Embed Size (px)
Citation preview
HOANG XUAN SINH
DAI SO DM CMG (Ti] IS Idn thit tim)
NHA XUAT BAN GIAO DUC
0th, Erich nhi&n xuat ban:
Chu tich FIDQT hem T6ng Giam ddc NGO TRAN AI Ph6 T6ng Giarn diic hem Tdng hien tap VU DUONG THVY
Bien tap kin ddu vd tat ban : TRAN PHUONG DUNG
Bien tap la thuat : BPI CHI HIED
Sad ban in : HOANG DIEM
Sip cha : PHONG Off BAN (NXB GIA0 DUC)
512 21/320 - 05 Me se: 7K108T5-TTS GD - 05
Lol NO! DAU
Cho xugt ban Ian Chit nh&t
Tap II hau nhu ride ldp ddi vai MI) I, man khOng ke den met s6 khai niem nhu Junin vi, pile!) the, ma trdn..., ma melt dOi khi de cap den. Thy (My, ve mat ngOn nga ua ki hieu cua ii thuylt tap hop, tap II trung thanh tun tap I.
O day cheat& tai khong lam vide thaylt minh tang chuong, ccic ban doc co the xem bdn thuylt minh chuong Dinh DO ad cao cap cim BO GM° dye. Sau mai § cita ding chuang, ban doe c6 nhang bhi tap de, hada hilu situ a thuyet hon va ran luyen hi nang tinh town, hod° hieu rdng li thuylt han, cal mat so khai niem mai dua vim trong bai tap. M6i van de duce nhac led thi duoc chit thich chuong, tilt,. mac can van a da duce dua vao, chang han ch V §3, 2 co nghia la van a do da nOi din 0 chuong v, tilt 3, mite 2. Nan van a duce nhdc Lai cling chuong cal van de clang xet thi chi' chit thich tilt ,va mac ;
cling tilt thi chi chit thich mac. Thing cling mot tilt, cite dinh nghia, bd de, dinh li duoc ddnh 36 bang 1, 2, 3...
Cu& cling dl xay Ming mat gulf, trinh Dai el cao cap tang ngity cang Jett han, chang tai rat mong ban dye viii long chi bait nhctng thilu sot_han !thong trinh khoi cart man sach nay. Xin cam an PTS Btu Huy Hien 6 td Dai ad elm khoa Than trttang Dai hoc SW pham Ha NO 2 dd c6 nhieu clang gap lie phan Bat tap.
HaHai ngay 13-1-1972
HOANO XI/AN SINN
3
LOI NOI DAU
Cho Inuit ban litn thu hai
Thy nhiiu Mn Nha xudt ban Giao dqc dP nght chung tot cho tai ban cuan sach Dai ad cao cdp tap II, nhung chung tot dd tit chili ui dd co cudn Dai s6 lib S6 hoc caa giao su Ngo Tittle Lanh. Nhung trong qud trinh day hoc, cluing tdi thdy coon Dal sd cao cdp tap II duce sink uien Mc trubng DO hoc Su phqm dung dl On chi, con sink Men cac trubng Cao clang Su phew' lai clang nhu tai lieu chink khda, cho nen chung tesi nhan lei Mt Nhit xudt ban Gieto due cho in lqi coon Bach nay.
Tong cudn setch tai ban chung tOi dd. lam Mc Idea sau :
1) Chaa lai mat s6 cluing minh hay phat biltt dinh li cho ngan ton han, hay khOng thita.
2)- Cho them §3 trong chuang I, nOi so luoc ui cite tien di elta thuylt tap hop, mat dieu can thilt cho ngubi gidng yet cling Mn thilt cho sink vien co tri to mb .khoa hoc.
3) Them vi du, bai tap ve vanh chink va vanh Oda, hai loci yank d6ng vat 'tre quan trong trong S6 hoc.
Chang tot Iran trong cam on cdc.bqn clang nghiep dd co ahung 9 kiln (long gdp va Nho xudt ban .Gido due dd nhieu Ian a ngh‘ cho tai ban.
Ha NO new 21-12-1994
HOANG XUAN SINE
4
CHUONG I
TAP HOP VA MAN HE
§1. TAP HOP VA ANH XA
1. Khai niom tap hqp
NhUng vat, nhung dai tuong twin hoc... duoc to tap do mat tinh chat Chung nao do thinh lap nhUng tap hap. Day khong phiti la mat dinh nghia ma la mat hinh anh true quan cita khai niam tap hop. IA thuygt tap hap trinh bay a day la mat It thuygt so cap theo quan digm rigay thc.
Ngttai to ndi : Tap hop the hoc sinh trong mat lap, tap hop the lap trong mat trtamg, tap hap N the so to nhien, tap hop Z eac so nguyen, tap hop Q cite s6 huu ti, tap hop R cac sti time, tap hop C the so phtte...
Cite vat trong tap hop X goi la the phan td eim X. Kt hiau x C X doe la "x la mat phgn tit tha X" haat "x thuae X P. Phit dinh cim x E X ki higu la x X.
'Pa him hai tap hap A va B bang nhau va vigt la A = B kbi va chi khi mai phgn tit thuac A thi thuae B va dim nghia la the quan ha x E A va x E B la twang throng. Nhu vay A = B khi va chi khi chting chtla the phan td y nhu nhau.
2. BO Win caa mat top hqp Dinh nghia 1. Gid su A va B la hai tap hap. Ta hi . hieu
A C B quan he sau day voi mai x, x E A keo theo x e B.
Ndi met each khac, quan he A C B cd nghia la mpi Win tit Ma. A dau thuec B.
Quan he A C B la quan bao ham, doe la "A chilit trong B", hoac chat* A", hoac "A la mot be phan cita B" hoac "A la mot tap hop con dim B" va ngudi ta cling vigt B 7 A. Phil dinh vim A C B vigt la A Q B hay B A A.
Dinh li 1. Quan he bon ham c6 the tinh chat sau :
(i) Cdc quan he A C B vie B C C keo theo quan he A C C.
(ii) Mudn co A = B can nit cla a 5 A C B va.B C A.
Cluing mink. (i) Ta hay lay met phan tit thy g x E A. Vi A C B nen x E. B. Nhung B C C nen x E C. fly vol mai x, x E A keo theo x E C, tut la A C C.
(ii) than nhien. n Thttang met be plan A Ma met tap hap B dutqc xac dinh
beri met anti chat C nao dd, ma mpi phan tit dm tap hop B thda man tinh chat C sa la phalli tit ciut tap hem A. Ta ki hieu nhu sau
A ={xEBI x cd tinh chat C
va. doe IA : "A la tap hop tat ca Mc phan t8 x E B ma x cO drib chat C ".
Vi du. - Xet tap hop Z Mc ad nguyen va be phan A the ad nguyen chgn, ta vie%
A = {xEZI x chia hgt cho 2 }.
3. Hiau cOa hai hqp
Dinh nghia 2. Cho hai tap hop A va B thy .P hIP A - B={xEAlx0B}
gel la higu cart tap hop A oh tap hap B. Ngu B c A thl hieu A - B g9i la phan bit dm tap hqp B trong tap hap A va can ki hieu la CAB.
Dinh Ii 2. Gid sit' A vd B lit nhung 60 phdn caa mot Sp hop X, the' thi
(i) X - (X - A) = A.
(ii) Cite quan h@ A . C B od X-B C X- A lit Wang duong.
Cluing mink. (i) Tap hop X - (X.- A) gem cAo phtin to x E X sao cho x e X - A, We la gam the phan tit x E X sao cho x e A.
(ii) Gia sit ACB.VI quan he x E A ken theo quan ha x E B, nen quan ha ± B keo thee x e A, hie la quart ha x E X - B keo theo quan he x E X - A. Dao lai gia sitX-BCX-A. The' thi bang If Juan twang to nhu tren ta cd X - (X - A) C X - (X - B), tilt la then (1), A C B. n
4. Tap hqp rang
Gia sit X la mat tap hop, X cling la mat ba phan dm X digu do cho phep ta xet tap hap 0 = X - X got la bo phan rang caa X x e 0 co nghla la x e X vb. x X. R6 rang khong c6 mat phan tit x nao cita X Lai c6 filth chat dd.
Tap hop X - X = 0 khong pho thuac vao UV hop X. Nth each khac, ta ed
X- X= Y- Y Arth so) X, Y.
non fly, to cd the' coi X - X va Y - Y chila cat phan tit y nhu nhau vi chting chano ed phan to nao ea (xin thing col day la mat chitng minh).
Tap hop X - X = 0 khong ph6 thuac van tap hop X, vi If do do, ta goi nd la tap hop ring. Tap hap nay khong c6 mat phan tit nao ca.
Ito rang ta c6 0 C X vai mot tap hap X va tinh chat nay dac trung tap hop rang.
5. Tap hqp mot, hai phOn bY Gia sit x la mat vat. Thg thi cd mat tap hap ki hiau {x} chi
gam c6 mat phan to la x. Mat tap hop thuac loaf de got la tdp hop mot phan S.
Bay gib giA sit x va y la hai vat phan Mat. Thg thi cd mat tap hop ki lieu {x, y} chi gam cd hai plan ta la x via y. Mat tap hop thuae lotti d6 goi la tap hop hai phan
Ngubi ta cling djnh nghta nhu vay tap hop ha ban... phan tit. Cite tap hop da ding vol Sp hop rang got la cdc top hap
han, con cac tap hop Mac goi la ale Op hap u(5 han.
6. Top hop cac 60 ph(in mitt mOt top hqp
Gia. sit X la mat tap hop, the t) phan ciut X lap thAnh mat tap hop Id hiau P (X) va goi la tdp hop cdc bd ph4n cda X. Tap hop thy bao gib cling S it nhat mat phtin tv , d6 la X.
'Pa S cluing mirth duac rang, ngu X la mat Sp hop him han gam n phan tit thl P(X) la mOt Sp hop him han gom 2" phan tit Nhu vay cac Sp hop 0, P(0), P (P (0)), P (P (P (0))), P (93 (P (P (0)))), P (P (P (P (0)))))... then the W S 0, 1, 2, 22 rt- 4, 24 = 16,- 2 16 = 65536... phan tit. Tit Sp hop 0 chitng ta da -thanh lap nhang tap hop cri nhigu pit to data mdc trong this° to ta Itheng dgm duac.
7. 71th de cat cUa hai top hqp
Gia sit x vA y la hai vat, tit hai vat nay ta thanh lap mat vat thtl ba ki Mau (x, y) va goi la c4p (x, y). Hai Sp y) va bt, u) la bang nhau khi va chi khi x = u va y = v. Dar Mat ta c6 (x, y) = (y, x) khi va chi khi x = y, digu nay Si Ian thd to ma ta vigt hai vat cim mat cap IA can thigt.
Ta ro thg mb rang khai niam cap nhu sac. Gia sit So ba vat x, y, z, ta Ott
(x, y, z) = ((x, y),
va got (x, y, z) la mat bd ba Milan S
(x', y', z) = (x", y", z")
can va da la
Thvg fly ((x', y'), z') = ((x", y"), z") trong during vdi
(x', y') = (x", y") va z' = z", fly Wring duong voi x' = x",
Gang vay, cho bOn vat x, y, z, t ta dat
(x, y, z; t) = ((x, y, z), t)
Ira ta ggi (x, y, z, t) la mat bt) bdn.
Dinh ngliia 3. Cho hai tap hop X NIA Y; tap hop cac cap (x, y) vat x E X vb. y E Y goi la tich de the tha X uti Y va ki hieu bang X x Y.
Khai them tich da the cci thg ma rang cho trtiOng hop nhigu tap hop. /46u X, Y, Z, T la nhitng tap hop, ngubi ta dinh nghia
XxYxZ=(XxY)xZ
XxYxZxT=(XxYxZ)xT...
Cac phan tii oda X x YxZ lit the be ba (x, y, z) vet x EX, y E Y, z E Z. Cling nhu vay, the phan to the. X x Y x Z x T la the ba bOn (x, y, z, t)voixeX,yEY,zEZ,tET. Cu& ding nen X la mat tap hop, ta dat
X2 =XxX,X3 =XxXxX,X4 =XxXxXxX,...
8. Hop va giao ctia hai tap hop
Dinh nghia 4. Gift sit X th Y la hai tap hop. Ta ggi la hop
mitt X ud Y tap hop ki hien X U Y g6m eac phan tit hoae thug° X hotm thuae Y, nghia la
z E X U Y trong during vbi z E X hoe z E Y.
Ta can e6 thg ndi X U Y gam eac phan tit thuac it nhat mat trong hai tap hop X va Y.
Dinh nghia 5. Gia sit X vet Y la hai tap hop. 1k ggi la .giao
tha X ut2 Y tap hop ld hiau. X U Y g6m the phfin ta vita thuac X vita thug° Y, nghia la
z E X fl Y Wong during veil z E X va z
Ngttbi ta bao hai tap hop X va Y la /thong giao nhau hay roi nhau khi X fl Y = 0, nghia la khi X vb. Y lcheng th phan tit chung nao.
9
Rd rang to eat the quan ho
XnYcXvkY,XUYD.XvkY.
Ngoai ra, gia sit Z la mot tap hop thy y , mu6n cho Z C X va Z C Y, can va du coz E X vb. z E Y voi Inca z E Z, nghia la z E X n Y, tilt litzcxn Y. Nhu vAy X n Y la tAp hop Ion nhat trong tat ca the tap hop Z vita chda trong X vita chile. trong Y. Cling vay, 'math Z chda ca X va Y can va du la Z chda X U Y ; nhtt the' X U Y la tap hop be Witt dada ca X 14n Y.
Dinh Ii 3. Poi cot AV lit(t) A, B, C tit( X tuy 9, to co :
(i) Tinh chat giao hoan ,
AnB=Bn A,
'A UB=BU A.
(ii) Tinh chat kit hop
A n (B n (A n B) 11 C,
A U (B U C) = (A U B) U C.
(iii) Tinh chcit phan phdi
A n (B U -» (A n B) U (A C),
A U (B n = (A U B) fl (A U C).
(iv) Gang that Do Mo6c-gang
X - (A U B) = (X - A) n (x- B), X - (A n B) = (X - A) U (X - B).
Ching minh. (i) va (ii) hidn nhien. 11k hay chdng minh tong thde that nhat cue (iii). Gia thx EA n (B U C), didu do c6 nghia la x E A va x Dandle it nhat mot trong hai tAp hap B, C, chAng hen x e B. Vtiy. x E A 11 B, tilt la x E (A n B) U (A n C). Dao lei gia sat x E (A 11 B) U (A fl C), lieu do co nghia la x thuOc it nhat mat trong hai tAp hop A n B, A n C, chinghanx EA n B, tae lax E A vax EB,vAyx EA va x e (B. U C) do do x E A n (B U C).
Ta ehting minh cOng tilde thd hai caa (Ili). Gia thx EA U U (B n C), dieu d6 cat nghia la x thuSe it nhat mot trong hai
10
tap hop A,BnC, chAng han x E A. \ray x E A U B Ira xEAUC,titclaxE(AUB)(1(AUC).NauxEBnC thi ta cdx€BviixeC, the la x E A U B va x EAU C,
\ray x E (A U B) n (A U C). Dao lai, gia x E (A U B)
n (A U C), diau do cd nghla la xEA U B va xEA U C. x EA U B cd nghia la x thuac it nhat mat trong hai tap hop A, B. NauxElithixEAU(BilC).NanxtZAthixE B, vavixEAUC,nenxEC.VityxEBDCvadoddxGAU U (B n C).
