 va 1=< [ v{ji m nguyen duong Nhl1v~y ta da m6 ta xong Calltruc cua ideal nguyen so D[x] Tuy nhien dq.ng cua ideal nguyen so Binh ii 3.1.4 chl1agQn, han nua rhea Binh ii 1.5.6 mQi ideal nguyen so D[x] la giao clh hUll hq.n cac ideal kh6ng the rut gQn c6 Cling can, nen tier rhea ta se 016 dq.ng clla ideal kh6ng the rut gQn 22 3.2 Cliu true eua cae ideal khong thi rzlt gQIltrollg D[l:] Trang nwc nay, ta se m6 ta c1c ideal kh6ng th~ rut gQn cua D[:!;].Do trang r , vanh Noether I11Qi ideal kh6ng th~ rut gQndell nguyen sa Hen l11Qiideal kh6ng nguyen so dell co th~ rut gQn VI the, trang nwc ta chI c:ln xet trang t?P dc ideal nguyen so thay VIt~p tat ca cac ideal cua D[x] Call truc cua cac ideal kh6ng th~ rut gQn vai can toi ti~u duQ'cm6 ta qua dinh ly sau 3.2.1 Dinh Ii Cho I ta ideal czla D[x] Khi do, neu I nguyen sa vai din kh6ng t6l dC;Zi thi I kh6ng the' nit g9n Chung mink Gia Sll I nguyen sa vai can kh6ng toi d?i co Rad(I) = J va I co th~ rut gQn Do I nguyen so nS'ican kh6ng toi d?i Hen rhea Dinh Ii 3.1.1 1= Y Do I co th~ rut gQn Hen rhea M4nh di 1.4.8, ta co Ii, 12 nguyen sa, I] =I I =I 12 thoa I = I] n h va Rad(I1)= Rad(I2)= J Theo Dinh Ii 3.1.1, co 1.:],k2 nguyen dliang cho Ii = Jkl va 12 = Jk2 Kh6ng mat tinh t6ng quat, ta co th~ gia su k1 ~ 1.:2,khi I] ~ 12, Sur I = h, mall thuan Do I kh6ng th~ rut gQn.Q Nhu v~y, rhea Dinh Ii 3.2.1, mQi ideal nguyen sa vai can kh6ng toi d?i d~u kh6ng th~ flIt gQn Tuy nhien, mQt ideal nguyen so vo1can toi d?i kh6ng nhat thiet la ideal kh6ng th~ flit gQn; ch~ng h?n, xet D = Z va trang Z[x], ideal < X2,pX,p2>=< X,p2 > n < x2,p > nguyen so vai can toi d?i va co th~ rut gQn f)~ m6 ta Call truc db cac ideal kh6ng th~ rut gQn vai can toi d?i, ta c:1n mQt so be) d~ sau 3.2.2 Bfl d€ Cho I ta ideal czla D[x], I =< [cp(xW,pS > vai T,s nguyen dl1o'ng Khi T,s ta eae s6~nguyen dl1ang nho nh{[{ thoG [ I:pn-l[x] = < [ip(x)]l,p> Chung minh Ta chung minh d~ng thCrcthu nhat Neu Tn= thl 1:< [ip(x)]rn-l> = I: R = I Mi;itkhac [y(xW E I nen rhea M~nh di 3.1.6, ta co I =< ip(x),pr > v6'i r ngllyen dLio'ng Ho'n mia, rhea Bo?di 3.2.2, T 1a 56 nguyen dlro'ng nho nhat thoa p, c I nen r = n \)y b6 de dlfQ'cchung minh v6'i Tn= Neu Tn > 1, ta co [ip(x)]rn= [ v 3.2.4 Btf di Cho I lil ideal czla D[x]{hoa [ ho?fc I: ps-l[X] =< [ip(x)]",P > Khi 1=< [ Chzlng minh Do [ip(xW E I, ps c I nen < [ip(x)Y, p5 >~ I Ta se chC£ng minh I ~< [ Suy fa t6n t~i E ~ thoa f(x) = CP(X)Ul(X) Gia Slt f(x) = cpk(X)Uk(X), k < f, d6 pk(x)udx) E [, d6 udx) E [ : cp(x)k Do k < l' nen [: c :< I [~(x)]m-l > :'\gu'Q'cl~ti,v6'i mQi f(x) E :< [, ta c6 f(x)[ hay ['P(.r)]m-l (f(X) f(x) - Hdn nua [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 cho [;(r)]k' I = [;(X}]k f(x) + PI g(x) m(x) n(x)' Suy fa 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)]k f(x)n(x) + P[x] Gian lu'Q'cn(x) + P[x] [;(x)]k' + P[x] ( n(x) + P[x], [;(x)]k' + P[x] =I-0 + P[x]) hai v~ clla dang thuc \-ua thu du'Q'c,ta co , m(x) + P[x] = [cp(x)]k-kf(x) + P[x], suy fa m(x) E< y(x), P >= AI, mall thu1n V?y k = k', Do vai tfO clh l va [' nhlt nen ta co th~ gia si't l ~ z' Ta chung mini) , l = [' Gia si't l > [' D~t I =< [cp(x)]k,pi >, J' = [, Khi do, D[x] = D[x]/pl[rJ, ta co Y=< [ l' =< [, < Po>= P[x] vai Po E P (do Rd dl 3.1.5), , - - VI I = I nen I = 1' Do d6 I M = l' M 7 Ta co Po E< [;(x)]k >.\1 nen Po 1 = [( )]kHdn mla [cp(xWE I c I :< Po> nen < [cp(XW,pg-l>c I :< Po > NguQ'cl~i neu f(x) E I :< Po>, b~ng qui n~p theo i (1 :::;i :::;s - 1),ta se chung minh f(x) = [ nen pof(x) E I, do pof(:r) = [:p(x)]rg(x) M~Hkhk [cp(xW tt< Po> nen ~ E< Po> (do < Po> nguyen to ), do g(x) = Puh(x) hay Suy fa pof(.c) = [.p(xWpoh(x) Po(7(X) - [~(:~Wh(X)) = (**) Ma Po =1= nen f(x) - [-;(x)]rh(x)E Rad(O) =< Po > Suy fa f(x) = [cp(x)]rh(x)+ P6g1(X), v~y (*) dung yO'i i f(x) = = Gia su (*) dung v6'i i = k < s - 1, nghia la [cp(x)]rh(x) +P§9k\X) Ta co f(.r) - [cp(x)]rh(x) = p~9dx), ket hQ'p v6'i (**) ta dltQ'c P~+19k(X) = O \1 k+ < s nen p~+l =1=0, suy fa 9dx) E Rad(O) =< Po>, hay 9dx) = PO9k+l(X),9k-l(X) E D[x] V~y f(.r) = [:p(x)]rh(x) + P~+19k+l(x) Theo giil thiet qui n:;1p f(x) = [:p(x)]rh(x)+ pb9i(X), v6'i mQi :::;i :::;s - V6'i i = s - ta duQ'c f(x) = [.;(x)]rh(x) + pg-19s-1(X) E< [cp(X)]r,pg-l> V~y ta da chung minh dltQ'c I: PIT] =< [cp(xW,ps-l > Tiep theo ta se chang minh I :< y(x) >=< [y?(X)r-l, ps >, hay chung minh 1:< y(x) >=< [:p(x)]r-l > D~ tha'y < [:p(x)]r-l,Fg>c I :< ~ > NguQ'c l=< [cp(xW-l,ps > 31 Theo M~nh di 1.4.9), (! : P[:r;])M = hf : (P[:r:]hf va (I :< cp(x) »M = hI :« y(:r) > Lu va do ta thu dltQ'Chai d5ng tht'tc l~li.Q Hai b6 d~ sau duQ'ctrich tu [9] (Dinh if 34 va Dinh if 35 ctla chlldng 4) 3.2.9 Bil di Cho R ld l)(lnhdia phuongNoethergiao hoan co don vi va f ld ideal nguyen srJcua R co Rad(1) t6l dc;zi.Khi d6 neu f kh6ng the' rUtg9n thi f: (f : J) = J vai m9i ideal J chua f 3.2.10 Bil di Cho R ld vanh dia phuong Noether giao hoan co don vi va f ld ideal kh6ng the' rUtg9n Clla R, J ld ideal cua R chua f Khi d6 J kh6ng the' rUt g9n neu va chi neu f : J chinh modulo f 136d~ saul;) chi~u ngltQ'ccua Bo?di 3.2.6 3.2.11 Bo?di Cho f ld ideal kh6ng the'rUt g9n D[x] vai Rad(I) =< -p(:r),P >= JI Khi d6, fon tfli m, n nguyen durJngsaGcho f =< [;p(x)]m,pn > Chung minh D?t S = { Q IQ kh6ng th~ nit gQn trang D[x], Rad(Q) =< tp(x),P >, Q i-< } [cp(x)]a,pb > voi mQi a, b nguyen dltdng Ta se chang minh S = b~ng phan chung Gia su S i- VI D[x] la v~mh Noether nen S c6 ph~n ttt toi d?i la fa VI Rad(Io) =< y(x), P > nen c6 m, n nguyen dvdng saD cho [,p(x)]m E fa, pn C fa va ta c6 th~ xem m, n la cae so nguyen dltO'ng nho nh{{t thoa tinh chat Ne'u m = 1, rhea M~nh di 3.1.6, fa =< cp(x),pn >, mall thuKn VI fa thuQc S 32 l\i2\1 f/, = 1, theo M41llz df 3.1.7, fo = [;(J,,)]TII, P >, mf1LlthLl~n Nell Tn > \':1 n > 1, rhea Blf df 3.2.3, ta co 10 :< [~(x)Jm-l > = < ~(x), pk >, (6) fo :< pn-l[xJ > = < [ip(X)JL, P >, (7) c16 ::; k ::; n va ::;l ::;117 Neu k = n ha?c l = m, rhea Bo?df 3.2.4, 10=< [~(x)Jm.pn >, mall thuan Gia stf k < n va l < m D?t S' = { hI );\1 /." =1=« hI kh6ng th~ n.'1tgQn trang (D[x])Ju, Rad(I1\1) = « [ipCr W pb ip(:r), P > > ~,'J voi mQi a b nguyen cllfo'ng } Ta co (Io)Jula ph:ln tv toi d~i ci.'taS' Th~t v~y, ghi Slfco (It)M thuQc S' saG cha (Iohl chua trang (IdA/ TnJoc lien, ta chUng minh II thuQC S VI (IdM kh6ng th~ flit gQn nen It kh6ng th~ rut gQn va II ~ M Sur fa RadIt ~ AI; m;)t khac, Rad((It)M) = (Rad(It)).u = « ~(x), p »M va Rad(Id nguyen to nen rhea Binh /f 1.4.10, ta co Rad(I1J=< Ho'n nua II =1=< [~(x)Ja,pb > vCJimQi a, b nguyen dVa'ng,VIneu co ao,bonguyen dvo'ng thoa It =< [ip(x)Jao, pbo> thl (It)M = « [r.p(x)]ao,pbo > L\1, mall thuan V~y It thuQc S Tiep thea, ta chung mint 10 ~ II Gia sv 10 :l It; ta co u E 10 \ It Sur fa '!.:E (IO)M; va da '!.:E (IdA! nen co E It,h rt !v! saG cha '!.:= -hg Sur fa 1 uh = E II VI It kh6ng th~ rut gQn, da nguyen so', va h rt AI = Rad(1d nen u E It, mall thuan V~y 10~ It VI 10 la ph:ln tlf t6i d~i cua S nen 10= It Sur fa (IO)M= (Idlvl V~y (10)1>1 la ph:ln tlf t6i d;;1icua S' Tiep thea, ta se chung minh co T,s, U,v la c:k 56 nguyen dvo'ng saG cha (f0: P[X])lvI (Io :< ip(x) »M = « [~(xW,pS »A/, = « [~(x)Ju,pv > hI (8) (9) TrVCJclien, da m, n la ck 56 nguyen dvo'ng nho nhat thoa [~(x)]m E 10,pn c 10 33 V~l 11/ I/ > nen (10)M ~ (10: P[x]),\! (/O)M ~ (fo:< :p(x) »M, (1ohJ (10) (/0 : P[.T])AJ, =I (/O)M =I (/0:< ;p(:£)»M (11) Do (fa)M kh6ng th2 rut g()n nen rhea Dinh 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 )M V?y (fo : P[:T])Afkh6ng th~ rLItgQn (D[X])M,Rad(1o : P[x])J',1= « :p(x), P > hI Ho'n nCta, (10).\[, ph~n tlTtoi d?i ct'Ia 5', chua va khac (fa : P[X])AInen t6n t?i T,S nguyendtTong saG cho (10 : P[xDA[ = « [rp(x)t, ps > )Af V?y ta co (8) Hoan roan tuong t\T, ta thu duQ'c (9) B~ng quy n?p rhea i(l :::;i :::;n - 1), ta se chung minh (10: pi[X])M =« [tp(x)t,ps-i+l 34 »Af Voi 'i = 1, tu i = j(1 :::; 11 :::; j (8), ta co di~l1 phai chung minh Gi!l Sli ding thlic Cling voi - 2), nghia la (10 : pJ [J;])M = « [:p(x)]'", ps-j+L »Al Suy fa = (ps+L[X])MC (10)M (ps-j+1 [X])M(Pj[X])M do pHI C 10 Soy fa + ;:: n Do :::;j :::;n - nen (8 + 1) - j =8- j + ;:: Ta co (10: ph-I[X])Al = [(10: pj[x]) : P[x]Lu = (10: pJ[.r]Lu : (P[x])Al = « = « [:p(xW, ps-j »M V?y d:1ng thuc dung ,"oi i [ ).\f = « [tp(x)t-m+2, PV >)M (13) Tu (6) va (7) fa co (10:pn-l[X])M (10 :< [tp(x)]m-l »M = «[hI, (14) = (15) 35 « , C/ Tv Bif dl 3.2.6,Bo?dl 3.2.11"J Bo?dl 3.2.2,ta co 3.2.12 Dink Ii (M6 D[:r] fa khangthi cac ideal kh6ng th~ rut gQn voi can toi d