Luận án Vành với lý thuyết Divisor

VÀNH VỚI LÝ THUYẾT DIVISOR TRỊNH NGỌC AN Trang nhan đề Lời nói đầu Mục lục Chương1: Sơ lược về ý thuyết chia hết trên nửa nhóm giao hoán và miền nguyên. Chương2: Vành với lý thuyết Divisor. Chương3: Vành đa thức trên vành với lý thuyết Divisor. Tài liệu tham khảo

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CBtTONG 2 VANHVOllY THUYETDIVISOR Trong16pcacv~mhnhu":v~mhEuclide,v~mhchinh,vanhGauss... m6i ph~ntircilano d~uco th~phantichduynhaithanhtichcacphilntlrnguyento . Hay noi cachkhactrongcacvanhnaydinh1yco bancila so hcvftncondung. Tuynhientrong1ythuyetsotal(;lihayphailamvi~cv6ivanhmatrongdodinh1y co banGuaso hckhongcon dungnfra. VI vi;ly nguaita duavaokhai ni~m "DIVISOR" d~com9tkhaini~mr9ngbongi1avanhv6i1ythuyetdivisor. Trongchuongnaychungta senghiencuum9tso vand~ ellavanhv6i 1y thuyetdivisornhu:Dinh nghiavavi d\l,h~liend~cua1ythuyetdivisor,m9tso tinh chatcuavanhdivisor,vanhcon,vanhthuong, vanhcac thuongcua vanh divisor. 2.1Dinhnghiava vi dl;l 2.1.1DINH NGHIA: ClIo D 1am9tmiennguyen. Khi do D duqcg9i13.vanhv6i1ythuyet divisor neut6nt(;lim9tnlranhomv6is1;1'phantichduynhatthanhcacnhantIt A A' 0/\ ' 1'" A' ? h / nguyento c:Z; vacong call nUan.om. (.) : D* (jfj .~ j ~ , / a H (a) saoclIocactieDd~sauthoaman: TieDde1: Ph~ntu a thuocD* chiaherclIo b thu9cD* khivachikhi (a) chiaher h (b) ? h/ (J)\C 0 trongnuan om '::LJ . TieDde2: Neua va bthu9CD chiaherclIoph~ntIt A E (jJ thla ::t b clingchiabet 15 choA ( a E D* duqcgqila chiahe'tchoA E qj) ne'u(a) chiahe'tchoA trong ? h ~ OJ)nUan om dJ ) TieDde3: Ne'uA va B lahaiphanttrcuaqj) saochot~pcacphtinttra E D* chia he'tcho A tIling v6i t~phqpcacphanttr b E D* chiahe'tcho B thl A =B. 2.1.2CHU Y: - CaephdntU:cuanltanhomqj) g9ifa divisorCllavanhDo - Caedivisordfmg(a) vai a E D* g9ifacaedivisorehinh - Ph(in tU: E E qj) g9ifa divisordunvf - Con caephdntU:nguyenta'cua qj) g9ifacaedivisornguyenta~ - VOiphdntU:a ED, (a)cosZ!phdntichduynhdt thanhtichcaedivisor ?lJ .. A A' ) nguyento P; E f aj ak aj ak (a) =Pjo 00 Pk daikhitacilngviet a =Pj 000 Pk V " ~ h A' ? D.t< ' d " ( ) ( ) (Jl\ - at caep an tu: a, ,..., an E ., cae lVlsor a, , 0. ., an EdJ ton tfli UCLN C =(( aj), . 0 . , ( an)) E qj) Ta nhie'ukhieilngnoi C fa UCLN ciia aj,..., an C=(aj,...,an) - M9i phdntU:cuavanhvOifythuyetdivisor chicohL7:uhfln uiYcsa'khang lienketo Th~tv~y Gia str a coVOs6uocs6 khonglienke't =>(a) coVOs6uocs6 16 al ak nhung (a) =PI ... Pk nghiala (a) co (a]+1)", (ak+1) uacs6 (mauthuAn) V~y a chicohuubanuacs6. 2.1.3cAe vi DU: Truackhi duaracacvi dt;t, tanhal~iding:M9tvanhduqcgila v~mhnhanta hoa(vanhGauss)neumiphilotadellphantichduqcduynha'thanhitchcac thuas6nguyent6(neukh6ngk~dencacphilntakhAnghichvath11tll cuacac nhanta) Vi do 1:- Vanhnhfmtahoalavanhvaily thuyetdivisor Changminh: Tren a ~ b GiAsa D lavanhnhantahoa D* xacdinh quanh~tuongduongnhusail: neu a=b 6 ; s khil nghich (a lienkefvaib ) Ta dat f!j) = D/ Kh ' 1" 0.1\1, " h" ". h" 1A 1, I A " l A' " d ' d' A1(0 c.:.tj anuan om VOlp cp n 1an an 1ancacp1antl1 ~ll- H(n. Nghiala neu --;;=~ , b = b' thi ab = a'b' Th~tv~y a = 61a' b =62b' SHYra ab =6162a'b' Hen ab = a'b' M~tkhac (~b)~ = ab ~ = abe =~ be =~ (b~) f!j) phantichdU"qcduynha'tthanhcacphilntanguyent6 : PhilntakhAnghichduynhfttla T , vi neu ~ E f!j) laphilntakhA 9lJ -.- - }. 'A' / ngh!chthl tont~l b E - J saocho a b = 1 SHYra ab = 1 suyra ab =6 khAnghich =>a khAnghich =>--;;= T. 17 Ph~ntlrnguyento'cua r!J) lacacph~ntlrd~mgp v6i p nguyento'trong D, Th~tv~y P = --;;ob p =ab [; ([; kh~lnghtch ) p nguyento'trong r!J)Bai v~ytathayngay p nguyento'trong D, Baygiav6imQi ~ E r!J) khacthong,khac1, Khi do a E D khac khong , thong kha nghichDen a = [;PIP 2 0 0 0 P n' Pi nguyento'trong D, Suyra a =---p; P2 ,.. Pn ,Sl;fphanrich duynhatcua ~ trong r!J) suyra tusl;fphantichduynhatcua a trongD. Xct d6ngCallnlranhom (.): D ~ r!J) a H a Ta chlrngminhthoa3tieDd~cua1ythuye'tdivisor Thatyay, . i) a : b khi a =b 0 e v6i e E r!J) dodo a = b, e nghia1a a =be. [; V~y a: b [; khanghtch NgU'Q'cl~i ne'ua: b thi a =bc =>- -- -- => a=boe => a: b a = boe H ) V /' ' b ' d " C 0/\ , . b 0 11 01 mQl a, E D va IVlsor E cZ/ ma a : C, : C Ta chungminh a :t b: C Thatyay, , GiaSlrC=~ Vi a : e vab : ;; Dentheotiend~1 - Suyra a:t b : e => a :tb : e a: cvab: c =>a :t b: C 18 iii) Di.it A =a thethl ~ a: B ~ a :B Nghiala A: B . a: A TUO11gtt! diItt B =b tad~dangsuyra B: A V~y A =B Vi d1;l1 chotathaymQivanhnhantuhoad~ulavanhv6ily thuyetdivisor chi~unguqcl(;licodunghaykhong?Saunaytasethaydingchi~unguqcl(;li khongdungavi d1;l3 .Tuynhientacodinhly sau: 2.1.4DINH LV: VimhD lavanhnhimtirhoakhivachikhi D lavanhvOiIf thuyetdivisor matrongdo mQidivisorcuaD la divisorchinh. Th~tv~y: (~) Vi d1;l1 dachUngminhmQivanhnhantuhoad~ulavanhdivisormatrong do r!lJ = D/. Ro rangtronglapnaymQidivisord~ula divisorchinh «=) DftulientachUngminh divisor P =(p) ladivisornguyentokhivachi khi Plaphiintunguyento cua D, Th~tv~y: Gia8UP nguyento va P =PIP2 suyra P =(P)=(Pl)(P2) Vi 1, d. . A '" [ (PI) = E 1 P a Ivlsor nguyento ~ (P2) = E ~ PI khanghichhoiItcP2kha nghich dodo P nguyento. Nguqcl(;lineuPnguyento thl p =(p) ladivisornguyento vineu P =PI P2 ~ (p) =(PI)(P2) ~ P =PI P2E: vi PnguyentonenPI hO~lCP2khanghich Suy ra [ ~ = E ~=E 19 BA ., '" * h' ( ) / hA / h d h'" 0/\ayglOlieu a EDt 1 a COslfp antic uyn at trong :;:L) (a)=PI . . . Pr (Cacdivisornguyent6 P, lakhongcilnkhacbi~t) P] = (p]), . . ., Pr = (pJ th~trongD chungta c6phepphantich thua s6 a = &PI' . . P r trongd6 & lauoccuaclanv~clla D . Do d6 c6 slfphantich duynh~ttrong D. valieu thanh Truockhi duaravi d~2 tac6d~nhnghiaphilntunguyend~is6nhusail: 2.1.5DINH NGHIA: Gia su K la truangmbr9ng b~chil'uh~nb~tky cua Q. a dugcgQilaphilntit'nguyend~is6cua K lieu IIT(a , Q) E Z[x] trongd6 IIT(a, Q) la dathacb~tkhaquidankhbi trong Q [x] nh~na lamnghi~m. Vi do2: Giasu K latruangmbr9ngb~chunh~mb~tky cuaQ ; D lavanht~tcacacphilntu nguyend~is6 cuaK. Khi d6 D lavanhvoily thuyetdivisor. Th~tv~y, niXanh6mcacdivisorchinhla Qj)={AI A laIdeal Voi phepnhantrong Qj) AB = { I a,b, hhan cua D } laphepnhancacIdeal a,EA,b,EB} Khi d6 Qj) Lanuanh6mgiaohoan,c6danv~(la D) voi slfphantich duynh~tthanhtichcacnhantunguyent6. Trongnuanh6mnay A : B A c B Va dod6 cac philntu nguyent6 chinhlacac Ideal t6idai D6ngc~u (.) : D ~Qj) a H (a) dugcxacd~nhbbi (a) la Idealchinh sinhbbi a . Khi d6 (.) thoabatieDd~cualy thuyetdivisor. VI v~y D lavanhv6i ly thuyetdivisor.Changminhchi tietcuacac nh~nxetnayc6the;xemtrong [4] . NhcJnxetrangtrongvi dlfnaykh6ini~mIdealvakh6ini~mdivisortri'tngnhau. 