• (iv) Gia sit x E X - (A U B). Dieu dd ea nghia la x e X
vfleAUB, tticlaxE2Cvax(I6Avaxit B. VayxEX-A
va x E X - B, tic lit x e (X - A) tl (X - B). Dtto lai gilt sit
x E (X - A) 11 (X - B), dieu do c6 nghla la x E X - A va xEX- B, tdclaxE/CvaxAvaxcEB. Vayxe.Xvit
x (it A U B, tile la x e X - (A U B). D61 via tang thdc thd hai dm (iv) ta co the" Ching minh Wong tp, hoac gp dung tang tilde thd nhat mitt (iv) va dinh II 2 (i) ne'u A, B C X. Ta xat, via A, B C X
A n B (X - (X - A)) n(X-(X- B)) =
= X- ((X - A) U (X - B))
Vity X - (A n B) = (X - A) U (X - B). n
9. Anh xa
Dinh nghla 6. Gilt sit X v& Y la hal tap hop da cho. Mat dnh xit f tit X din Y la mat quy Sc cho Wang ting vat mai phtin tit x dm X mat phan tit 'tan dinh, Id hieu f(x) caa Y. Ta viat
f f : X Y hayX—>Y
x f(x) x f(x)
Tap hop X gni la ngu6n hay mi.en the d nh va tap hap Y g9i la dich hay mien gid tri coa anh xit f.
Vi du. 1) Xet tap hop N cat s6 to nhien va tap hop Z s6 nguyen khang am nhe han mat sic nguyen during da cho m.
Voi myi x E N ta hay chia x cho m vA (Woe mat s6 du kJ bleu la f(x), S6 f(x) thuOc Zm Turing ling
x J—n f(x)
xac dinh mat anh xa
f : N
2) Xat tap hop the s6 thvc R. Tutting ling
x 1 n x2
xac dinh mot itnh xa td It d6n R 3) Gift sit X = (1, 2}, Y = (a, b, c}.
Thong dng
1 I—. c
2 1—n a
xac dinh mat anh xa tV X dgn Y.
4) Gift sit X = Y = (1, 2, 31.
Taring ling
1 "-.3
2 2
3 I—. 1
xac dinh mot anh xa tit X dgn
Qua dinh nghia mkt anh xa, cluing ta they rang khai niam Emit xa la khdi niam m6 rang cue khai niam ham s6 ma ta gap
trvang ph6 thong. Cac ham sd ma ta gap 6 trttang phd thong la nhfing anh as ma - ngugn Ira dich la tap hop Mc sd thtte It hoac nhf.tng b0 phan cue lt, va 56 ?Tx) Wang ling yeti s6 x lit mat bigu thdc dai ad hay mat bigu tilde throng gist , chAng han :
f(x) = 21c2 — x + 3 hay f(x) = 5sinx.
Prong dinh nghia anh xa ta that' cite tap hap nook' va filch khOng nhat - that lit nhitng tap hop s6 va phan to f(x) toting ttng voi x lei cang khong phil la mat bigu thdc dal s6 hay luting giac !
12
Trong Giat doh cluing ta thuting co railing bai tam ye ye d6 thi eta met ham se. 6 day doing ta cling hay dinh nghia
da thi eta met anh
Dinh nghia 7. Gia su f : X -• Y. BO phan l eta X x Y
gem. Se cap (x, f(x)) vet x E X goi IA do 'thi oda dnh xe
Nhu vay, cho met anh xa f : X -• Y, ta dune met ybe phan
I' eta X x Y cri tinh chat : vbi moi x E X, c6 met vit chi met
cap, co phan to that nhat IA x, thuOc r. Dim lai, cho met be
phan P eta X x YS tinh chat d6, thi r cho ta met anh xa
f : X Y ma d6 thi lit F. Cho nen lived ta d6ng nhat anh
xa f vat d6 thi eta S la met be phan eta doh de St X x Y.
10. Anh MTh
Dinh nghia 8. Gia au f : X -> Y la met anh xa da cho, x
la met phan tay f eta X- A la met be phan thy 9 eta X,
B la met be phan thy g eta Y. TM thi nguoi ta goi :
- pa) la dnh mitt x bdi f hay gid tri can anh xa f tot di dm x.
- f(A) = { y E Y I tantaixEAsscho f(x) ei- y} la
dnh Cela A bdi f.
(B) ={ x E XI f(x) B ) la too dnh todn phan caa
B ben f.
Dac HO vet b E Y, r ({b}) {xEXI f(x) = b }.
don gian ki hieu ta vidt f I (b) thay elm ft ({b}) va goi la tao
anh town phan eta b bbi f. Mei phan tit x E f (5) goi la met
tao (ink eta b bel f.
Ki hieu f(A) la met diau lam clang vi f(A) chi S nghia khi
A E X. RO rang ta c6 f(0) = 0 vdi moi f Ta chdng minh da
Bang the quan he-:
- A C hi(A)) obi moi b0 pia A act X
- B t(rim)) ubi moi b0 phGn 3B oda Y.
Nhung ta khong cci quyen, trong cat quan he ay, thay the
dilu bao ham bang dau citing Unit. Chang han, trong vi do 2)
13
ctla muc 9, neu lay A = thi ta cd ritiotil = {- 1, 1} va B = {-1, 1} thi ta co fir l (B)) = {1}
11. Dan anti - Toan anh - Song anh
Dinh nghia 9. Anh xa f : X —> Y la met don anh nau Vol moi x , x' E X, quan he f(x) = f(x') keo theo quan he x = x' hay x m x' keo theo f(x) # fix') ; hay vai moi y E Y e6 nhigu nhgt mot x E X sao cho y = f(x). Nailed ta can goi met don anh f : X --, 17 Ia met anh xa mot ddi mat: .
Vi du. 1) Xet anh xa
f R —> R
x x3
Ile rang f la met don anh, vi ngu. x Ira y la nhitng s6 time thi quan he x3 = y3 keo theo x = y.
Thing le lay R, ta lay C thi anti xa
f : C C
x x3
khang phat la don anh nita, vi got to , s2 la ba gia tri can bee ba cua don vi, ta cd
2) GM sit X lit met tap hap, anh xa
—n
x x
got lit anti xa dong nheit cita X H hidu 1,, hose ex Hign nhien lx la don
3) Cho X C Y. Anh xa j : X Y
x t—*j(x) = x goi la don anh chinh hic tit X den Y. Ta cd thg cd nhieu don Ards tii X den Y, nhung don anh j goi la chinh Sc vi n6 duce fly dung met each to nhien.
14
Dinh nghia 10. Ta bao mot anh xa. f : X --s Y la mat tam anh niu f(X) = Y, nci mat each line, Mu Si moi y E
it nlifit mat x E X sao cho y = fix). Nguai ta con goi mat than anh f: X —. Y la mat drift xa fit X len Y
Cac anh xa trong cac vi du 1) va 2) la nhung than anh.
Dinh nghia 11. Ta bao mat anh xa. f Y la mat song anh hay mat anh xa mat d6i mat tit X len Y, Mu ud vita la don anh vita la town anh, ndi mat each 'Khoo Mu vdi mai y E Y co mat S. chi mat x E X sao cho y = fix).
Chang hon anh n (tong nhat 1, la mat song anh via mai X.
12. Tich anh xa
Dinh nghia 12. Gib. sit cho
f: X—>Yvag Y Z.
Anh xa X Z
x gr(ln)
goi la tich oda anh xa f ud (nth xa g, Id hibu gof, hay van ta. 0
Dinh li 4. Cid sit cho
f:X—>31,g Y Z, h Z—>T
Thi thi
h(gf) = (hg)f.
Ta bdo phep nhan ode arch sa c6 tinh chit kit hap.
Chzing mink. Ta cd Si mai x E X :
(h(gf)) (x) = h(gf(x)) = h(g(f(x)) =
= (hg)(f(x)) = ((hg)f) (x). n Do do ta ki hiOu h(gf) = (hg)f bang hgf va goi la tich ena
ba anh xa f, g, h.
Chia y rang Mu f : X —> Y la road anh xa bat ki thi ta
fix = 1Y1 ft
15
Dinh nghia 13. Gia sit f : X Y va g Y X In hai anh xa sao cho
gf = lx Ud fg = l y
Th6 thi g goi la mOt dnh xr) ngucc clic) f.
Tit dinh nghia ta suy ra f cling 121 mOt anh xa %edge ohs. g.
Dinh sau day cho ta bi6t khi nao mot anh xa cd anh xa tigurele
Dinh li 5. Anh xa f : X —> Y cO mOt anh xa ngitqc khi va chi khi f la mr)t song dna.
Chang mink. GU sit f cd mOt anh xa nguroc g Y X.
Theo dinh nghia 13, ta cd
gf = lx va fg = 1 y .
tdc la
g(f(x)) = x ved mot x
Xat quan hO
fix) = f(x), ta suy ra x = g (f(x)) = g (f(x)) = x'.
f la mOt don anh. Bay gio gia sit y la mOt phan td thy g cala Y. Dht x = g(y) C X trong clang thdc f(g(y)) = y, ta ddoc y = f(x). V4y f la mOt to/in anh.
Dito lai gia sit f la mOt song anh. Quy Sc oho thong ling v6i m6i y e Y phan tit duy nhat coa f l (y) xac dinh Wit anh xa g : Y X va ta thay ngay
gf = lx va fg = l y. •
Nhu vityf:X—n Ycci mOt anh xa notoc khi va chi khi f la song anh, va trong trddng hop dd ta cd mOt anh xa ngdgc g : Y —*X cua f xac dinh b61
y 1-r g(y) = x, sao cho fix) = y.
Ngoai anh xa nmierc nay, f can co anh xa ngutoc nao khac khOng ? rlh co
16
Dinh li 6. GM set g : Y X uh g' Y X Zd hai anh x¢ nguye cast f: X Y. The thi g g'.
Cluing mink. Ta c6
gf = 1 x fg' = 1 y .
Tit do
g = g y = g(fg) = (gf)g' lxg' = g'. •
Nhv yay nab f : X —1, Y co anh xa nguac thi anh xa ngttac la duy nhat, xac dinh bed
y x, arta x la phtin td duy nhat caa. f 1 (y).
Do lam clang ngathi ta cling ki hieu phan td duy nhat x cita 1 (y) bang f ya do dci ngmai ta Id hieu anh xa
la anh xa ngttoc oda f, bang f 1 . Vi f la anh xa ngtotc oda
ri nen f= (f 1 ) -1 Ta cd 1X1 = lx .
He quit. Cho hai song anh f : X Y u4 g Y —1. Z. The thi gf : X —> Z Id mkt song anh.
Cluing mink. Ta cd
(gf) (f'g') = pi) = glyg_i =
(fig-i) (gf) = = = .50
fly = (gf
13. Thu lima va ma rOng anh xa
Dinh nghin 14. Gib. sd f : X Y la mot anh xa va Ala mQt be phan dm X, anh xa
g : A Y
x 1—n g(x) = f(x)
goi la cat thu hep ella anh xa f Mt° kJ, phkti A ytt Id hieu la
g = f , can anh xa f goi la cat and rOng elk g tren tkp hop A
X.
17
14. Thp hqp chi s6
Gia sit / la mat tap hop toy g khge Tong ma cat phan to dole ki hieu lit a, /3, y... va f la mat anh xa
f: I
'Pa Id hieu
f(a) = x,,
tip) = xfi
= xy
Ta bac the phan tU xa , x13 , xy thanh lap mOt hp nhung
phan td cda X dupe dank s6 bdi tap hop I, Id hiau la (xa)a E can tap hop I gal la tap hop chi s6. Nau the xa , x13 , xy . la nhiing tap hop thi ta gal (xa), c la nt9t hp tap hqp dcinh s6
bdi tap hop I. Ndu cac phan tit dm X la nhUng ba phan cim mat tap hop E, tire la ta ed xa , xp , xy C E, thi ta got (xa)a e / la met hp nhang by phan mia tap hop E.
Thoc re cluing ta da thay vide danh ad trttoc day rat. Trong &tat dal chung ta thuong xet nhung day s6 Utile no, u 1,. u2, Digit &I co nghia lit ta da danh sd bang clic s6 to nhien 0, 1,
Vi du. Gia sat I = { 1, 2, 3 ), X = { a, b } va do dci 93 (X) = { 0, { a }, b }, a, b } }, va
f : r —P(X) 1 •—n A i = f b 1
2 A2 = { a, b
3 A3 = a, b }
/chit vay ha (Ai ), e i gam ba tap hop A 1 , A2, A3 trong do
Az = A3 .
18
15. Dap, giao, tich de cite mot hp tap hqp
15 day, chang ta hay mo rang the phep toan hop, giao, tich ra mat s6 thy 3: flitting tap hop.
Dinh nghia 15. Gigt sit (2(a)aei la mat h9 tap hop. lb Ail
la hop cua hp d6, va kt lieu bang U Xa tap hop the x sac a E
cllo x thuac It nhat mat tap hqp cua 119 (Xa)aef.
Dinh nghia 16. Gigt sii (21a)aci. la mat h9 tap hop. Ta goi
la giao edit hp c/6, va Id hiau bang r) X , tap hap die x sao ezel
cho x thuac tat ca cite tap hop cart h9 (Xa)aei
Dinh nghia 17. Gilt sit (20aei lgt mat 119 tap hop va
X = U X la hop cna ho dd. Ta goi la tich d8 cac cua hp a E /
(Xa)ae 1!va U hieu bang 11 Xa tap hop can 119 (xa)aer nhilng ael
phan to cart X sao cho xa E Xa via mot a E L N6u cdc tap
hop .2c, den bang mat tap hop A, till tich d0 cite cua 119 (Vac/ goi IA idy thita 'de cot b4c I cua hlp hap A va ki lieu
la
Trong trueng hop I = (1, 2} ta lai tim thity hop, giao, tich de the cua hai tap hop.
Vi du. 1) /Cat ho tap hop (In)ne N danh s6 bat cite s6 tO
nhien 0, 1, 2,... v6i
In = (0, 1, n},
tha thl U In = N n E N
(1 I = In = {0} .
neN
2) Lay thita de cite RN la tap hop clic day s6 Rule (n o, , 14 )
19
BAI TAP
1. X& tap hap {A 1 , A2,..., ma aid phan to A t , A2,..., la nhitng tap hap. Chang minh ed it nit& mOt tap hop Ai kliOng chtla mOt tap hop nao trong cac tap hap con lai.
2. Cluing minh to S A - (A - B) = B khi va chi khi B C A.
3. GiA sit X 14 mOt tap hop cd n phan tit va r la mOt s6 to nhien, 0 C r C rt. Tinh :
a) S6 cac b6 phan Se. X ed r phan
b) S6 cac phan to cila
4. Bi4u than hinh hoc cac tap A x B voi
a) A= {x ER I 14X4 3}
B = R, tap hap cac s6 thitc
b) A = B = Z, tap hop the s6 nguyen.
5. Bigu din hinh hoe tap hop X em the diem (x, y) caa mat pang dg cac S clang (x, x) ved 0 x C 1 hoc S clang (x, x + 1) v6i x 3 0.
6. Chdng minh •
a) A UB =A khi va chi khi B C A.
b) A f1 B = A khi va chi khi A C B.
c) A UQ = A.
d) A n 0. 7. Tap hop X 6 bai tap 5) S pilaf la d6 thj cua mOt anh
xa tit R dgn R ? 8. Tap hOP G = {(x, 4.1 x < U {(x, 0) I x 3 0} CO
phial la d6 thl atla mOt anh xa tit R den R ? Bigu (Ban hinh hoc tap hdp dd.
9. Tap hop
1 G= {( x. x —1)1
x R, x
20
cd th6 coi nhu da thi caa mat anh xa the Tao ? Bleu diOn hinh hoc tap hop dd .
10. GM wit f : X -> Y la mat anh xa A va B la hai bQ phan dm X, C va D is hai bQ phan cua Y. Chung minh
a) f(A U B) = f(A) U f(B).
b) f(A n B) C f(A) fl f(B).
c) U D) = ft(C) U riay.
d) f'(C n = ri(c) fl rkm. e) f(X - A) D f(X) - f(A).
ri(Y - C) = X - r{ (C).
11. Gia sit n la mat sd tp nhien cho triteic, f la mat anh xa tit tap hop cat se to nhien N den chinh nd duoc xac Binh bdi
n - k new k < n f(k) = {n + k nen k an.
f ad phdi la don anh toan anh, song anh kitting ?
12. GM saf:X-.17 vag:Y->Z1a hai anh xa vah = gf la anh xa tich oda f va g. Chung minh :
a) INI6u h la don anh thi f la don anh, lieu them f la than anh thi g la don anh.
b) Neu h la toan anh thi g la than anh, nen them g la don anh thi f la toan anh.
13. Cho anh xa f : X -> K Chung minh f la mat don anh khi va chi khi cd mat anh xa g : Y -> X sao cho gf = 1x. (X m 0).
14. Cho anh xa f : X -> Y. Chung minh f la mat than anh khi va chi khi cd mat anh xa g : Y X sao cho fg = ly .
15. Cho ba anh xa f : X Y va g, g' : U -> X. Chting minh : a) Neu f la don anh va fg = fg', thi g - = g'.
b) bieu veil moi g, g' ma fg = fg' keo theo g = g', thi f la mat don anh.
16. Cho ba anh xa f : X -> Y va h, Y -> Z. Chung minh Yang neu f la mat than anh va hf = h'f thi h = h'. Nguac lai
21
n5 veil moi h, to th h f.= h'f keo theo h = h' la mat
than anh.
17. Chang minh nau c6 mat song anh tit X d6n Y va mat song anh tit X dgn Z, thi th mat song anh to Y dan Z
18. Cluing minh rang mu6n cho mat ba phan G tha tich 66 the X x Y la d6 thi dm mat anh xa tit X deli Y thi can va du la anh xa (phep chit') :
G X (x, x
la mat song anh.
19. Gia sa (Adaei ut mot ho !Ailing Ith pilau tha mat tap hap X, B la mat tap hap thy y. Chang minh
a) U Aa D A„ vbi moi a e I . a el
b) n /la C vbi MO a E I. aef
a) B n AG) U (B n Aa).
aEl aEl
d) B U (n A„) = n (B U Ad.
a E a el
e) X — (U .4a) = n -
aEl aEl
0 X (n = U (X —Aa)
aEl aEl
20. GM siti:X—>Ylit mat anh xa (Aa)ae la mat ha nhfing ba phan efia X, (Bfl)fi e j IA mat ha nhftng ha phan cila Y. Cluliag minh
a) AU Aa) = U Ma). ael aEl
b)An Aa) C n f(A„). aef aEl
22
c)ri(u B1) = U ri(sfl).
fie/ fie/
d)r(n = n
fie/ 13E/
21. Cho hai tap hop X.vA Y . Ta ki hieu bang Hom (X, Y) tap hop tat ca the anh xa tit X den Y. Chung mirth
a) C6 met song anh tit Horn (X, Y) den YX .
b) Neu Y chi c6 hai phtin to thi e6 met song anh tit Horn (X, Y) den P(X).
c) Ttif a) va b) hay suy ra neu X ad n phan tif, thi P co 2" phtin tit.
§2. QUAN HE
1. Chian hai ngoi
Trong §1, 9 thong ta da dua vao khai niem anh xa. Met anh xa f : X Y cho thong dng voi mei phan to x E X met phtin t0 y = f(x) e Y. Nhu vay,cac phtin to x, y c6 met.quan he viii nhau, quan he dri la y = f(x), hay ndi met each khae, quan he do IA (x, y) e G, yeti G la dd thi tha f. Vag plat hon, ngt.tai ta nghi den met each ghep cap nhang phtin to curt X yea nhung phan to caa Y de thanh lap met be phan caa X x Y, va goi each ghep cap do la met quan he hai ngei.
Dinh nghia 1. Gia sit X Ira Y la nhilng tap hop. Met quan he hai ng0i t0 X dgn Y la met be phan S cua tich de the X x Y. Ta bao, vai hai plain to a E X va b e Y, rang a c6 quan he S yea b ngu va chi neu (a, b) E S. Ta viet aSb.
D5 thi G aim met anh xa f X Y cho ta met vi du ve quan he hai ng6i, va tong throng hop nay ngutoi ta vial b = ;(a) dui khOng vie% aGb. Met anh xa cho ta met quan he hai ng6i, nhung dao lai khang dung (§1, bai tap 5).
23
Nhu \Tay, mat anh xa cho ta mat quan ha hai ngbi dac Mat. Ngoai quan h@ hai fled quan trong nay, Wan h9c con cd hai loaf quan ha hai ngdi quan trong nita, do la quan he tudng during va quan he thti to.
2. Quan ha Wang Wang
Dinh nghia 2. GM sit X la mat tap hop, S IA mat b0 pan oda X x X. The thi S goi la mat quan he Wang duong trong X ngu va chi ngu cac digu kien sau day Woo. man :
1. (Phan xa) Vdi moi a E X ; aSa.
2. (D6i ming) Vdi moi a, b E X ; ngu aSb, thi bSa.
3. (Bac cam) Vdi moi a, b, c E X ; ngu aSb va bSc, thi aSc.
Ngu S IA mat quan he tuong duong, thi nguoi ta Hwang ki hiau S bang - va thuong doc aSb (a b) la "a Wong during veil b".
Vi du. 1) Dgu bang thudng dung trong 86 hoc thong Uniting We s6 thtic lit mot vi du quen thuac ye quan he Wong throng. 'Prong trudng hop dd tap hop S la dyeing thAng y = x cua mat phAng de can R2
2) gh xet quan he &Mg du mod 5 chang han trong s6 hoc. Hai so nguyen tn, n gal la clang du mod 5 ngu m - n chia hgt cho 5 RO rang quan he nay la mat quan he Wong during trong Z. Ta hay ki hiau bang C(i), i = 0, 1, 2, 3, 4 tap hop cac s6 nguyen tag duong vol i
C(i) = {5x + i I x E X}, 4
thg thi moi s6 nguyen thuac U C(i) va C(i) fl C(j) = 0 voi o
i # j, i = 0, 4 ; j = 0,1,..., 4, 'lb. se Wily We digu kin taring to nhu vay cho mat quan he Wing during thy Y.
3) 1k xat tap hop X cac vecto trong knang gian coa Hinh hoc giai tich. 1k Wm mat vecto a co quan he S voi vecto p khi va chi khi a ding hudng , ding chit, ding m6dun vdf j3. Quan he S r6 rang lit mat quan he Wong duong. rIlt cling ki hiau
24
bang C(a) tap hop cac vectd tudng during veil a, the thi C(a) chang qua la mat vectd to do.
Dinh nghia 3. GM sit S la mat quan he toeing during trong X Va a E X. Tap hop
C(a) = Ix E X I xSal •
gal lit lap Wang duang cast a den obi quan he Wong duang S.
VI S la phan xa nen a E C(a).
Ta thy tac khac rang C(a) co cac tinh chat sau :
(I) C(a) r 0. (ii) x, y E C(a) keo theo xSy.
(iii) x e C(a) va ySx keo theo y E C(a).
136 de 1, Vbi hai phan tit. bat Id a Os 5, ta den co hails C(a) C(b) = 0 hods C(a) = C(b).
Cluing minh. Gia sit C(a) n C(b) # 0. Ta se &sang minh C(a) = C(b). Goi c mat digm thuac C(a) fl C(b). Ta co cSa
va cSb, va do Grill chat clai ring in bac cam, nen a E C(b). Do do vai moi x e C(a), tdc la yea moi x Mang duang poi a, to dew ed x E C(b), tue la C(a) C C(5). Thong to, ta cluing minh C(b) C C(a). fly, ta c6 C(a) = C(b). n
Tit be de trail ta suy ra ngay C(x) = C(a) vai moi x e C(a).
Dinh nghia 4. Ta boo to 'awe hien mat sit chia lop tren rnOt tap hop X khi ta chia nd thanh nhiing Ina phan A, B, C,... khan 0, r ii nhau tang del mat, sao cho moi phan tit ciao X thugs mat trong cat 60 phan do.
Dinh fi 1. Gid sit' X la mat tap hop, S la mot quan he Wang duong trong X. Thd thi cite lop Wong duang phan biOt aria X dee vOi S thanh lap mot sit chia lop tren X.
Oning minh. That vey, yea moi x E X, ta ed x C C(x). Con hai lap tieing dung phan biet la rbi nhau thi do 136 de 1. n
Nhu vay cho met quan he Wong dining S trong mat tap hop X, ta dude mat sit chia lap tren X, do la vise chia X
thanh cac lap Wong &tong. Dinh li sau day cho ta thay dao lai sung
Dinh li 2. Gid su ta co mot sy chia 16p tran mot tap hop X ail A, B, C,... la cac ba phlox caa X do su chia lap. The thi co met quan ha twang duong duy nhat S trong X sao cho cac lop twang duong caa X clai adi S la Sc ba phan A, B, C,...
Viec chUng minh dinh li tren, xin dash cho doe gia, xem nhu bai tap.
Dinh nghia 5. Gia atl X la mat tep.hop, S la mat quan he Wang duong trong X. Tap hop cac lop twang duong phan biet cila X ddi vol S gni la tap hap thuang act X trail quan he twang duong S va duac ki hieu la X/S.
Vi du. Xet quan he &Mg du mod n trong SO' hoc (n la mat s6 nguyen throng oho trtidc). Hai s6 nguyen x, y gal la dang du mod n ndu x - y la bai coa n Rd rang quan he nay la mat quan he tuang diking trong Z. Tap hap thuang cua Z Hen quan he ddng du mod n ed n Idp Wang during :
• C(0), C(/),... , C(n-1) voi C(i) = fnx ilxe = 0,1, n - 1.
3. Quan h0 thii tir
Dinh nghia 6. Gab. sit X la mat tap hap, S la mat ha phan cfm. X x X The thl S dune goila mot quan he thil ta trong X (hay ngubi ta con gal S la mat quan he thit to gida cac phan tQ mia X) ndu va chi ndu cac didu Man sau day thda man
1. (Phan xa) Vol moi a E X : aSa. 2. (Phan ddi ring) Vdi mai a, b E X ndu aSb va bSa, thi
a = 6.
3. (Bac eau) Vol mai a, b, c E X ; aSb va bSq thi aSc.
NgUdi ta bao mat tap hap X la 84 this to ndu.trong X co' mat quan he thu to.
Vi dy. 1) Quan he E trong tap hop cac s6 to nhieh N la mat quan he thu to.
26
2) Quan he chia het trong N : ngded ta Id hieu al b va dee
"a chia het b".
3) Quan he bao ham C gifta cac be phan eta met tap hop X
Trong vi du 1), vol a va b try y to luon cd a g.. 6 Ileac
a. NgUdi ta gel met quan he that to nhu thy la town phan.
Trong vi du 2) khong plied ta hien luen cd al b hoac It( a ved a,
b thy y, clang Ilan vai a = 2 VA. b = 3. Dieu dd cling thy ra
vat vi du 3). Ngudi ta ban I vat C la tinting quan he thd to 60 ph4n.
Neu S la mot quan he thd tut trong X, thi rived ta thudng
Id hieu S bang g.. (bat chub° ki hieu quan he thil to thong tinging eta s6 nguyen hay s6 flute) va doe a g b la "a be hon
b". Ngubi ta coi b .?•.- a la clang nghia vet a g 6 va doe la "6
Ion hem a". Ngdoi ta con viet a < b (hay b S a) quan he
va. a # 6" va doe la "a thtte su be ban b" hay "6 thAlc
ski lon hon a".
Dinh nghia 7. Gia. su X lit met tap hop sap thd W. Met
phgn tit a e X gei lit phdn to t6i ties (phan to t61 dal) eta X
neu quan hex # a Ix a) kat) theo x = a.
Vi d,. 1) Trong tap hop cat se to nhien thve stt len lam I, sap that to theo quan he I (quan ha chia het), the Olin tit tOi tigu la the s` nguyen
2) Trong tap hop cat he theta doe lap tuyen tinh eta khong gian vecto IV sap tint W theo quan he bao ham C, cac ha
\recto gem rt \recto la t61
3) Tap hop cac se thuc, ved quan he that to thong thudng, khong cd plign to t6i dai cling khong cd phan to tei tidu.
Dinh nghia S. Girl sit X la met tap hop sap that tu. Mot
phgn tit a E X get la phlin tti be nit& titan to lan nhgt) coo.
X nen, vet mot x E X, to cd a E x( x g a).
Neu met tap hop X sap thd to cd met phtin tit be nhat a thi a la phtin to be nhgt duy nhgt. That thy gib sit 5 6 la phtin to be nhgt, ta suy ra a < b va 6 # a, We la a = 6.
Cling nhan get nhu thy del veil phan ta len nhgt.
27
Vi dzt.• 1) Tap hop the s6 to nhien sap thit tp theo quan he eq phtin tit be nhgt IA 1 va OHM to Ion nhat IA 0. Neu sap
• OM W theo quan he this to. th6ng thuiting, tap hop eac so to nhien ea phgn to be nhgt IA 0 va khOng c6 phan tit lon nhgt.
2) Gia sit X la mot bq phan khac rang thit P (E), tap hop the ba phan elm mat tap hap E. Mu X act phan W be nhgt A (Ion nhgt) ddi obi quan he bao ham, thi A ellang qua la giao (hop) eim the tap hop thuac X. DM) lai, neu giao (hop) the tap hop the. X 1p.i thuiic X, thi dO la phtin to be nhat (Ion nhat) eaa. X. DO.c bi0t, 0 IA phg.» tit' be nhgt va E la phan to len nhat cua P (E).
3) Tap hop the s5 thoc, vbi quan ho UM to thong thuong, khOng ca phan to be Mt) Ming khOng ca phan tit IOn nhgt.
Dinh nghia 9. Ta bao mat tap hop X IA sap Mit tit t6t trau nti la sap thu to va nau moi ba phan khae rang ella X ca mat phan tit be nhgt.
Vi dp. Tap hop the so to nhien N vei OM to thong thuOng la sap Hui to tot.
BAI TAP
1. GM sb f : X —> Y IA mat anh xa, S la ba phan aim X x X Om the cap (x, x) sao cho f(x) = flx ite
a) Chung minh S la mat quan he Wang during trong X. b) Xet tap hop thurang XIS va anh xa.
p : X XIS x C(x).
Chung minh ea mat anh xa duy nhat X/S —> Y sao cho bigu do sau la giao hoan
/ X Y
P XIS
nghia la f = TP •
c) Chang minh f la don anh va trong twang hap f la toan anh thi 7 la song anh.
2. Xet tap hop the s6 nguyen Z va tap hop N' cac s6 tor nhien khac 0. GQi S la quan ho trong Z x the dinh bai
(a, b) S (e, d) khi va chi khi ad = be.
Chang minh S la mat quan ho tuang thong.
3. Gia th S la mat quan he hai ng6i the dinh trong tap hop cac s6 nguyen Z bai cac cap (x, y) vol x, y nguthn va x + y 16.
Chang minh :
a) S khOng phai la 116 thi Mut mat anh xa tit Z dth Z.
b) S khong phai la mat quan Wang (Wang.
c) S khang phai la mat quan ha dui to.
d) Nth d61 gia that mat ehtit bang each cho x + y chain, thi the' nth ?
4. Gia sit X la mat tap hop va T la mat qtian ho hai ng6i phan xa, dot 'tang trong X. Th hay xae dinh mat quan ha hai ng6i. S trong X nhu sau : thy khi va chi khi cd x t = x, xa = y sao cho x 1 Tx2 , x2Tx3,... , xn_ i Tx n . Chang minh
a) S la mat quan he Wang diking va T C S.
b) Val moi quan he tuang dttang H sao cho T C H thi S C H.
5. Xet quan ha hai ngai trong tap hop cac s6 thtic R xae dinh bai tap hop X (*1, bai tap 5).
a) Hay b6 sung X d6 cd mat quan he hal fled phan xa, d6i xang T (tat nhien b6 sung it nil& !). Mau din hinh hoc T.
b) CO T, hay xay dung quan he Wang throng S nhu bai tap 4). Brett din hinh hoc S.
6. Gia th X la tap hop cac ham s6 kha vi xac dinh trail tap hop cac s6 tithe R. GiA sa S la quan ha the dinh bai ySz khi va chi khi dyldx = dzldx vai mai x E R. S cd phai la quan ho Wang ding khOng ?
29
7. Cho X la khong gian ba chigu thong thuang va 0 la mat digm c6 dinh dm X. 'Prong X ta xac dinh quan he S nhu sau : -
PSP' khi va chi khi 0, P, P' thAng hang. a) S cd phai la quan he thong (thong trong X khong ? b) S ad phai la quan -ha tuong <Wong trong X - {O} khangl
Ngu phai, the dinh Mc lop Wong throng. 8. Gia sit X lit mat tap hop va S la mat quan ha thil to
trong X. Chung mirth rang quan he T trong X the dinh bed khi va chi khi bSa thing la mat quan ha thil to trong X.
9. Gia sit X lit mat tap hop va S la mat quan ha tuong diking trong X. Chung minh S khong phai la mat quan ha tilt/ ta.
10. Xet tap hap X = Nn (n ?- 1), vai N la tap hop the s6 to nhien. 'Prong X ta xac dinh quan he S nhu sau :
(a 1 ,..., O) khi va chi khi (a 1 ,. ...; = (b 1 , host cd mat chi se i (i = 1,2,... , n) sao cho ¢ t = b i , a i-i bi-1, a.< b i - -
Chung minh S la mat quan h@ Ulu to toan phtin. 11. Gia sit f la mat don anh tit mat tap hop X cten tap hop
the se tg nhien N va S la mat Tian he trong X xac dinh nhu sau :
xSx' khi va chi khi f(x) s fix).
Chung minh S la mat quan he tht/ W than phan. 12. Cho hai tap hap X va Y. Gal ft. (X, Y) la tap hop the
anh xa tit the ba phan elm X den Y, nghia la neu f E t 0C, thi f IFt mat anh xa cd ngudn la mat ba phan dm X va cd dich la Y. Xet quan ha. S trong 'V (X Y) the dinh nhu sau :
fSg khi va chi khi g la ma rang cim anh xa f.
a) Chung minh S la mat quan ha thu tg.
b) Tim to phtin to UR tido, tai dai, be nhgt, ldn nhgt cda (1• (X, Y) ddi vai S.
30
13. Chung minh nth a la phtin tit be nhat (len nhat) cua mot tap hop X del thi mot quan he thti td S, thi a la phein to tei lieu (t6i dal) duy nhat eim X
14. Cluing minh nth X sap Hill ttt t6t thi X sap thu toan phan.
15. Cluing minh cac tap hop trong the bid tap 10) va 11) la sap tint to tot.
§3. SO LUOC VE CAC TIEN DE CUA Li THUYET TAP HOP
1. Mb dgu
Nhu ta da ndi a. dam chuong, li thuyet tap hop ma ta da trinh bay la mat If thuyet so cap theo quan diem new this. Bay giO chting ta hay gioi thigu so Moe the tien de dia li thuyet tap hop. Ban disc mthn tim hieu ki ea the tham khao nhieu sach, ehang ban "Li thuyet tap hop" cua Bourbaki.
2. KW nidm nguyen thily
Chung ta khong Binh nghia cha tap hop : ta goi nd la khai niem nguyen thdy. Khdi niem thuec yap &toe ki higu nhu ta da biet bang e, cling la met khai niem nguyen they.
Thy hai khai niem tap hop va thuoc vao kheng doge dinh nghia, nhung ehting to lai td an dinh the quy the : dieu gi ta ed the lam duoc va diet' gi ta khOng the lam dude vai hai khai niem dd DO la bay tign de &la Zermelo - Fraenkel, ma chting ta them vao cal thtl tam, tien clg Chan.
Theo iruyen thong, cac ties de dd throe cho dtith ten va tint to sau : tien de awing tinh, tien de tuyin 1ga (hay con goi tien de nOi ham, hay tien de chi re, hay tien de tach), tien de cap, tien de hop, tien de tap ling Sc b0 ph4n, tien de ue han
31
(hay tien de the so tg nhien), tien de ehon Ira sea Ming la tien da thay the. Tien de mei cimg con g9i la tien de thay chi chi tham dv vao li thuyet tap hop khi dua vao kited niem quy nap sieu han va s6 hoe tin/ tut. Tien de dd disc Fraenkel dua vao nam 1922. He thong tien de (khong ad tien de chant thanh lap li thuygt tap hop eua Zermelo - Fraenkel. Mat vai trong can tien de dO n6i len nhiing tinh chat it nhieu hien nhien khi ta phat bigu cluing trong ngen ngit thong thyong.
Triton het cluing ta hay xac dinh the So lit met vat ban toan hoc. Bang the chit (16n, nhe, la tinh, hi lap, v.v.:), the ki hieu tha logic kinh dign, hai ki hieu E va =, va the Su ngoac, chMig ta cci the vigt bat ki van ban toan hoe nao (ma chting ta cling gel lit Gong thus , phat bleu, menh de, khang dinh, v.v...) can thuygt tap hop.
Chung ta se gel la ph& bigu toan hoc hinh that cal ma ta dude bang each 4 dung the quy the sau
1°) x E y, A = B, trong dO x, y, A, B la Mating chit thy y, la nhang phat bigu. Cac plaid Mu del g9i la Sang tong thdc so cap hay nguyen ta.
2°) Mu P vet Q da la Sang phat bleu, the thi
(1P), (P A Q), (1) v Q), (P Q), (P ti Q) la nhang phat bigu.
3°) Mu P la met phat bigu, the thi (B x) (P(x)), (Vx) (P(x)) la nhang phat
3. Tien de queng tinh
(VA)(VB)(A. B (Vx)(x E A x E B)]
True giac, dieu do mu6n nhi met tap hop he/in toan doge the dinh boi Mc phdn to caa no ; hai tap hop A va B bang nhau vi chi nS mei phan tit tha A la phan to eita B ve nmitic lai.
Chzi y : - Gia sit X, A, B la nhfing tap hop. Chung ta da Si rang quan he X e A dot la X thuee A, hay X la met phan tit Ma tap hop A. Thong thuang nivel ta hay vigt quan he do
32
dual clang x e A, nhung chi chi la met att lam dung each vie% trong khi x la met tap hop cling nhtt A.
4. Tien de tuyan Ipa hay nai ham
(VA) (3B)(Vx)ix E A A P(x)) x E B]
True gide, dieu dd cd nghla cho met eeng that P(x) trait met tap hop bin x, ton tai met tap hop B, ma the phan tit la cat ph&n tit dm A, ed tinh chat P(x) (nghia la lam cho c6ng than P(x) dung). Tap help B Mc dd ducic xac dinh duy nhat bai tion de quang tinh. Trot& khi phat bleu Hen de tuyen leta duoi clang da cho, nhieu nha town hoc da nghl rang chi can met cang that la do de xac dinh met tap hop :
{x I P(x)} (4.1)
la tap hop the vat lam cho thug thile P(x) dung.
Nam 1901 Bertrand Russel khaim pha noel ta cd the suy ra met mau than t8 (4.1) bang each xet the vat hi:tang thuec chinh nd. That tray, xet cting thite x x KM dd ehting ta mu& xet tap hop the tap hop khang thuec chinh nd Gift sit
JO hop dd ten tai. fly ta ed the vial (3 BRVx)(x E B » x & x)
Lay x = B, ta di den thug thite : BEB4-13 (4 B.
ye mat lich sit, chinh nghich li dd da d&n tai Ong thtic hien nay tha Hen de tuyen Ma.
Nghich 11 B. Russel doa trait thing thtic x x, dua cluing ta toi dinh li eau day
Dinh li 1 - Khdng ton tot tap hop ma cdc vOt act n6 14 cdc tap hqp.
Chang mirth. That fly, gia sit met tap hop A nhu tray ten tai. Theo tien de tuyen Loa, ta c(5 the xac dinh tap hap B ?is; :
B=jseAlx0x)
Theo dinh nghia dm A, tap hop B la met phfin tit eim A. The thi ta c6 B E B nen va chi neu B B, diet' nay man
oni sb.... 03 83
than vi P va 1P khong the thing thoi dung (nguyen tge khAng mAu thugn),I♦
Th vita they Id hieu clang (4.1) khOng MI& thief chi ra mOt tap hop. Nhung H hieu d6 lai rat Hen. De mixt chits. Mt den do, ta dva tho mOt tit nguyen fluty m6i, do IA khai niem ChOng ta Me (4 1) kf hieu lop the tap hop x thee man tfnh chat P(x). Vi du :
fx I x = x}
la lop tat cd cat tap hap. Lac do ta se n6i den 16p cac nh6m, 16p cite khang gian vecto tren mat trubng.
5. Tien de cop
(Va) (Vh) (3 c) (Yx) [x E c 4-4(x =aVx=b)] Trite ghic dieu, do co nghia cho hai tap hop a va b cd met
tap hqp c co Olen tit IA a vb. b Ica chi Chung. Tap hop do la duy nhgt theo flan de quang tinh.
Dinh nghia 1 - Gi4 sit a va b la hai tap hap. 111 gel la cap (khong xep tint thanh lap beli a va b, tap hop Id hieu fa, b} xite Binh MI Hen de cap. Ngtati ta gal la don to thanh lap Mi tap hap a, tap hqp 4a, a} = {a} chi cd tap hop a IA plain to duy nhgt.
Dinh nghia 2 - 01a sit a va b la hai tap hop. 'Pa gel IA cap sap dui hi the. a (thing thtt nhgt) va tha b (dung thd hai), cap thanh lap bai don to {a} va cap {a, b}. Cap d6 Id hieu :
(a, b) = { fat, ta, b}
6. Tien de hqp
(VA) (3 B) (Vx)fx E B »(3 C) ((C eAnxe C))]
nye gide diet' cO nghia cho met tap hop A ma the !Men tit gam nhung tap hop ki hieu C, c6 mOt tap hop B ma the phgn tit IA the tap hop the van met trong the tap hap C ciut 130 (= tap hap) A. Tap hop dcl IA duy Hi& theo Hen d@ quang filth ya goi la hop eim the tap hop cim be A va Id hieu
U C hay U A hay U ,{C I C e CEA
34
7. Tien de tap hqp hoe be phen
(VA) (3 B)(VXUX E B »X C A)
Trirc Mac ed nghia cho mat tap hop A, cci mat tap hop B ma the phgn tit la the ha phan ciaa A. Tap Imp &I la duy nhgt theo nen de aiding tInh va goi la Op hop Sc b0 plz4n can t6p
hop A, ki hiau P (A)•
8. Tien de chqn
(VI) 0), [(Vi E I, # 0 ) 11X1 z 0)1] e
Trate gist digu d6 cci nghia neu (Xi)jel la mat he kitting rOng
(I z 0) nhung tap hap kitting rang, the thi tich Descartes cart
he dci is mat tap hop khong reng.
Dieu do tong c6 nghia rang mei ho khong rang (Xi ); e
nhitng tap hop khong reng c6 mat ham cher', nghia la mat anh
xa p xac dinh trong I sao cho voi e I, T(i) E
Paul J. Cohen da cluing minh rang flan de chum la clac lap del yea cac Hen de coo. li thuygt Zermelo - Fraenkel. Truck d6 Gadel da chting minh nett II thuyet tap hap cart Zermelo -
Fraenkel la khong matt than, n6 se khong mau thugn neu ta
them van den de chon.
'Prong bail giang Dai sg, chting ta hay dung b6 de Zorn va dinh II Zermelo (thu tar tat). Tien de ellen Wong dttong ved cac
phat bieu d6.
B6 de Zorn - Moi t4p hop E khOng rang sap thzi ikt quy
nap co it nhtft mot phan td t6i dot.
Djnh li Zermelo - Mei 14p hop co the sap tint tot t6t.
9. Tien de ye hen
Djnh nghia 3 - VOi MO tap hop x, ta get la eel ke tiip
ctla x, tap hqp co the philn tit la cac phgn tit eua x va tap.
35
hap x. Vay dd la hop cua tap hop x va don to {x}. Nd &roc Id hien x+ :
x+ = x U ixi. Vol ki hiau dd, a cd
1 = 0+ , 2 = 1+ ,3 = 2+ ... vi 0 = 0, 1= 0 }, 2= { 0, { 0 }]...
Theo each My dung moi s6 to nhien dau c6 the' vial Wong minh. Van de dal ra IA cd ton tai mat tap hap chda tat ca cac s6 to Mien do ? Tien de vd han se tra ]di eau MI dd.
Tien de u6 hen la nhu sau :
(3 A) [ 0 E A A (Vx) (x E Aix` e
Troc giac dieu dd c6 nghia din tai mot tap hop chda 0 va chda cal lid tiep cita m61 phan tit cua no.
Ta cd thd do ding chting minh dinh sau day : Dinh li 2 - Co met tap Op be nhat, ki hieu N, cd the tinh
chat sau
(1) 0 E N,
(ii) Voi moi x e N, x+ E N.
Tap hop N goi la tap hop the s6 to nhien.
10. Tien de thay the
[(Vx) BY) (Vy) ((x E A) A Six, y) e y G Y)]
1(3 B)(Vydy E B •:=:. (3 x)(x e A A S(x, y))i Noi mat each khan, gia ad S(x, y) la mat phat bidu pho
thuac hai bign x va y vi A la mat tap hop. Ta gia su voi moi x E A, lap {y S(x, 3/)} la mat tap hop. Tha thi ten tai mat tap hop dada dung Mc phan to y sao cho S(x, y) la dung vii ' it nhat mat x e A.
Tien de nay dirge stir dung khi to dim van khai niam quy nap sieu han va se tht?
36
CI-WONG 11
NCIA NHOM VA NHOM
§1. NIJA NHOM
1. Phap twin hal ngOi
Lam Dai sd, chu ygu la tinh WEIL ma vi du dign MS la ban phep town dim S6 hoc s cap. Tilde that ma Si lam mat phep toan dai s6 tren hai phan td a, b cua Ming mat tap hop X IA cho Wang dug Si cap (a, b) mat phdn tit the dinh eta tap hop X. Ndi mat each khac, cho mat phep toan 41 sd la cho mat anh xa to X x X dgn X.
Dinh nghia 1. Ta goi la pile], tam hai ngOi (hay con gal St IA phep San) trong mat tap hop X mat anh xa f to X x X
dS X. Gig tri f (x, y) eta f tai (x, y) goi la edi hap thttnh cita
x Ott y.
Cal hop thanh cua x va y thutmg ddoc ki hiau bang each vigt x va y theo mat thd tut nhat dinh Si mat dau 4c trong chc phep toan dat gifta x, y. Trong the dflu ma ngdgi ta hay dung tot nhigu nhgt, la cac dau + va . ; dal vai dgu . ngurdi ta thliong quy tallc be di ; vgi cac dgu do, cal hop thanh eta x va y duos vial x + y, x . . y hay xy. Mat SS toot hai ng6i ki hiau bAng dau + goi la phep tong, cal hap thanh x + y lac dd goi IA tong oda x va. y ; mat MIS toan hai ng6i ki hiau bhng dau . goi la phep nhan,cai hop thanh x y hay xy lac dd goi la tech eta x va y. Ngubi ta can dung cac ki hiau x o y, x * y, x T y, x 1 y... dg chi cai hop thanh cua x va y.
37
Vi du : 1) Thong tap hap N the s6 4t nhign, phep ph4p nhan la nhUng phep toan hai ngtoi ; cal hop thanh eta x
va y E N bai the phdp toan dd ki hieu theo tint to bang 'x + y, xy. Phep hop thanh xY, la mot phep than 2 ng61 trong tap W = N— {0}.
2) Phep trit khong phai la mot phep San hai ng8i trong N, nhung la ma phep twin hai ngti trong tap hop Z the sd nguyen.
3) Tich anh xa la mOt phep toan hai ngti trong tap hap the anh xa tit ma tap hop X don chinh no.
4) Tich ngoai the vecta a A ft cua liinh hoc giai tf ch la mot phep toan hai ned trong tap hop can vista S ding met didm g6C 0 dm khong gian 3 Sian thong thubng.
5) 'rich ma trail la ma pile') toan hai ngti trong tap hap the ma tran sting cap n.
Sau day, trong the If luan tang 'pat, ta se via cai hop thanh tha x va y la xy n8u khong cd li do nao khign ta phai via khan.
Dinh nghia 2. MOt 130 phan A chit X goi la do dinh (d6i vbi phep toan hai ngoi trong X) nat va chi Su xy E A Si moi x, y E A.
Phep toan hai ngai * the dinh trong 130 pan do dinh A bai quan 110 x * y = xy voi moi x, y E A got la eat thu hap van A tha phep than hai ng6i trong X Ngubi ta can Si rang * la phep Loan cam sink troll A 1361 phep than . cua X. Ngabi ta thuang ki hiau phep toan earn sinh nhu phep than dot X.
Dinh nghia 3. MOt phdp toan hai ng6i trong mat tap hop X goi la kit hop ndu va chi n8u ta cd
(x y) z = x (y z)
vai moi x, y, z e x ; la giao hoar Mu $ chi Su to S
xy = yx
x, y E X.
Phep °Ong va phdp nhan trong tap hop the s8 to nhien N la kat hop, giao hoan ; nhdng AO mu hda Ichtong kat hop :
38
(252 # 2(12) , cling kh6ng giao hoiin : 2 1 # 12 . Tich anh xa trong.
vi du 3) la ket hop, kheng giao hoiin (nett X cep nhieu hon mat
phgn tit).
Dinh nghia 4. Gia sit de, cho mat phep than hai ngOi trong mat tap hdp X. MCA phan to e caa X goi la mat don t4 trai
eau phep toan hai ng6i neu va chi nen
ex = x
voi rayi x e X Thong tu, mat phan tit e dm X goi lit mat don
ui phdi cua phep toan hai net nett va chi nett
xe = x
vdi nagi x E X. Trong trudng hop mat phg.n tit e cita X vita la
mat don vi trai vita la mat dan vi phdi, thi e goi la mat don 0,
hoar mat phtin to trung lOp dm pilot) toan hai ngdi.
Trong vi du 1) s6 0 la phan to trung lap cua phdp Ong, so` 1 la phan to trung lap ciut phep nhAn va IA don vi phAi cua pile)) mu hda. Trong vi du 3), anh xa thing nhat lit phan tit trung lap. Tich ngoiti edc vecto trong vi du 4) khong cd don vi tram ciing khong cd dan vi
Dinh li 1. Niu mot phep town hai ngdi trong mat tap hop X co met don vi trai e' ties mat don 0 phdi e", thi e' = e".
Cluing mink. Xet tich e'e" trong X. VI e' lit don vi trai nen e'e" = e". MM khac, vi e" la don vi OW nen e'e" = e'. Ta
suy ra e' = e".
Mat ha quA tdc khAc la
114 qua. Mot phep town hat- ngoi co nhiM nhdt mot plain to trung lap.
2. NI.In nh6m
Dinh nghia 5. Ta goi lit nua Whom mat tap hap X eimg val mat phep toan hai ng6i ket lid!) da cho trong X. Mat nint nhdm
cd phAn tit' trung lap goi la mat vi nhdm. Mat nua nhdm la
giao hodn nett phep toan caa nd la giao
Cac vi du 1) (trit phep mil Ma), 3, 5) trong 1 la nhAng vi du ve nita nhdm, va hon nits vi nhdm. My) be phan do dinh
39
A ciut mot nita nhdm X eting voi phep than cam sinh trot' A la mot nita nhdm g9i la nita nhOrn con cos rola nhdm X
Trong mot nit°. nhdm X, not:* ta ki hi6u gia tri chung eiut hai v6 cda tang tilde
(xy)z = x(yz)
bang ki hiOu duy nhat xyz, g9i la tick eta ba phan tit x, y, z lay thee dui to dd. Cling nhu vay, ta clat
xyzt = (xyz)t
va goi la dell eta bon phan td x, y, z, t lay thee thu to dd. MOt each tang quirt
x„ = (x1 xn _ 1)x„
va g9i lit tich eta n phan t* x i , , xi,.
Tinh chat chinh eta eac phep Wan hai ngei k6t hop la dinh li sau day :
Dinh Ii 2 ((huh II k6t hqp). Gta sa xi , x2 , xn lit rt (n a 3) phan td (phan biet hay khong) mia mat nue: nhdm X, the thi
xix2 xn = (x ... xj)(x xi) Cluing mink. Vi X la mQt nua nhdm nen mOnh de la thing
ved n = 3. Ta gia sit manh de la thing cho k nhan td , WO mai 3 e k < n. 1h ad, theo dinh nghia tick eta nhieu phan tit va theo gia thier quy nap
(x 1 xi)(xi x)) (xm
=[(x 1 ... xi) (xe+1 x,„)] (xm =
= (x 1 ... x,„)(xm x,,) = (x1 ... xiii )[(x„, x„ _ 1 ) =
[(x i xm )(xiii _ 1 )1; = (x1 .. x„ _ i )x„ = x1x2
Dinh nghia 6. Trong rant naa nhdm X, lay thha n (rt la mot to nhien khac 0) caa mot phan to a E X la tieh cat) n phan tit bang a, ki hidu a". Ta co" the quy tae (do dinh li 2)
a" = +", (an)" = arm
40
Trong trvong hep phep town hai ngdi dm X ki hiAu bang clau Ong +, thi tdng cem n phan td deo bang a gal la bei n caa a, Id hieu na. Quy the tren luc dd vidt
ma + na = (rn + n)a, n(ma) = mna .
Trong tadt rida nhdm giao hoan, doh cua ba phan tit khong plat thuds vao thu ttt the nhan tit, ea the'
xix3x3 =x ix3x2 = x2x 1x3 = x2x3x i = x3x iiz = x3x 2x i .
Thug quat hdn ta ed dinh li
Dinh li 3. Trong mdt nem nhom giao hoan X, tich
x ix2
khOng pha thuoc oho tha trt to nhan tn.
Cluing mink. Vi X la giao hoan nen menh de la dung vdi n = 2. Ta gia sit menh de la dung cho k nhan tit, vdi mod
k < n. Ta hay chdng minh
x ix2 xn = xi I 2
trong dd (i 1 , i 2, ..., la met hot vi ena {1, 2, ..., n}. Neu
Xn / it
x= • ta ed the' vigt hai mitt clang Ulric trAn nhu sau theo
Dinh li 2 va tinh giao hoan dm X :
•X• • • Xi Xi Xi
I t - 1 k +
= (X1 ... xi k_ ) (Xi k (Xi 12.1
... xi ))
= (xi, . .. xi , _ I k +1 n
) ((xi ... xi tx„)
= ((xi ... x, k -1
) (Xi k 22 I ... Xi DX„
... X )X theo Dinh li 2 st zk-] 't+1
,,, n
= (x 1 x2 ... xn _ i )xn theo gia thidt quy nap
hay
= x 1x2 xr,. n
41
Vi du. 1) Tap hop CEIC s6 to nhien N vai mat trong the phep town hai nged sau day
phep Ong,
phan
phep lay 1.1CLN (uric chung lan nhat),
phep lay BCNN (bai chung nhO nhat),
la mat luta nhdm giao ham. D61 vdi phep Ong, N con la mat vi nhdm goi IA vi nhdm Ong the s6 to nhien. (Dieu d6 th dung yeti aftc phep than con lai kitting ?)
2) T(tp hop P(X) the ba phan dm mat tap hop X la mat vi nhdm giao hotim ddi vai m61 phep town hal ng6i giao va hop.
Bia TAP
1. Gia sat a va b la hai phiin t8 cfia mat mita nhdm X sao cho ab = ba. Chdng minh (sib)" = eb n vai ITO s6 to nhien n > 1. Ngu a va b lit hai phan tii sao cho (ab) 2 = a2b2 thi ta cd suy ra chive ab = ba khang ?
2. Ggi X la tap hop throng cita Z trait quan ha (tang du mod n (ch I, §2, 2, vi do),
a) Val mai cap (C(a), C(b)) ta oho thong ling lap Wong during C (a + b). Cluing minh nhu thy ta c6 mat tinh xa ta X x X (Ian X.
b) Chling minh X la mat vi nhdm giao loan d6i vai phep than hai nged the dinh a a).
c) Nati vol m6i cap (C(a), C(b)) ta cho thong ling lap Crab), Chdng minh lfic do X cling la mat vi nhdm giao loan.
3. Gib. set X la mat tap hop thy j. Xet tinh xa
X x (x, y) n-• x.
42
Chang minh X la mat tiara nhdm dal voi phep than hai ngoi tren. Nita nhOm dd ed giao hoan kheing ? cO phan tit don 1.1. khOng ?
4. G91 X la tap hop thtiong cita Z x N . trail quan ha tudng
(Wong S xac Binh bbi
(a, b) S (c, d) khi va chi khi ad = be (ch I, §2, bai tap 2). Ta Id hiau cat phan tit C(a, b) dm X bang alb, (a, b) E Z x
a) Vol mai cap (a/b, cld) ta cho thong ang lop Wong during ad + kik!. Chang minh nha vay ta co' mat anh xa tit X x X
den X.
b) Chang minh X lit mat vi nhdm giao hoan dal vai phep
than hai ngOi b a).
c) Neu vbi mai cap (a/b, cld) ta cho Wong dug lop tuong
during aclbd. Chang minh lac &I X cling la mat vi nhdm giao
holm.
5. Xet tich de the N n (a 3 1) yeti N la vi nhOm tang giao hoan cat s6 to nhien.
a) Chang minh cling voi phep than (a 1 , ..., an) ± (b1 , Eon) = (a 1 + b t ,..., + la mat vi
nhdm giao hoan.
b) Trong Nn ta da xac dinh mat quan he that tit (ch I, § 2, hal tap 10). Chang minh n6u a, p e sao cho a < i(3
thia+y<p-tyvoimoiyeN n .
§2. NI1DM
1. Nh6m
Dinh nghia 1. Ta goi la nhom mat nits nhdm X cO the tinh
chat sau : l e . cd phtin to trung lap e ; 2° . voi moi x E X, cd
mot x' e X sao cho x'x = xx' = e (phan tit x' goi la mat phan tit tied rung hay nghich dao cim x). Nhu vay, mat nhdm la mat vi nhdm ma mai phan to deli cd nghich dap.
43
Ngu tap hop X IA hau han thi ta ban ta ed melt nhom ham hum va s6 phan tit cilia X gpi IA clip caa nhcim. Nau AM) toan hai tied trong X IA giao hoan thi ta bao ta th mat ahem giao hoan hay nhom aben.
Vi du. 1) Tap hop the so" nguyen Z Ming vat phep tong thOng thttong la mat nhdm giao hoan ma ta goi la nhdm Ong cac s6 nguyen. Cung vay, ta ed nhdm Ong the so. hau ti, nhdm cang the s6 thuc, nhdm tong the s6 phde.
2) Tap hop the s6 hau ti khac 0 Ming vdi phdp nhan thong thuong la mat nhom giao hoan ma ta g9i IA nhdm nhan Sc s6 hau ti khac O. Cung vay, ta ed nhdm nhan the so time khac 0, nhdm nhan the s6 phcie khac 0.
3) Tap hop S cac phep thg cim I, 2, ..., n} Mang via deft the phep thg 1a mat nhcim him han, bile/1g giao hoan voi moi n > 3.
4) Nita nhdm cOng the s6 to nhien N khOng phial IA mat nhdm (tai sao 7).
Ngoai the tinh chat him met vi nhom, mat nhdm con ed mat s6 tinh chat co ban sau day.
Dinh 11 1. Mai phan td cua met nhom chi eó met plan hi dai
Chang mink. Gia sir x', x" IA hai phan tit Mai cuing oda x. Th ed
Nhan hal veal ye hen trai mai x', ta dove
x'(xx") = x'e, vay (x'x)x" = x'e
hay
tile IA
'Prong trdong hop phep toan hai ngOi. cua nhdm Id hieu bang datt (dau. Ong +), thi phan to c161 xdng duy nhat eda x ki higu IA x-I (-x) va con ged la nghich dcio cua x (d61 eim TU dinh nghia cim phan tit nghich dao (phan to dal) ta tuiuglUch
44
(s 1) 1 = x, ( - ( -x) = x). Ngu nhdm la aben va phop town cua nhdm ki hieu bang dgu . (Mu +) thi phtin to xy-1 = (x + (-y) = (-y) + x) ki hieu la xly (x - y) ya goi la thitong caa x teen y (hieu the x va y).
Dinh li 2 Gnat gian uerc). Trong mat nhom, thing thzic xy -Y xz (yx = zx) keo they clang that y = z.
Nhu vey, ta bau trong met nhdm luet gian be thue hien dirge yid moi Agri tit
Ch'ing mirth. Nhan ben trai hai ve cda tang xy = xz yeti x-1 , ta cd
t (xy) = x 1 (xz)
hay (x 111 x)y = (x l x)z
hay ey = ez
tilt la y = z. n Dinh li 3. Trong mat nhom, plutong trinh ax = b (xa = b)
to nghiem duy aka x = a lb Ix = ba l 1.
Chang mirth. Ta thtly ngay gia tri x t= a -lb la nghiem ena
phvong trinh vi a(a -15) = (aa 1 )1, = eb = b. Dd la nghiem duy nhat, vi ngu c la met nghiem cita phttong trinh nghia la ac = ax = b, ta guy ra c = x theo het glen ado. Chung minh toting tv cho phuong trinh xa = b. n
Dinh li 4. Thong mat nhom ta to
(xy) 1 = y lx 1
udi x, y C¢ hai pit hi bat ki caa nhom.
Cluing mink. Ta co
(xy)(y-lx-i) = x(yylx 1 = xx t = e
lx 1 )(xy) = y 1 (x lx)y = y ly = e
c I
(xy) 1 = y lx -1 . n
45
Dinh li nay m6 rang the khac cho tich m re thy y nhan
to 1 (x ix2 xn) -1 = x- x -1 x-1 Mc Met
(ar) l
vai ince se tii nhien khac 0 ; nguai ta quy the vigt Oen to do
dual dang a', va mat khdc dat a" - = e, nhu thy ta da loan
thanh vie xac dinh a cho m9i se nguyen A. 1n van cd (dec gia hay chting minh diet' do)
al ai = a?+”, (a))/' =
vek moi A , p E Z. Clni g : ki hien lit dung cho phep nhan, trong triamg hop phep Ong ta vigt Aa.
DU6i day ta hay dua ra met s6 dinh nghia thong diking cha nhdm.
Dinh li 5. hide nem nhom X ta met nhom ndu va chi nits hai &Su kien sau duce than man :
1° . X co mat don vi &di e.
2° . Ved moi x E X, co mOt x' E X scro.cho .x'x = e.
Cluing mink. Hign nhien hai dieu kith tract la can, ta hay Chung minh cluing la du. Gia sit cd cac Mau kith 1 ° va 2° . Lay met phein tit thy y x E X. Theo 2 ° cd met ,x' E X sao cho x'x = e. Van theo 2 0 cd met x" E X sao cho x"x' = e. Ta co
xx' = = x"x'xx' = x"ex' = x",(' = a
Mat khac, ta lai ea
xe = x(x'x) = (xx')x = ex = x.
Ket luan e 14 don vi caa X va x' la nghich dais tha x. fly X la met nhdm. n
'Pa thing ditgc met dinh li Wang tit nth ta thay the' don vi trail bang don vi phai va phan to x' trong dieu kith 2 ° dang le 0 ben trai tha x thi vigt 6 ben phai tha x.
46
Dinh li 6. MN nOa nhom khcic rdng X la mot nhom niu ud chi niu cac phuong trinh ax = b vb. ya = b co nghCkm trong X Si moi a, b E X.
Cluing mink. Dinh Ii 3 oho ta biet dieu Mgt la can ta hay chiing minh no la du. Vi X la khac rang nen cd mat plign to a E X. Gia sV e la mot nghiem eta pinning trinh ya = a. 'hi hay cluing minh e la don vi trai eta X. Gia sic b IA mat phgn to toy y eta X. Goi c la mat nghtem eta phuang trinh ax = b. 'Pa cd
eb = e(ac) = (ea)c = ac = b.
Vey e la mat don vi trai caa X. Bay gib ta hay cluing minh vol moi b E X, cd b' E X sao cho b'b = e. Mutin fly ta xet phuong trinh yb = e. Sat ten tai nghigm eta phuong trinh nay cho to b'. Nhu \ray the dieu Alan 1 0 va eta dinh II 5 dttoc them man, ta cd X IA mot nhOm. n
2. Nh6m con
Gia su A la mat 60 Olen eta mat nhdm X, ta cd thg ea xy E A voi moi x, y E A, tile la A la mat bb phan an dinh eta X (§1, 1, dinh nghia 2). Nhu fly phep toan hai ng6i trong X earn sinh mat Olen toan hai ngid trong A. Nhu A cling yen phdp toan cam sinh la mat nhdm, thi ta It A la mat nhom con eta nhdm X. Chang han ta cd 140 Olen the s6 nguyen Z lit mat nhdm con eta. nhdm Ong the s6 him ti Q. Ta chn y bio phan {-1, 1} tuy lap thanh mat nit deg vOi phep nhan, nhung khang phai la mat nhdm con eta nhcim cOng cat s6 nguyen Z.
Dinh nghia 2. Mat he Olen an dinh A eta mat nhdm X la mat nhom con eta X neu A ding vai phdp toan earn sinh IA mat nhdm.
Dinh ii 7. Mat 60 phOn A dm mot nixes X la mOt nhom con czict X niu tta chi niu cac diku /den sau day than man :
1) Vat moi x, y E A, xy E A.
2) e E A, Si e lit phan to trong ldp cita X.
3) Vai moi x E A , x 1 E A.
4 7
Cluing mink. At cd. Gia sit A lit met nhdm con dm X. Theo dinh nghia nhdm con, ta cd ngay 10. Goi e' m phan to trung lap oda nhdm A. Ts co
e'a = a yea mai a E A
Mat khac ta Ming cd
ea = a
vi e la pilau to trung lap cu. X.
Do dd
e'a = ea
hay e' = e
vi luat gian dee thoc hien &toe vdi moi phtin td cim nhdm X. Vity ta cd 2°. Cue' cang goi x' lit nghich dao trong nhdm A cua mat pilau td x E A. Vay ta cd
xx = e = 1 1 x.
Thoc hien Mat gian doe, ta dime x' = x i . Dieu nay thong minh 3° .
DA. Gia sit A la mat be phan cim X thda man the diet; kien da cho. VUi 1 °, A lit mot nita nhdm, vi phep toan hai ngti da ciao trong X la ket hop. 2 ° va 3° not len nda nhdm A co phtin to trung lap va moi pilau tit dm A co nghich dito trong A. Vay A la met nhdm, do (Id A la met nhdm con cda X. n
H9 qua. Gid set A la nay 60 plain khac r6ng cart met nhom X Cac &On kin sau clay la Wang daring :
a) A la mot &Wm con &La X.
b) Vdi moi x, y E A, xy e A Mt E A.
c) Vdi moi x, y E A, xy t E A.
Cluing mink. Theo dinh li 7 ta cd ngay a) keo theo b). Hien nhien b) keo theo c). Ta chdng minh c) keo theo a). VI A x 0
nen cci mat x E A. Theo c) ta cde=xx l eA. Vay diet' Men 2° cda dinh li 7 them. man. GM sd x E A. Van theo c) ta co
98
x -I = ex I E A. Ta duge dieu kien 3° can dinh li 7. Duel ding,
gia sa x, y E A. Vi y -I E A, nen c) cho x(y-I) -1 = xy E A. Nhu fly dieu kien 1 ° Ma dinh Li 7 them man. Do do . A la met nhdm con eim X. n
VI du. 1) Be phan mZ gem the sic nguyen la bei dm met se nguyen m cho trubc la mat nhdm con etla nhdm Ong the se nguyen Z. That fly, tip dung diet' kien c) Ma he qua trail, ta ce mZ let nhdm con vi mx - my = m(x - y) E mZ, tUc la hieu ctla hai phan to thuec mZ lai thfle mZ.
2) 'Prong met nhdm X, be phan A gem the luy thita nit eim met Oen tit a E X la met nhdm con ctla X That fly,
as (eta) = al a = a1-8 E A. Dec biet the bOi Ma met se nguyen lit met nhdm con Ma nhdm Ong the so` nguyen Z. Nhd fly, vi du 1) chi la met trubng hop ciac biet cua vi du 2).
3) BO phan { e 1 chi gem c6 phan tit trung lap va 130 Olen X la hai nhdm con dm met nhdm X. Nglibi ta goi chting la the ?them con tam thabng cart X.
Dinh Ii 8. Ciao can mat ho btft ki nhung nhdm con can mat thorn X la mat nham con caa X.
Chang mink. X& mot hg Mt ki (A,), e I nhfing nhdm con
caa X va goi A la giao cua cluing. Trude Mt ta ce A s 0, vi phiin tit trung lap e caa X thubc A veil 1110i a e I, do de
e E A. Bay gib ta lay hai phan to bet Id x, y E A. Vi x, y E A, nen x, y E Act voi MOI a E I. Vi the 21a la nhung nhdm con
nen xy 1 E A veri 1110i a E 1, do de xy 1 e A. He qua caa
dinh Ii 7 cho ta Net A la nhdm con caa X. n Gift sit U la met be phan Ma met nhdm X. The thi U chtla
trong it nhat met nhdm con cat*. X, cu the X. Theo dinh Ii 8, giao A cos. tat ea the nhdm con Mut X chtla U la met nhdm con cart X dada U. Do la nhdm con be nhat cita X chtla U
(quan he thd to la quan he bao ham).
Dinh nghia 3. Gia. sit U la met be phan caa mot nhdm X. Nhom con A be nhat elm X chile. U goi la nhOnt con sinh ra
bei U. 'Prong traimg hop A = IC, ta ndi rang U la met he sinh tha X vA X aye sinh ra bed U.
Neu U = { a ), a a X, thi de ding they rang ahem con A sinh ra bid U cd the phin tit la the luy thaa all, A e Z. That ny, then v1 do 2) be phan gem eac•lay thita ce l la ma nhdm con coil X. Han naa, d6 lit nhdm can be nhtit tha X chtta {a} vi m9i nhdm con can X ehda { a thi deo chda the lily thiia tha a Ngubi ta goi A lit nhdm con sinh ra bit a.
Dinh nghia 4. MOt tilt X goi la xyclie neu v.& chi Su X dam Binh ra b&i mOt phi]) tit a E X. Phan tit a goi la mOt phan to sinh tha X.
Nhu tray ma nhdm IL la xyclie nett vb. chi Su the phin tit cAa n6 la cac lay thiia a1 , A a Z, caa met phtin tit a a X.
Vi dy„ 1) X& nhdm the phep the 83 ma the phin tit la
e =
fy =
1 2 (1 2
1 2 (3 2
3 3)
1)
' s. 1 1
13
3 =
(2 1 2
3
(1 3
3 1.)'
2
4.
)
= 1
(3 2 1
3 2)
e "3 =
(1. 2
2 1
3 3)
lb hay nghien cdu the phan tit can nhdm con shah ra bbi ft, c6
ft =fZ,fl = e
Ta hay chang minh rang can lily tiara fit khi A chay khap Z
chi cho ta ba phin tit phan bit la e, fl, f2. Thht tray ta hay ley ma thy thita thy y . Chia A cho 3 ta throe
A= 3q + r, 0 4 r< 3
r la du cia phap chia A cho 3. Ta 'cat
[At = = • fr, = (fb q irc
Nhung f = e , nen
fi = Oc bg fri = e g et =
50
Vi r lay met trong ba gia tri 0, 1, 2, nen fl la mot trong
ba phan to f = e, fl = , ff = 12 . Nhu fly nhdm con A3
sinh ra bat /3 gam ba 'align tu, d6 la e, f va f2 . Nghien mitt fp ta cd
fi = = e. Bang If Wan twang tu, ta cuing c6 nhdm con sinh ra bdi 12
gam ha phan tit e, fl , fr Vay nhdm con sinh ra bdi 12 truing vai nhdm con sinh ra bat /3.
Con fp 14, 15 la nhang chuygn tri nen ta c6 ngay
T 23 . T 24 . e.
Cling If Wen nhu tren ta talc e, 13 1 la nhdm con sinh ra bai /3, {e, /4 } la nhdm con sink ra hat 14, {e, 15 } la nhdm con sinh ra bai /5 .
Cu6i sung, vi = e vdi MO A e Z nen nhdm con sinh ra beg e la nhdm con tam uniting {e}.
Nhdm S3 khang doe sinh ra bat met phan to nao cast n6 ca, trey 83 khang plied la xyclic. Con cat nhdm con kg tren cita 53 dell la xyclic.
2) Nhdm Ong cat s6 nguyen Z la xyclic vi cat phan tit obi no la ate bet dm 1, fly 1 la met phgn tit sinh aim Z. Ta cuing cc' thg cot ate phgn tit aim Z la the bet aim -1 cho nen -1 Ming la mat phgn t& sinh cda Z
Ngoai 1 vb. -1 ra, Z khang can phgn tit sinh nao khac. Z la met nhdm xyclic ve ham, can cat nhdm xyclic bong vi du 1) la hau han.
Gia sit X let met nhdm, e la phgn tit truing lap aim X va a la met Win tit aim X. Neu kitting c6 met s6 nguyen during n
nao sao cho a" = e thi nhdm con sinh ra bat a la v6 han, vi a2 x d' vai A m p. 'Prong trtiang hap trail lai, goi m la se nguyen
during be nhat sao cho am = e, thg thi nhdm con sinh ra bat
a :co m phgn tit : = e, al = a, a2 , , (clang minh trong tu nhu trong vi cht 1)).
51
Dinh nghia 5. Gia sit a IA met phan tit bat ki cim met them X va A la nhem con sinh ra bai a. Phan ta a g9i la co cdp ut5 hqn ndu A va han • trong trueng . hop nay khang 06 met s6 nguyen duang n nao sao cho an = e. Phan tit a goi IA cd cdp in nett A cci cap m ; trong truang hop thy in /a se nguyen duang be nhat sao cho = e.
Mat phan tiictEX co cap 1 khi va chi khi a = e.
3. Nh6m con chugn tic va nhOm thuang
Gia sit A la met nhcim con aim met Mt X, ta hay dinh nghia quan he trong tap hop X nInt sau : mei x, y E
A, x y ngu va chi ngu fl y E A.
B6 de 1. Quan trong X /2/ mat quan he Wang duang.
Chang minh. la phan xa vi f i x = e e A. - la dal sting vi the x I y E A thi y l x = (fl y) 1 E A. Cudi env - la bac cau vi neuz i y va y l z E A thi x I z = (fl y)(y-l z) E A. •
Vol mai phan to x E X, ta Id hieu lip thong duang chla x IA Fe va Id hieu be then cim X gam cac 'Man tit co clang xa vii a chay khdp A la xA, the la x4 = xa I a E A I.
B6 d6 2. 1 = xA.
Chang mink. Truac hat ta hay chdng minh x C xA. GM, sit y e x. fly x y, tdc la z 1 y la met phan tit a nao de dm A, do d6 y = xa E xA. Bay gib ta chtIng rainh bao ham ngttac lai : xA C x. Mugn vay ta lay met phan to y thy g tha xA. y cti dang xa vie a la met phfin td nao d6 ciut A. Tit' y = xa, to suy ra E A, fly x y, tdc la y n
Dinh nghia 6. CO.c be phan xA gpi la Mc lop trai cua ahem con A trong X Thong to, Mc 16p phdi Ax dm A trong X la cac be phan ma the phan tit co dang la ax vie a E A.
Cling nhtt dai vii cac lop trai, ta co thg cluing minh cac lap phal. cim A la can liqttudng duang theo quan he thong duang :
x y ngu NA chi age E A.
52
Tit cac b6 de 1 Ira 2 ta suy ra
Hg qua. (lid sit x ua y la hai pan ta tiny y caa nhom X thl thi :
(i) xA = yA nen utt chi Mu y e A.
(ii) xA fl yA = 0 Mu va chi Mu x y 0 A.
Tap hop thud:1g dm X tren quan ha tilting dtiong goi la tap hop finking dm nhdm X tren nhom con A, ki higu la XIA. Caa pilau to caa XIA la cac lap trai xA.
Bay gib ta hay lip dung cac kat qua trail de" nghien ca) quan hg gills) cap cim mgt nhdm hUu hat va cap oda mat nhdm con ena nd. Ts. cd
Dinh Ii 9 (Dinh II Lagranggio). Cap caa mot nhdm X has hgn la bpi caa cap caa moi nhdm con caa no.
Chang mirth. Gift sit X co cap la n va A, mat nhdm con bat ki aim nd, cd cap la m. Trade hat ta hay cluing mink MO lop trai xA, x E X, dau cd s6 plias' tit la m. Milan vay ta hay icdt
A = {x i , x2 ,..., xm } va cac phan t6
Cac phiin to nay la phan bigt, vi nth) cd qu i = xx2 °hang
han, thi xi = x2 . Dd la tat ca cite plain tit dm lop trai xA.
Nhu vay xA cd m pilau tit Vi X la hitu him nen s6 cam lop trai xA la him han, goi 1 la s6 cac lap trai xA. Do cac lap trai la rbi nhau nen ta cd n = ml.
'Nogg to ta cling cd I la s6 cac !Op pilaf. Ax. n S6 I cac lop trai xA (hay Cop phiti Ax) goi la chi .96 caa
nhdm con A trong X.
VI MO Oat) tit x cua mgt nhdm X sinh ra mgt nhdm con cd cap bang cap cim x, nen
He qua 1. Cap caa mot phan to thy S caa mot nhom Mu hon. X la Mc caa cap clia X.
Vi min plign tit x # e Goa mat nhdm X dela sinh ra mat nhdm con cd cap Ithimg nh6 hon 2, nen :
53
Ha qua 2. Moi nhOm hau han co cap nguyen t6 dau ld xyclic ad duce sinh ra ben mat phan td bat ki, khac glean hi trung lap, cda ;thorn.
Vi du. Ta hay lay nhdm cac phgp thg S 3 cd cap 6. Vi du
rave 2 oho ta bit the plain tit coa nd ed cap hoac la 1, hoac la 2, hoae la 3, do la nhUng uac etla 6. Bay gib ta hay lay mat nhdm con dm S3 chang han nhdm con A = e, f3 1 Dinh li
Lagranggia cho ta bigt s6 the lap trai tha A la. 3. Ta hay thanh lap the lop trai dd. Ta co
eA = fee e, ef3 = f3 1
f1 A = {fi e = 11 13 =
f2A = {
{/fie
= 12, 1213 = f5}
f3 A = f3 E eA
f4A vi [4 e ft A.
f5 A = f2 A, vi 15 E 12 A.
Bay gib to hay xet cac Cop phai eim A :
Ae = ee d e, f3 e = 131 = Af3
Aft = {eft = ft , 13 11 = f5 ) = Afs
Aft = (eft = f2, f3f2 = fs} = Afx
Chti y ta cd
fi A m Aft , f2 A m Aft .
Tth ye tthang hop tang quat, gia set X la mat nhdm va A la mat nhdm con cita nd. Ta dude tap hop thuong X/A ma cac phan to la the lap trai xA, x E X. 'Pa mut% trang N cho XIA mat phdp toan dg nd tra thanh mat nhdm. Nhung ta lai mugn phdp Oen de khang phai la thyg ma suy ra tic phdp Wan cim X, au the ta mu6n cho trung ging via cap (xA, yA) lap trai xyA, thg thi thong ling dd co phai la mat anh xa tit X/A x X/A clan X/A khang ? De tra lei eau hai do, ta 'cat vi du tram Chang
54
han ta cho tuong dug voi cap (ft A , f1 A) lop 4A. Vi = Jr, ,
nen ffel. = f2 A. Mat khdc vi f1 A = f4 A, nen cling vitt quy tax
Wong ling ta c6 lop tral eA = 124 A Wong Ong vet cap
(f4 A, f4 A) = (ft A , f1 A). Nhung f2 A m eA, vay quy tem Wen
khang cho ta mot anh xa. Mu6n quy Sc 416 cho ta met anh xa, ta can an Binh cho nhdm con A eta X milt then kiOn, ma ta got la diet' hen chutin tic.
Djnh nghia 7. Mat 'them con A rem mot nhdm X gyi la chudn
Mc thu oh. chinaux l axEAvoimoictEAvaxEX.
Djnh 0 10. NeU A la mat nhdm con chudn Mc cact mat nhdm X, thi :
(i) Quy Mc cho tuong ring v6i cap (xA, yA) lop trai xyA la mat anh m tit X/A x X/A din XIA
XIA cling vol phep town hai mg&
(xA, yA) •—n xyA
la mot nhdm, goi Ia nhdm thuong elm X trOn A.
Cluing mink. (i) D4 cluing minh quy tac dd la met anh xa thi ta phai chting minh rang nail xi A = xA va yi A = yA thi
x i yi A = xyA, hay theo he qua oda bit de 2, thu x 1x 1 e A Ira
y-1 y i e A thi (xy) 1 (xi yi) = y 1 x 1 x i y i e A. Bat x = a,
ta co a E A theo gia thiat. Xet tich y 1 x 1 x1 y 1 = y-1 ayi . Ta
c6 the vier : y 1 ayi = (fl ay)(y-1 Nhung y-1 ay E A vi A
la chuan the va y l y i e A theo gia thiat, vay (y -1 ay)(y-l yi )
E A, the la (xy) (xi yi ) E A. Ta ki hiOu cal hap thanh eta xA
va yA la xA.yA.
(ii) D4 thit nghtem tinh chat kat hop eta phep than hat ngOi trong XIA, ta hay lay ha phan tit toy y x, y, z cua X The thi
(xA.yA)zA = xyzA = xA.(yA.zA).
Do c16 phep toan hai ngai da cho la kat hop.
55
, Do thd nghiem su ton tai cua met don vi trai , ta hay set lap eA = A, trong d6 e la phan trung lap. Ta cd
eA.xA = exA = xA
ved .M9i lap trai xA e X/A. Vay eA = A la met dam vi trai.
Cugi ding vie m9i xA E X/A, ta cd
= x I xA = eA.
Theo dinh li 5, X/A Ming veil phop teen hai ngOi cla cho trong X/A la met nhdm, vii eA lit Olen tit trung lap oh'. x I A la nghich den cua xA. n
Dinh li sau day cho ta bit met dinh nghia thong during clia nh6m con chugn the
Dinh li 11. GM' sit A Itt mat nhom con Giza mat nhom X. Cdc dieu kien sau dtty /a Wong duang
a) A la Malin Mc.
b) xA = Ax vai mai x
Cluing minh. Ta hay chdng minh a) keo theo b). Gia su xa, a E A, la met phan til by y cua xA. Vi A la chugn Mc nen y -I ay E A WO mei y E X. Lay y = x-I , ta cd xax-1 E A. Dat xax I = a' E A, to cd xa = a'x E Ax. Vay xA C Ax. DA° lai gia su ax, a e A, la met 'Man ta thy y mia Ax. Vi A la deign the nen x I ax E A. Dat x 1 ax = a' E A, ta cd ax = xa' E xA. Vay Ax C xA. Kgt ban xA = Ax.
Bay gib ta clung minh b) keo theo a). GM sit a va x la hai phan tit thy y theo thil tu dm A va X. Ta cd ax E Ax. Theo gia thigt Ax = xA, nen ax e xA, tile la ax = xa' voi a' la met phan to nao dd dia A. Ta suy ra x-I ax = a' e A. Theo dinh nghia 7, A la chugn Sc. n
Do dinh li tram tit gib ngu A la chugn the thi ta khang phan Net lap trai, lap phai oils. A va goi met lap trai (hay met lap OM() cilia A la met lap dm. A.
56
Vi dt.t. 1) Trong met nhdm X, the nhdm con tam thtiOng {e} va X la chugn the
2) Trong mot nhdm aben rnoi nhdm con la ehugn Me.
3) Trong nhdm the phep the 83 ta hay xet nhdm con A3 gam
the phth the than (2, vi du 1) Ta cd
eA3 = A3 ={ e 11 , 12 }
Vi ft , f2 E eA3 , nen
eA3 /1.3 = f2 A3 .
Vi eac lop trai mitt A3 la the lap tuang duang, nen °hang thanh lap met chia 1th dm 83 , Sy ngoai lap trai eA3 ra
ta chi On met lap trai gam the phan ta can lai 13, f4, 15 Ta
suy ra f3 A3 = f4 A3 = f5 .43 = f3, f5).
Cling bang 11 luan Wang to, ta them
A3 = A3 e = A3 f1 = A3 f2 = { e , 11 , 12}
A3 f3 A3 14 = A3 fs = {f3, f4, fgl
Do dd, A3 la ehugn the theo dinh Ii 11.
4) Xgt nhdm tong the s6 nguyen Z va nhdm con nZ cua Z gam Sc so nguyen la bed dm met s6 nguyen it da cho. Vi nhdm Ong the s6 nguyen la then, nen nZ la chugn the, va do dd the lop trai, pliai cua nZ bang nhau. Cac lap aim nZ duce ki hieu la x + nZ, x E Z, vi phep toan 0 day la phep cling + Quan he Wang cluang xac dinh bai nZ la : x N y nth va chi nth x - y E nZ, tut la hieu x — y la met boi etia n Quan he nay chang qua la quan he clang du mod n. fly Z/nZ gam it lap, dd la 0 + nZ, 1 + nZ, (n 1) + nZ (Chuang I, §2, 2, Vi du) T4p hop thuang Z/nZ cling Nth phep toan
(x + nZ) + (y + nZ) (x + y) + nZ
goi la nhdm tong the s6 nguyen mod n. Ki hieu x + nZ bang va lay n = 4, ta ed bang Ong caa Z/4Z nhu saw
57
o 1 1 2 3 2 2 3 0 3 3 0 2
4. thing au
Dinh nghia 8. Wt ding edit (nhdm) la mot anh Ica f to Tr* nhdm X clan mot nhdm Y sao cho
;Tab) = f(a)
vol MO a, 6 E X. Ngu X = Y thi dting can f got IA mpt tt ang eau cita X.
MOt clang cau ma la mat don anh thi goi la mot don cdu, mot clang eau toan anh got la mot Man cdu, mot (long cdu song anh goi la mot deeng coy, WA tor (Ring cau song anh goi la mot to thing eau. Ng u f : X —> Y 7 la mot ding cgu to nhdm X den
nhdm Y thi ngubi ta vigt f : X => Y. (Trong throng hap X va Y la nhung nits nhdm, ta cling dinh nghia (tong eau (mia nhdm) nhu trail va cling c6 cac khai nipm tuang
Vi du. 1) Gia sit A la mat nhdm con cua mot nhdm X. Don anh chinh tac
A X a 1—n• a
la mot (tong eau goi la don cdu chinh Mc.
2) Anh so citing nhat cua met nhdm X la mot (tong eau goi la to thing colt (long nhat ono X
3) Xdt anh so to nhdm nhan cac s6 time during It+ Ten nhdm Ong the s6 thuc R
log : R R
x logy
58
trong dd log x la logarit eo so 10 So. x VI log(xy) = log x + log y, nen log la mat clang Su. Bang eau nay eon la mat song anh nen la mat (tang Su.
4) Gilt S A la mat nhdm con chuan tac cila mat nhdm X.
Anh xa
h : X XIA
x h(x) = xA
la mat dang cau to nhdm X den nhdm thuong Th4t fly,
h(xy) = xyA = xA. yA = h(x) h(y). Eking Su nay can la mat
toan eau, goi la toes °du Minh tat.
5) Gal th X vit Y la hai nhdm thy g, anh xa
Y x e,
Vol e lit phat tit trung lap caa Y, la mat clang eau goi la Bong eau tam thetemg.
6) Nall f : X —> Y la mot clang cau to nhdm Alf clan nhdm Y
thi anh xa nguoc : 12. cfmg la mat Bang eau That fly,
ta cd mat song anh, ta chi con cluing mink f I la mat d6ng cau. Gia th y, y' la hai phan tif thy g eim Y. Dat
x = r(y), x' = f-1 ta ed f(x) = y va f(x') = y'. Vi f la
mat clang cau nen f(xx') = 1(x) f(x) = yy'. Do do
(1 (yy) xi = r' (y)r'(y?). fly r" la mat dOng cau. Ta Joao hai nhdm X va Y la clang eau vat nhau, va ta vial X = Y, nau co mat clang eau tit nhdm nay ten nhdm kia.
Dinh nghia 9. Gia th f : Y la mat (tong Su to nhdm
X clan nhdm Y, the phtin trung lap ciaa X va Y duoc ki hieu thee the to la ex va ey . Ta ki hiau
Imf = f(X)
Keel x E f(x) = ey = ji (ey)
va goi Imf la dnh cua clang cau f, Kerf la hat nhdn aim dOng
eau f
Sau day, ta se dtta ra mat s6 tinh chat eda tong eau.
59
Dinh li 12. Gid szi X, Y, Z la nhung nhOm vez f : X —> Y .vd g : Y Z Ca nhung clang c6u. The thi anh xe tick
gf : X Z
tong la met dong cau. Dec Wet tech cda hai clang eau Id met demg colt.
Cluing minh. Gia oft a, b la hai phan td thy y caa nhan X. 'Pa cd, do f va g la nhung &Ong cal; gf(ab) = g(f(ab)) = g(f(a) fib)) = g(f(a)) ef(b)) = gf(a)g f(b). n
Dinh Ii 13. GM sit f : X Y Id met ang cdu tit met niter) X den met niter?) Y. The thi :
f(ex) = ey
(ii) f(x I) = ff(x),F I me mai x E X.
Cluing minh. (i) Gia sit x la ma phan to thy 3 am X.
Ta cd f(ex)f(x) = ftexX) = f(x) = ey/c(x)
Vay Be x) f(x) = ey f(x) hay flex) = ey sau khi that hiOn luat OM) vac.
i) Ta cd
f(x -1 )/(x) = f(x-lx) = f(ex) = ey = B(x)1 (x).
soy ra
fix = Kx/1 -1 n Dinh li 14. Gici silt f : X Y Id met ang cclu tit met nhent
X din met nhOm Y, A la met nhom con am X vet B la met nhOm con chudn Mc cad Y. The thi :
(I) f(A) Id met nhom con elm Y
(ii) r 1 (B) let met nhom con chuein Mc ata X.
Chang mink. (i) Tratdc ha f(A) x 4) vi A la ma nhOm con nOn ex E A, do dd ey = f(ex) e f(A). Ta hay lay hai phan hi thy 3 : y, y i E BA). Vi y, y i E f(A), nen cd x, xt E A sao cho
y = f(x) va yt = Fr i ). Xet tich yyji Ta cd yyT1 = f(x) f(x i r l =
60
= f(x)f(x1 1 ) = f(xxi-1). Vi A la nhdm con nen xxT I E A, do dd
yy1 1 = f(xx1 1 ) e f(A). Ta suy ra f(A) la nhdm con cim Y.
f1 # 0, vi B la nhdm con nen ey E B, do do
1(ex) = ey E B, ta suy ra ex e f 1 (B). Bay gip; ta Idy hai phAn
tit thy y. x, x 1 e f l (B), ta cluing minh xx.T 1 E f l (B). MuO'n
vay ta 'Mt f(xxi i). ah co f(xxi l) = f(x)f(x1 1) = f(x)f(xi) -1 . Nhung
f(x), f(x1 ) E B va f(x)f(x 1 ) -1 E B vi B IA nhdm con. Vey
f(xxT 1 ) e B, tdc IA xxi 1 E f I (B), do dd rl (B) IA nhdm con
MIA X. Cu6i cimg ta chiing minh nd IA chudn tac Muen fly
gia sit a e [ 1 (B) va x E X. Xet
f(x-l ax) = f(x I) f(a)f(x) = f(x) -1 f(a)f(x) E B vi f(a) e B va B
la chudn Sc. Do dci x 1 ax E ft (B) v6i moi a e r1 (B) va moi
x E X. Vey f1 (B) chudn Sc. n Tit dinh h. 14 ta cd he qua tat khac
H5 qua. Gid szi f : X —. Y la mot Ong ceiu tit met nhom X den mot nhom Y. The thi Imf mot nhom con cart ,Y pa Kerf la mat nhom con chudn tar ceta X.
Dinh li 15. Gid sti f : X —.le la mot dung edu tit met Ahem X deit mot nhom Y The thi :
(i) f lit mot town anh neu yet chi niu Imf = Y.
(ii) f la mot don anti neu utt chi /Mu Kerf = ex}.
Ch'ing mink. (i) Suy ra tit dinh nghla caa town anh.
(ii) Gia su f la mat don anh. Voi mei phan to y E Y co nhieu nhdt mot phAn tit x sao cho f(x) = y. Vey Kerf = { c}. Dad lai gia su Kerf = Xet hai phdn tit.x, x 1 e X sao cho
f(x) = f(x 1). 'llt suy ra f(x) f(x 1 )1 = ey. Nhung
f(x) f(x) 1 = f(x) fix 1 ) I = f(xx 1-1). fly f( ) = ey,
61
E Kerf = ex ). Do chi -t = e hay x = xr NM( tray f la don anh. n
Djnh Ii 16. Gid set f : X -sit mot d6ng cdu tit met nh6m X din mgt nham Y, p,: X X/Kerf la todn cdu chinh the (VI du 4) to nhOnt X din nhom thuong am X trip hat nhdn cda f. Thi thi :
(i) Co mot d6ng cdu duy nloit XIKerf Y sao cho tam gide saw
PN jf XIKerf
Zh giao hodn, tete Ih f = ip
(ii) Bong cdu 7 ld mgt don cdu oh Itnf = f(X).
Cluing mirth. (i) Wit Kerf = A ve. cho Wong Ong vol mei Olen tit xA cua XIA phan to f(x) dm Y. Quy tee cho Wong ling nhu tray la mat anh xa. That fly, gia = thg thi ta
cd x 1x 1 e A (he qua mitt lid de 2). Nhung A la hat nhan caa
f, nen fix txt7 = fix ')f'1) = ftx) -I f(x1) = ey , We la fix) = f(x 1 ). Ta dat
: X/A Y
xA 1—• 7(xA) = f(x).
Ta cluing minh f la mat dung eau. Ta c6, vdl xA va yA la hai phgn to tily g cua X/A,
MAYA) = T(xyA) = f(xy).
Nhung f la mat deng eau nen f(xy) = f(x) f(y), do do
f(xA.yA) = f(x) f(y) = fixA)7(yA)
then dinh nghia cua T. Vay T la mat clang Call Tit Bang thtic T(xA) = f(x) vat mot x E X, ta cd the vigt
fix) = f(xA) = T(P(x)) = TP(x)
62
veil mgi x E X. fly f = fp. Cuai cang gia sit S mat clang cau : Y sao cho f = fly, vol moi xA e XIA, ta co.
lo(xA) = = 49/3 (x) = f(x) =
do do p = T (ii) xA e X< sao cho f(xA) = Theo dinh nghia cita
T , f(xA) = f(x), v0.37 f(x) = ey. Do cki x e A = eA, We la
xA = eA. Ta suy ra Kerr = { eA ), vay f la mat don cau (dinh li 15). Vi p la toan cau va vi f = fp nen ta suy ra Imf = Imf
= f(X). n He quit. 176i mgi d6ng cdu f • X Y to mOt nhOm X din
mOt nhdm Y, ta co
f(X) = X/Kerf.
Vi du. 1) Gia sit A la mat nhOm con chuiin tae cim X h la toot Su ch{nh tac W X den XIA. Ta c6 Ker h = A. That vay, ta c6 h(x) = xA = eA Mu va chi Mu x E A (I* qua aim lad de 2). Trong troong hop A = {e} thi toan cau h eon la don anh, do do h la mat dang cau, vay X = Xlt el (dieu nay ta cd the °hang mink true Trong twang hop A = X thi nhdm thin:mg X/X la lilt chi c6 mat phan ta, do la lop eX, vi ta cd xX = eX ved mai x E X. Than Su ch{nh the h hit del la tam thubng vi Nen" m9i phan to tha X thanh phan tit trung lap.
2) Gia sit f la mat anh xa to nit Ong Z the s6 nguyen den nhdm nhan C the s6 phde khae 0 xaa dinh nhtt sau
f : Z C*
Vat k p—) cos 2hTh + isin
n IL
trong do n la mat s6 nguyen dung cho fru& Rd rang f la mat don can vi
f(k + h) = cos 2(k +h)n
+ isin 2(k
+h)7E
2kar 2kn 2ha 2ha ( cos—
. n n (cos n + isin n ) f(k)f(h).
63
Mat Ichat vi
2ka 2ks n ( cos— + isin-1 = cos2kn + isin2ka = 1
it n
nen Mc plias tit' cua f(Z) la the can bae n caa don vi. '1h suy ra tap hop the can be it call don vi cling vai phep nhan Mc se phtic la mot nhdm. Bay gii) ta 'tat tai Kerf. De la b0 phan cua Z gem the se nguyen k sao cho
21= cos— + isin— = 1.
fly k la bai ells n, do de Kerf = nZ. Theo he qua eaa dinh Ii 16 ta co
f(Z)=" Z/nZ,
Bic la nhdm nhan the can bac n caa don vi clang tau veri nhdm Ong cac se nguyen mod rt.
5. D61 )(Ong Ma
'Pa da Wet rang trong mot nhara, clang auk ab = ac (hay ba = ca) Iteo theo dang this b = c. Dieu nay do so ten tai
cua clei :cling a 1 cur' a. Nhung neu ta 'cat mat nact nhdm, thi ab = ac chua chile da ken theo b = c ; chang hsn trong naa nthm nhan the sti to nhien N ta ed clang tilde 0.2 = 0.3, nhong 2 # 3. Td day ta th khai niem :
Dinh nghia 10. Cho mat tap hop X cling vet melt phep than hai nee trong X. Mat pilau ta a E X goi la chinh guy ben trai (ben phdi) nth vai mai b, c e X sao cho ab = ac (ba = ca) thi b = c, a goi la chink quy neu no la chinh quy ben trai va ben phAi.
Nhu fly, trong mat vi nhem, mai phen to th clei xiitng deli la chinh quy, nhong nguoe Isi khong thing (icet vi nhdm nhan N thc so tv nhien). Dieu kien chinh quy chi la die.' kian can de th dei xing.
64
Bay gid ta hay scet van de sau day : "nhung" met vi nhdm giao hoan X vao met vi nhdm X "Ong' hart sao cho mpi pilau td chinh quy et X e6 del siting trong X.
Bs de 3. GM sti X la mOt 'Ma nhom giao hoan co phan to trung lap e ua X* la bO phan cua X Om the glean to chink quy cua X. Thd thi
e e X* .
r to on Binh.
Cluing mink. (i) Vi ex = x voi ring x E X, nen tii ea = eb ta co a = b.
(ii) Gia sit a, b E X* . Tit abx = aby ta suy ra bx = by vi a IA chinh quy. ?dining b thing chinh quy nen bx = by keo theo
x = y. fly ab la chinh quy, We la ab C X. • Dinh li 17. GM sit X lts mOt thorn giao hoCut, 1Ca b0
phan cua X Om thc phdti to chinh quy cua X. CO mOt ui nh6m giao hocin X va mOt don cdu f tic X din X co the tinh eh& sau :
1°. Cite phan to dm fir) co dot sting trong X.
2°. Cite plidn M min X co clang [(a) f(b) I uoi a e X, b e
Cluing mink. 'Pa hay xet quan he S sau day trong tap hop X x :
(a, b) S(a', b') neu va chi ngu ab' = at.
Quan he S la mot quan he tuang dyeing : that fly, re rang no la phial xa va del sting ; no la bac eau vi neu ta co ab' = a'b va a'b" = a"bc ta suy ra ab'b" = (ebb" = a"bb' nhung b' la chinh pay, nen ab" = a"b. Gia ad Tr la tap hop thuang aim X x X* Wen quan he Wang &fang S. Cdc piths to cua X la the lap Wang during C(a, b) ma ta ki hieu la (a, b). Ta dat f(a) = (a, e) thi mai a E X, e la phan tit trung lap cua X. Ta hay trang bi cho X met phep Wan dd X le .met vi nhom giao hoan va ta se they rang cap (Tx, f) them man bai toan dat ra.
°N W.. - os
Gia sit x = (a, b) vb. y = (c, d) la hat phan tit tha X. Phan ta (ac, bd) chi Ow thuae vao x va y ; that thy gia sa x = (a', bp, vay ab'=a'b hay nhan hal-ve' vBi ed, ab'ed (= abed, ta suy ra (ac, 6d) = (a'c, b'd). Th. thdy ngay rang X cling vat phep town (x, y) ,-)xy = (ac, bd) la mat vi nhorn giao hoan vat int tit don vi la (e, e). Ta chit 3 la ta ed (e, e) = (a , a) vitt mot a E X. Ngoai ra ta e6 ngay f la mat don eau va, vitt moi a E xang tha f(a) = (a, e) trong X la a). Dual cling mot phan tit x = (a, b) E X co the \gat
x = (a , b) = (a, e)(e , b) = (a, e)(6 , e) 1 = f(a)f(b)1 . n
Cap f) caa dinh li 17 la duy nhgt (sal kern mat clang eau) nghia la nen th mat cap (Y, g) khan tilt man bat toan thi c6 mat rang eau co : X Y va g = That vay, ling vdi
mai plain ttr x = f(a) f(b) 1 , ta xdt phan to g(a),g(b) -1 . Phan tit
g(a)g(b) 1 chi Ow thuae vao phan ta x, vi n6u ca
f(a) f(b) -1 = nap f(b') -1 We la f(a) f(b') = f(a) f(6) hay f(ab') = f(a'6) (f la citing eau), ta suy ra ab' = a'b (f la don anh), do do g(ab') = g(a'b), hay g(a)g(b) = g(a') g(b) (g la doing
thu), the la g(a)g(b) 1 = g(a')g(!)') 1 ; vay ta c6 anh xa. to
f(a) f(6) 1 1-+ g(a)g(b) 1
ti X dgn Y. Dg dang chting minh rang do la mat dAng eau va g = sof
'Prong dinh It 17, vi f la mat don eau nen ta hay (tang nhat a va f(a) vdi mot a E X. Do do the phan S cua X vidt cladi
Bang ab 1 yen a E X, b E
HO quit. Neu tdt cd Grit phan to dm X au la chinh quy thi Mt cd cac phan to ena X deli to (Mt xling, do do X La mot
nheint.
-Ong dung. 1) S6 nguyen. Ta hay lay X la vi nhdm the ea to nhien N, phdp toan la Olen tang fining: tinging : moi phan ti N la chinh quy nen X la mat nhcim. Ta ki lieu. X bang
66
Z ; the phan tit tha Z goi la the s6 nguyen ; phgp toan trong Z xay dung nhu trong dinh li 17 gig la phep Ong the s6 nguygn va thing ki higu bang dau + nhu phep Ong the sd to nhien. Mai phAn tit tha Z vigt dudi clang in - n vOi m, n E N. Mu m 71 ta th m - n = p, p so tit nhien sao thorn = n + p. N6u m < n, to th in - n = -p, p lit s6 to nhien sao oho m + p = n. Vay cac phan tit tha Z la... -3, -2, -1, 0, 1, 2, 3,...
2) 86 hau ti &tong. Lay 81 ° tap hop ode s6 ter nhign klitte 0 lam X, phep toan Ian nay la phep nhAn thong thtibng cac s6 W nhign ; m9i Oran tit tha N* la ehinh quy nen X la mat thorn. 'Prong truong hap nay X ki hiau la Q * va car phan td
cue n6 goi la the s6 Mu ti duong ; phep toan trong tte xay dung nhu trong dinh If 17 g9i la phep nhan the s6 him ti duong va ki lieu bang x.y hay xy eat hop thetth cua hai ad him ti
during x va y. Mai phan ti/ tha Q + vigt dudi clang po -1 vbi
p , q E Ns, ta con quy tree vigt &tan ta pq-I cua Q+ la p/q
pl P2 P1P2 hay - ; tich curt — va — nhu fly la — • s6 to nhign m
q1 q2 q1q2 dude (tong that vOi m// = mnln E
Chu 3r. sao vOi Z, ta ed
Z = N U -1, -2, -3, ..., ...}
(xem tYng dung 1)), trong khi dal thi (1+ ta khdng c6
I 1 1 1 Q+ = bir U 2' 3' 4' n'
(xem Ling dung 2)) ? Dg tim higu ta xat cac dinh II sau day.
X trong hai dinh li sau chi mat vi nhdm nhan giao hoan (tat nhign ta cOng e6 the' lay mat vi nhdm thug), va .21+ la 130 phan the plan tit chinh quy tha X ma ta gia thigt t = X. Theo IQ qua caa Dinh II 17, X la mat thorn, ma the phan tit
dang ab-I voi a, b E X (Dinh II 17).
Dinh fi 18. