VÀNH VỚI LÝ THUYẾT DIVISOR
TRỊNH NGỌC AN
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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.
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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