20 Vi do 3: (Vi dl;lv~v~mhvai1)'thuyetdivisormakh6ngphaivanhnhantvhoa) Giasv D ={a_+bF la,b E Z } Khido D lavanhvai1)'thuyetdivisornhU'Ilgkh6ngphaila vanhnhantuhoa. +DAulientachUngminh D Ia t~phqpHitcaca.cphAntunguyend~isocua Q(FS) Th~tv~y a +b0 Ia nghi~mcuadathucX2- 2ax+a2 + 5b2=0 M~tkhac X2 - 2ax+a2 + 5b2 E Z[x] lienD chinhla t(lptatcacacphAntv nguyend~isocuatru'OngQ(FS) lientheovi dl;l2 D lavanhvai1)'thuyet divisor;Nuanhomcacdivisorcua D chinhlanuanhomcacIdealcua D + D khonglavanhnhantuhoa. Th~tv~ytru'aclien taco nh~nxet ranglieu a = a + bF5 E D lauac cua fJ E D thl 1 a 12 =a2+5b2 lauaccua I fJ 12 . Baiv~ycacphAntukhangh~chuaD phaila cacphAntu CO1 a I =1 tucla :t 1. * Taco 3, 7, 1 + 20 ' 1- 20 lacacphAntunguyentocuaD; Vai chungkh6ngcomQtuacthl!csl!nao. Th~tv~ylieu a =a + b 0 lauacthl!cSl!cuamQtrongcacsoay thl I a 12 =a2 +5b2phailauacthl!cSl!cua 9,49, 21. Noicachkhaccacphuangtrlnh a2 + 5b2= 3 phaiconghi~mnguyen. Di~unaykh6ngth~du<;1c. a2+ 5b2 =7 *Tal;;tico 21 =3.7 =(1 + 20)(1 - 20) V~y 21E D cohaisl!phantichthanhtichcacphAntunguyentokhacnhau. Suyra D kh6nglavanhnhantuhoa. . Baygiatrongvanhvai I)' thuyetdivisor D. Ta nghienCUllsl!phantichcuacac phAntu 3, 7, 1 + 20 ' 1- 20 thanhtichcacdivisornguyento. + Dat PI =( 3 , 1 + 20 ) P3 =(7 , 1 + 20 ) P2=( 3 , 1 - 20 ) P4=( 7 , 1 - 20 ) 21 Khido PI, P2 , P3 , P4 lacacdivisornguyent6 (6 dayPI =( 3 , 1 + 20) la kyhi~uIdealsinhb6ihaiph£intir3,1+20). Th~tv~ytachUngminhnola cacIdealt6id(;licuavanh D, Tnroclientachungminh PI laIdealt6id(;li. Gia sir B <JD , B ?- ~ :J a =a + b0 E B a~~ =>(b-2a)(N) =a-a(1+2N) thu('>cB makhongthu('>cPI => b - 2 a khongchiahe'tcho3 =>b - 2 a chia cho 3 dLJ' :t 1 =>(3k:t 1)F5 E B => F5 EB => 1 EB =>B=D ChUngminh tU'0ngtU'taco P2, P3, P4 la cac Ideal t6i d(;li Dodo PI ' P2 , P3 , P4 lacacdivisornguyent6 +X6tsl;1'phantfchcua (3), (7), (1 + 20), (1- 20) thanhcac divisor nguyent6: Taco (3) =PI P 2 (7) =P3P4 (1 + 20) =PI P3 (1 - 20) =P2 P4 Th~tv~y 3=(1+2[=5) (1- 2N - 3 . 6 , v ' '--y ' ~ ~ E PI EP, EPI EP, => 3E~P2 =>(3) C ~P2 N gU'Q'cl(;line'u a E ~ , fJ E ~ Ta chUngminh a fJ E (3) 22 a =3u + (1+2 f=5)v fJ =-3r + (1 - 2 f=5)s u,v,r,s E D Suyra afJ =3A' + 21vs E (3) Vi A, v, sED V~y (3) =PIP2 TaO1lgt1!taco (7) =P3P4 (1 + 20 ) =PI P3 (1 - 20)= P2P4 Va nhav~y (21) cos1!phtmrichduynhatthanhcae divisornguyent6 nhasail: (21)=(3)(7)= P\P2P3P4 =(1 + 20)(1 - 20) = P,P3P2P4 Ngoairataconcom9ts6vanhvOi1ythuyetdivisormakh6ng1a vanhnhanttrhoachingh(;ln: . 2(P) = {a+bPI a,b E 2 } 2 (p) 1avanhvOi1ythuyet divisor vi 2 (P) 1avanhcaes6 nguyend(;lis6cuatrl1Ong Q(p). Tuynhien 2 (p) kh6ng1avanhnhanttrhoa vi trong 2 (p) phfinttr 6 cohaieachphanrichkhacnhauthanhrichcaephanttr nguyent6: 6 = 2.3 = _(F6)2 ; 2,3, P d~u1acaephanttrnguyent6. 23 (1 + F23 J { I + F23 }. Z 2 = a + b. 2 I a,b E Z [ 1+~-23 J - 1,-, h ~. 1~ h " d" , ( 1 +F23 JZ 2 cling a v~n VOl y t uyet IVIsor VI Z 2 la v~mhcacs6nguyend<;lis6cuatruang Q(F23). Tuynhi~n Z (1 +f23J kh6ngla vanhnhanttrh6a VIph<intu 6 ( 1+~- 23 J ~ h . ~ h hA ~ h kh~ h h' h ~ 1 ~ hA' ? E Z 2 co al cac.. p antIC acn aut an tIC1cacp antu nguyent6: 6 = 2.3 = 1+ F23 . 1- ~- 23 2 2 2, 3, 1+~-23 2 1- ~- 23 d6ulacacph<intunguyent6cua2 z (J +F} 2.2H~tiendecuaIythuyetdivisor Trongtie'tnractadiltrinhbay dtnhnghia,cacVIdl;!v~vanhv6ily thuye't divisor.Tuynhiend~IDQtvanhlavanhv6ily thuye'tdivisortachIdin D thoahai liend~ 1 va 3 Cl)th~tacodtnhly sail: 2.2.1DINH LY CO BAN: TroDgh~tieDdecuav~mhvOiIy thuyetdivisortieDde2la h~qua cuah~tieDde1va3 Tnrackhi chUngminhdtnhly naytacanmQtsob6d6saildayvatrongsuot tie'tnay taluaugiasit D lavanhclingd6ngCali nitanhom (.): D* ~9JJ a H (a) thoaliend61valiend63 trongdo 9JJ lanitanhomvais\!phantIchduynhathanhtIchcacphantit nguyento 2.2.2BODE: - Cho a, bED Khi do (a)=(b) ne'uvachIne'ua =b&trongdo & la uaccuadanvi . Th~tv~y +Ne'u(a) =(b) thi a : b va b : a Nena=b &) va b =a &2 Dodo a =a &].&2 Nghiala &).&2 lauoccua1 V~ya =b& +NguQ'cl~i ne'ua =b&v6i & lauaccuadonvt The'thi(a): (b) va (b) : (a) Do dot6nt~i (c) , (c') saGcho (a)=(b)(c) va (b) =(a) (c') Cholien (a) =(a) ( c) (c') Tu do ( c) ( c')=(e) Dodo (c)=(c')=(e) V~y (a) =(b) 25 2.2.3BO DE: Ne'uA, B thuQc9lJ vaB "*E thlco it nhatphftntv a E D* ma a chiahe'tchoA nhU'nga kh6ngchiahe'tchoAB . Chitngminh: D~t A ={aED I a :A } AB ={aED I a :AB } Theotiend~3 A"* AB VI A"* AB - @ ,. ,/ / A .? / AB ~ A (Do B "*E va c . co lu~tgIanvac Theonh~nxct 1.1.6) Nenbaagioclingt6nt~iphftnta a E A , a ~ AB Tucla a:A va a: AB V~y 2.2.4HEouA: V6'ibfltkyA E 9lJ baagioclingt6nt~iphantv Th~tv~yne'uA =E thlhi~nnhien Ne'uA "*E apdl!ngb6d~2.2.3 v6'iB=A thl :3a E D* saDcho a: A, a: A2 a E D* saDcho a: A ,:? ~, 2.2.5BO DE: Tiend~3trongdinhnghialy thuye'tdivisortu'angdl1angv6'i: V6'ibatky A E 9lJ d~ut6nt~ihliuh~nphantv aI , . . . , an E D* saDcho A la VCLN cua (a) ) , . . . , (an) Chitngminh: Giasacotiend~3 tachUngminhv6'ibatkyA E 9lJ d~ut6ntai a) ,...,an E D* saocho A=( (a]),...,(an) Theoh~qua2.2.4 \/AE9lJ d~u:3a ED* saDcho a : A 26 Dod6 al an (a) :=API, . . Pn (*) Theob6d~2.2.3 thi :Jai E D* saocho a; :A va a, : APi TasechangminhAla VCLN cuacac(ai) va (a) Th~tv~y Ala uacchungcuacac(ai) va (a) rheacachxaydlplg. NeuB la VCLN clla(a),(aj) , . .. , (an) TachUngminh A ==B Vi Ala uacchung,B laVCLN nen B: A Ma B I a va a c6d~ng(*) Nenneu B "*A thi B chiahercho Ap vai 1~ io ~n10 Nhungtaco B I a; nen a; chiaherchoA p;0 0 0 Mfmthufulv6'icachchon a,' cua ta chung to A ==B. 0 Va A laVCLN cua(a), (a;). . NguQ'cl~igiasi'rvaibittky A E Qj) d~ut6nt~i aj ,...,an E D* saocho A=( (a}),...,(an)) Ta chUngminh c6 lien d~3 Nghialaneu A,B E Qj) sao cho t~p a E D* vai t~pa E D* chiaherchoB thi A =B. TnraclientachUngminh A: B Th~tv~yt6nt~icac ai E D* saocho A=( (a]),...,(an)).Khid6cac nencac a; clingchiahercho B . Dod6 VCLN A clingchiaherchoB Tuangtt! B : A. V~y A=B chiaherchoA trung (i= l,...,n) a. E D* chiahetcho A, ;? ,,' 2.2.6BO DE: Qj) lani'ranh6mvaisl;Tphantichduynhilt thanhcacnhanti'rnguyent6, B\' B2'.. ., Bn thuQCQj) saocho VCLN cua B i' B2' .. . , Bn la E. DatB=B] . . . B n Khi d6 B la BCNN cua £, . . ., B . Bl Bn 27 Chftngmink: Giil SLr aj] Bj =P] aji p . .I a'kJ Pk anI Bn =PI . . ani p.I ank Pk i = 1,k j = 1,n voi a > 0J I - Khido al ak B=Bj...Bn=PI",Pk n voi aj = La;, ;=1 a 1- ajl a j - aji a k- a jk Ma ~ B - ; P] PI Pk f31 f3k Do do BCNN cua B, . . ., ~ la PI", Pk B] Bn Voi f3,= max{a,- a/,} ::::;a, 100:;00:n . Songvi VCLN cua Bj,B2,...,Bn la E Do do voi illQi i c6dinh t6nt(;litnhilt j E {I,..., n}dechoa j j =0 Vi ::J j dea j i =0 Den max{ai-a; ,}=a,. ]oo:;oo:n a, ak V~yBCNN cua B B - p] ... Pk =B, . . ., B] Bn 28 all a]i a Ik B\=:P\",Pi",Pk a21 a". a2kl B2 = PI . . . Pi" Pk ChitngminhdtnhIf CO'ban: Gia SUd6ngCalinuanhom c.):D* 01) ~ c;!) thoamantieDd~1,tieDd~-3taphaichungminhthoatieDd~2. . DautieDtaconhanxetsail: V6ibatkyA E Ql) dlu tOnt{licaephii'ntil: a, aj , . . . , an E D* saocho (a).A lit BCNNcua (al),...,(an). ThiiltviilYtheoh~qua2.2.4thlt6nt(;tib ED* saocho b : A ~ (b)=AB Theob6d~2.2.5 B la VCLN cuacacphfinti'rd<;tng (bj),(bJ...,(bn) b,ED' D~t a = bl b2 . . .bn ab a, = b, Khido (a) .A la BCNN cua (a] ) , . . . , (an) ThiiltviilY Dat H' =(b,) . A' =B I . . .B I = (a) . , B' ] n Bn (B]',...,Bn') = E Theob6d~2.2.6 taco [ ~,...,~ ] =A' B' B'] n Dodo [ BnA' BnAI ] = BnA' =(a) CTfnhchat 1.2.7) ,..., B IB]' n Suy fa [ (a)B (a)B ](bl) ,..., (bn) = (a) DoviilY [ (a)BA (a)BA ](b]) ,... , (bn) = (a).A Nghiala [ ~,...,~l b] bnJ = (a).A 29 Dodo [a"""an] =(a).A V~y (a)A la BCNN cua (ai). Nh~nxetdl1Q'cchungminh. .Hemnuagiiisu a,b chiahetchodivisor A. TaphiiichUngminh a:!: b cilng chia hetcho A. Th~t v~ytheo nh~nxet tren ::Jc , Cl' . . "c nE D * (c).A =[(CI)" . . '(Cn)] saDcho Vi Ca, cb chiahetcho(c).A lien ca,cb cilngchiahetcho m6i c i . Do v~y Ca :!: Cb cilng chia hetcho m6i c i . Nghiala ca:!: cb chiahet cho [(cj),...(cJ] =(c).A Dodo ca:!: cb chiahetcho (c).A Thethi (c)(a :!:b) chiahetcho (c).A V~y a:!: b chia hetcho A. D!nh 1ydfidl1Q'chungminh. 2.2.7DINH LY: ChoD litvitnhvOily thuyetdivisorkhidoIy thuyetdivisortren D litduy Dhat. ()l\ ()l\ , Tuc la lieucocacll\lanhomcosl!phtmtfchduynhftt';:2)va ';;2/ clinghai (.)' : D* ~Q/) ~ Q/)' d6ngc~unuanhom (.): D* h? - / .~ d'" ? 1/ 1 '" d" h'~' , d2 ~/ 0)\ ~)I\'t oamancachen ecua y t1uyet IVlsort 1tont~llangcall rp: ';1/:::;~.;v saDcho V a E D* thi rp(a)=(aY Chungminh: 30 . GQi P={aEDla:P} d / P ()~ ~ Ntrong 0 E:;:L/ nguyento 7={bEDlb:P'} , d/ P ' 01) ~ t~/trong 0 E ,:tJ nguyeno D~' o~ h / onh ,(i1\'au tIeDta c U'Ilgml V P E ';;Z/ (,11\:J P E:;:Lj saocho PcP' Th~tv~ygilt su P ct- P' VPE flj Theoh~qua2.2.4thi :J b E D* saocho b: p' a] ar va b=P1 ...Pr trongdo Pj,..., Pr 1 , / 1. 0 ~ ~/ 'I ') I / 0/\a cac (lVlSOr nguyen to cua nna n 10m eX Do ~ ct-p. (giathiet)=> Voim6i i =1, 2, . . ., r comOtphantu C; ED saocho c; :~ va C;: p' a] ar c=c]... Cr b = a] ar PI . 0 0 Pr trong D (do tieDd~1) => C : p' (mauthu~n c;: P') Tnangt1;f ( . ~T1\ P Eo:!) :J Q' E flj' saocho Q' c P Ta sechUngminh Q'= p' vatli'do QI = P = P' Thatyay. 0 Theobe)d~2.2.3 taco :J h E D saocho h: Q' va h : Q' p' Giasu Q' * p' =>h : p' (mauthu~nvoi Q' c P') V~y Q'= p' flj ' ,/ ' jTndo V P E, / ::11 P (J1\ h:J. E ';;Z/ sao c 0 P = P' X / / I ()1\etan1xa rp: e':2) 0'-+ e2/) nhLfsail: 31 rheachUngminhtren, v P E Ql) nguyent6 3! p' E Ql)' nguyent6 saocho P = P' Ta dilt cp(P)=P' TuangUngtrendedangmaf(>ngthanhm(>tdltngCall cp: Ql) 0""-+ :;:L! a 1 a r cp(PI' . . Pr ) a 1 a r = PI' . . .P,' . TaconphaichUngminhcp(a)=(aY @ - @ -' ,? ? , " A A' () , ' () Gla sU a E D* vadIVIsornguyento P E - I va P E . I vagiathiC'trangchungcom~tfongph6pphantichthanhthLras6cua (a) va (a)' v6icacs6mil cua P la a va s6mil cua p' la j3 Tir h~qua2.2.4 =>:3r ED* saocho r: P va r ~ p2 - - => r: P' ( do P = P') =>(r) =PB trongdo (P, B)=E ChQn sED sao cho s: Ba va s: BaP =>s ~ P va p' (do P khongchiahC't Ba) X6trich as Ta co a: pa va s: Ba => as: paBa = (ra) Nhung r :P' Vi s : p' => a s = rat => as: p'a , t E D =>a: p'a Di~unayconghiala trongslfphantich(a)' thanhcacdivisornguyent6, divisornguyent6 p' com~tvms6mil khongit han a, dola fJ ~a nhungbmd6ixUngtaclingco a ~ j3 Vi v~y a =j3 Vi v~ychungtadachUngminhduqc P va p' thalligiavaoslfphantich cua(a) va (a)' v6i s6milla nhunhautucla alar (a)= PI", Pr . conghiala a I ar thi (ay = PI'", P,' cp(a)= (a)' 32 2.3Motsotinhchatcuavanhdivisor. 2.3.1MENHDE: N€u D lavanhv6iIythuy€tdivisorthlbatkyA,B E Qj) taco AIB ~ Be A A={aED*la: A}va 13={bED*lb: B}V6i Th~tv~y (=» A I B => 13c A hi~nnhien «=) N€u 13 e A taphaichUngminh A I B NghlalachUngminh n€u pa I A thi pa I B trongd6 P laphfult11nguyento cua Qj) Th~tv~y pa I A =>A e pa lien B c pa ma B e A Nhuv~ytacanchUngminh n€u Th~tv~y B e pa => pa I B , , al a ak a G'? ?lasu B=p\...P...Pk = P.A va B : pa khido 0 ~ a I <a Tub6d~2.2.3 :] a E D* a :pa 'A a : paA Suyra a : pa (Vi a: A va (A, P) = E )- - Do d6 a E B ma a ~ P a ngh'ial B <:Z. P a (Mauthu~n) V~y p a I B a'~O,(A,P)=E 2.3.2MENH DE: ChoA E Qj) la divisorbatky, PI , P2' . . ., Pn E Qj) lacacdivisornguyento tilyy.Khi d6t6nt';libED saocho b: A va b ~AP, V i =1, 2,. . .,n Ch{mgminh: V6im6i i=1,2,.. "n lientheom~nhd~2.3.1 - API, . . PH PHI' , . Pn : AP, API",P,-1Pi+""Pn <:Z.AP I 33 nent6nt(;li bi E API",Pi-IPi+I",Pn va bi ~ APi tucIat6ntal bi ED, -bi :API, . .Pi-I P,+I. . . Pn va - bi : AP, n D~t b =I bi i=1 Tacong~yb: A va b : APi Vi=1,2,...,n ,:? ~, 2.3.3BO DE: MQidivisorlaVCLN cuahaidivisorchfnh. Chimgminh: VOimQiA E 9J) tuh~qua2.2.4 :3a ED* saocho a: A Dodo al ak (a) = A. PI . . . Pk (*) Tum~nhd~2.3.2t6nt(;libED saocho b : A va b : APi Vi=1,2,...,n Ta chUngminh A =(a,b) Th~tv~y A Ia l1acchungcua a va b theoeachxaydl!ng Neu B la VCLN cua a va b tachUngminh A ==B VI A la aacchung,B la VCLN Den B: A Ma B I a va a cod(;lng(*) Denneu B :;t. A thl B: AP; val 1::::;io ::::; k0 Nhungtaco Bib Den b:AP;omallthu~n ChUngto A ==B V~yA =(a , b) 2.3.4DINH NGHIA: Mi~nnguyenD g9ilavanhdongnguyenellmQiphantv c;E K (K Iatrl1Ongcaethl1ongcuaD) langhi~mcuadathac f(x) = x n + an-I X n-I + . . . + a I x + a0 E D[ x] thl c;ED. 2.3.5MENH DE: D la vanhvOi1:9thuyetdivisor thl D dongnguyentrongtraOngcac thl1ongK cuano. 34 Chirngmink: Gii sir ; E K thoa ;n + an-I ;n-I +. . . +a]; + ao =0 ( a0' . . . , an~IED) nhung; ~ D Dilt ;: =~ . '=' b vaphanrich (a), (b) thanhcaethuas6nguyent6 Do ; ~ D Den a ~b trong D ~ (a) ~(b) (doliend~ 1) Co nghiala t6nt~idivisornguyent6 P com~ttrong (b) v6'is6mil16'n hoo trong (a) . Gii sir P com~trong (a) n b n - I~ a = - an-I a .crongdo a , bED v6'i s6 mil a 2::0 ~ (b): pa+] bn . an + I (d .A d;:: 2)- .. . - ao : P 0 hen e M~tkhac P com~ttrong (an)= (a)n v6'is6mu an ~ an ~pan+l. MauthuAnchUngto ; ED. Dinhly duqcchUngminh. 2.3.6MENU DE : D lavanhv6'ilythuyetdivisorcohuuh~ndivisornguyent6 thi D lavanhnhantuhoa. Chungmink: Gii sir D lavanhv6'ily thuyetdivisorv6'ihw h~ndivisornguyent6 PI ,P2,. . .,Pn .Daulientaco neuA ladivisorbatkykhacE, Tirm~nhd~2.3.2thit6nt~ibED saocho b : A va b ~APi Vi =1,2,. . .,n BaygiatachUngminhmqidivisor Pk ladivisorchfnh.Thi;ltvi;lyapdl;lngnhi;ln xettren v6'i A= Pk. T6nt~i PkE D saocho Pk: Pk va Pk : ~~ Vi= l,n. Khido (P k ) =Pk vi neu (Pk)=Pk B thi B : ~ Vi DenB=E. Cu6i clinglieU A ladivisorbatky cua D thi A ladivisorchfnh. Thi;ltvi;lygilt sir al an A =PI . . . Pn a] an a] an a] an Khi dov6'i a = PI ... Pn thi (a) = (PI) . . . (Pn) = PI", Pn = A. Vi;ly A ladivisorchfnh.Theodinhly 2.1.4 D lavanhnha.ntuhoa. 35 2.4Vanhcon,vanhthu'dngvavanhcaethu'dng Trongtie'tnaychungtase'nghiencoovanhcon,vanhthuong,vanhcacthuang cilavanhv6i Iy thuye'tdivisorc<?Iavanhv6i ly thuye'tdivisorhaykh()ng? D~c bi~ttasechangminhdvqcvanhcacthvongcilav~mhv6i ly thuye'tdivisorIavanh v6i Iy thuye'tdivisor. 2.4.1VANH CON CUA VANH VOl LY THUYET DIVISOR Vanhconcilavanhv6i Iy thuye'tdivisorchuachacdffIavanhv6i ly thuye'tdivisor. Chimgminh: Th~tv~yQ(0) IatHrong,HenQ(0) Ia vanhv6ily thuye'tdivisor. Xet D =Z (0) = {a+b)=31a, b E z} c Q(0) TachungminhD khongIa vanhv6iIy thuye'tdivisor Th~tv~ytachungminhD khongIavanhdongnguyentren trvongcacthuO'ng cilano,dodotheom~nhd8 2.3.5 D khongIavanhv6i Iy thuye'tdivisor. X " h'" ? a + b F3 ( " b b 1 " ) K ' I ' d 'etp antv a = VOla, E Z a, e 11 02 r6 rang a E Q (0) Iatruangcacthuangcila D, a2+ 3b2 x2-ax+ 4 nhungr6 rang a ~ D. V ~y D khongdongnguyen. a Ia nghi~mciladath(rc E D[x] (VI a2 +3b2 : 4 ) 2.4.2VANH THDONG CUA VANH VOl LY THUYET DIVISOR Vanhthuangcilavanhv6iIythuye'tdivisorchuachacdffIavanhv6ily thuye't divisor. Chimgminh: Tacovanh z[ x] Iadivisor.NhungZ[x]/ / (X2+3) Th~tv~yxetanhx~ cp:Z [x] ~ Z (0 ) lex) H cp(/(x))=1(0) = z(0) D~thay cp Iam9td6ngCallvanh,banl1l1anoIato~mCallVIchAnglwnphantu a + bF3 cot£.loanhIa f (x)=a +bx . 36 Ta changminh Kerrp= (p(x») trongdo p(x) = x2+ 3 , Ker rp = {lex) E Z[x] I Th~tv~y p(x) E Kerrp vi p(R) = 0 Den (p(x») c Kerrp NguQ'Cl<;tihi~nnhien lex) E Kerrp => I (R) = 0 1(F3) =0 } => lex) E p(x)Z[x] Do do Kerrp=(p(x») apd1;lngdinhly Noetherthunhift Z[ x){errp ==z(R) ma Z(F3) Z[x]/ j(x2+3) khongladivisor V~y khongladivisor 2.4.3VANHcAe THUONGeuA VANHVOl LY THUYET DIVISOR Dinhn2hia: A vanhgiaohoancodO'Ilvi s lat~pconnhancuaA (lat~pconchua1 vatich xy E S '\j x,y) Trong t~p A xS = {(a,s)1 a E A , SE S }taxacdinh quailh~hai ngoi sau: (a,s) ~ (a',s') ::Js]E S: sl(as' - a's) = 0 ~ la qUailh~tUO'IlgdUO'Ilgtrong A xS a a' -+- s S' = as' + a's ss' a a' aa' ". 1 - . - =- (co dO'IlVI - ) S s' ss' . 1 Khi do S-( A lam9tvanhduQ'cg9i lavanhcacthUO'Ilgcua A. 37 Ax kh {: laEA, SES }G9i S-IA = = ( = a' (a,s)-(a',s'») s s' Tren S-I A xacdinhhaipheproan DinhII: I Vitnhcaethuongcuavitnh.,vOiIf thuyetdivisorlitvimhvOiIf thuyetdivisor I Th~tv~ygiaSlr D lavanh.v6i1)'thuyetdivisortuclac6m(>tnlranh6mr!/) v6i st;fphantfchduynhatthanhtfchcacph~ntvnguyento vad6ngCalinl:ranh6m (,): D* 01\ ~ c:ZJ a H (a) thbatieDd61vatiend63 Gia Slr S la t~pconnhancua D. D~chUngminh S-ID lavanhv6i1)'thuyet divisortal~nluqtchungminh theocacbu6csail: 0 Xay d1!11gnlranh6mcacdivisorc\ia S-ID. ~ ()1\ . s { ( ) I Cl\ }Tren ;;LI x( ) = x,(s) X E :;;v , S E S Ta c6 quanh9tU'O'ngdU'O'11gnhu sau:(X,(s)) ~ (X',(s')) <=? X (s') = X'es) Khi dod6tMy (Sylf!l) = f!l) x (Sf = {(~I lanlranh6mv6iphepnhan: ~,~ = AB (s) (t) (s)(t) X 01\ s}E ,1/ , SE Baygiatrongnlranh6m (StIr!/) tadinhnghiaquailh9tU'ongduO'11gnhusail: A B - ~- (s) (t) A B <=?- =-'u (s) (t) u khanghichtrong (St1r!/) ( u c6dang (s') . (t') s' , t' E S) T - d;-t 0/\ -J =(S)-J(]))/ , ,a <:t ,1)s 0 cZ/ Khi d6d~thay r!/)S-I0 lanlranh6mv6iphepnhanlanhancacph~ntlrd<:tid 9n C h:! 1/ h / hA' " A k / h' ~ 1, [ A ] ute: rm c ITaP antu' - . y lCU a - . '-/1" (s)' (s) thi r!/)S-IDclingphepnhan [ ~ ] , [ ~l = [ AB l lanl:ranh6m. (s) (t)j (st)j 38 Ta chUngminh Qi)S-1D la mxanhomv6i sl;!phfmtfchduynhatthanhcac nhanttrnguyent6. . Dau lientaconhgnxetsau: A A E ,E kh ? h . h t (S)-l(j/\, 7:. h A' t ? h ' (i/\ -1- = - ,- va - ang lC rong ';;L) nenmozp an 11t uoc :;:LIS D (s) E (s) (s) . . , diu cod{lng[~]vOi A E Qi) . Philnttrdonvi cuaQi)S-1D:Dethayphilnttrdonvi cua Qi)S-l D la [ ~] N ,. ,/ h A , AO/\ n (8) d.gOal ra fa con COn (In xet : ';;L) =1=If' ~ [~]~[;] Th~tv~y AQi) n (8) =1=rjJ q 3B E Qi) d~AB=(s) E (S) q 3B E qj) d6 AB E ~) q ~ khanghichtrong(Sylqj) ~ [~]=[:] . Philnttrnguyent6 cua qj)S-1D Cacphilnttrnguyent6cua qj) S-1D la cacphilnttrco d<;lng [ ~] , v6i pqj) n (8) = rjJ va p nguyent6trongqj) ThA t A NA' [ P ] A l A' t ()/\ -I h' <;lV<;lY: eu E nguyen 0 rang ;:LIs D t 1 [~] =I=[~] lien p qj) n (8) = rjJ.M~tkhac giasix P khongnguyent6trongQi) thi p =B.C Suyra v6i B =1=E, C =1=E Khi d6 [ ~] =[ ;] [ ~] va 39 Bq})n (8) = rp,Cq})n (8) =rp (Vi P q}) n (8) = rp)Hen [~]*[~], [~]* [~] D d/ [ P ] khA A '" ()1\] /""1 b oA' 0 0 Ii . ongnguyentotrong c::J)S- D tralglat let V~yP nguyentotrongq}). N 1 ., 1 1 01\ n ' A A' ()I) h/ " hgHQ'e<.1-1gla sH p;:Lj (8) = rp va P nguyentotrongc::l/ tae ungmm [ P ] . A t'" t. 01\-IIi nguyeno rang c:,/j S D' T h'" , (,11\ n 8)a t ay VI P ';;.I) ( = rp Hen [~]*[~l GiaSlr [~]=[~][~] thi ~ = BC "~ E E (t) Suyra P(t) =BC(s) Vi P nguyento va P khongehiabet (s) (Vi pq}) n (8) = rp)Den (p,(s)) =E =>BC: P =>B:P ho~eC:P S P h " h'" B h - C (S) 1{)1\ / 1 , [ B ] [ E ] uyra - e la et - oae - trong - ~ tHe a - = - E E' E E E ho~c [~]~ [;l V~y[:]nguyen16trong@s '0' .M6iph~ntlreuaq})S-I D d~uphantfchduynha'thanhtfchcaephilntlrnguyento. GiaSlr [ ~ ] E 0-1\ -I E ;:L/s D Vi A coSlfphantfch A =PI " " "Pn ' A A' {j1\ A / [ A ] " [ ~ ] [ P ]Pi nguyentotrong;:L/ Hentaco : Ii = Ii """ ~ 40 V6i rn6i i ta c6 ho;j.c P, [!j) n (8)" ~Khi d6 [i] ~ [ ;] (Iii phin tv dO'nvi) 1 - fjI\ n Kh' d/ [ P ] I' h"'? ~ "'? (J/i-l 10(;lC P,dJ (S) = rjJ l. 0 If: ap antunguyentocua ';;LJs O' V~y[~] phiin tich thanh lich cae phin tITnguyen 16. Giii sir [~]co hai s,!,phi\ntfchkhacnhau: [~]~ [~].D]~ [~][Q~] ~...~ = QI...Qm~ E E (t) ()i\ ~. ( ) ( ) . ( )Do Q{'Z! n (S)=rjJ nen (t): Q => QI . . .Qm,(f) =E => s : t tUO'Ilgtv (t): (s) => (t) =(s) => => ~...~(t) = QI...Qm(S) Dod6 PI" .Pn = QI" .Qm"Do svphantfch duynhattrong Q/) tac6 m=n va P =Q;I i =1,2,. . . ,n. @ ChLrngminh S -1D clingvai anhx(;l [.] : S-ID (j)\ -1 ---+ ';;'1/s 0 a H [ ~ l (s) u:n=[ (;J ]S Hivanhvaily thuy6tdivisor D~thay[.] la d6ngCallmlanh6mtachLrngminh [.]thoatiend~1vatiend~3 elmIy thuy6tdivisor: . ChLrngminhthoatiend~1: N '" a. b Ieu -: - trang S- D a b c S t S t r /. C I VO'l - E S - D r => => [ ~ ] = [ ill ][ ~ ](s) (t) (r) => [ ~ ] : [ ill ] trona rlJ) ,-1 (s) (t) b S 0 41 Ngugcl~i:Giil sir co ; , ~ E S'D d6 [i:;]. [i~;]trang @S-IO Khido [ (;) ] ~[(~)][ ~] v6'i CE9}) Suyra (a) = (b)C . (Sj) E E (sz) (a)(sz) =(b)(s))(c) => (a)(sz) =(b)(sl)'C => C =(c) => => asz = bSICE => ~ =!!.-.cEts, s t s Sz Tucla ~ : b.- s t trangS-]D . ChUngminhthoatieDd~3: a'? ? [ A ][ B ]E (J)1 -1 h A / h~ 'I h A -I h . hA' la sU E' E ;;:vS D saoc 0 t(;lpcacp antu t u9c S D c la et cho [ ;] (trong9lJs-ID)tIlingv6'it~pcacph~ntuthuQcS-ID chiahetChO[~ ] tachUngminh[ ; ] =[ ~l Th~tv~ytheob6d~2.3.3Taco A = (a] , az) v6'i at, az E D cho lien al=AC), az=ACz v6'i C),CzE9}) ,(C"Cz)=E V, . A ( O" ) A 1 a; : trong;;:V lien [ (~)] chiahetcho [ ;] (trong9})5-1D)' [ ~ ] tucla (a,) = ~.~.(S' . ') S tE S E E EE(t,)' 1'1BOiv~y[(~)] chiahetcho => aJ; =BCs; , (i =1,2) => (a]tpaztz):B => (ACl1, ACiz) : B => B I (At) tzCpAtl tzCz) = At)tz (CpCz) tuc la B I At, tz'-v---' =E A (t)tJ : B t (S) -l/ffi h [ A ] : [ B ] t ()" -1=> -.- . - Tong ;;:v ay - . - rong;;;LJs DE E E E E Tuang Il!, taco [;] :[:] !Tong fl)s'ID tucla[~]=[;] 42

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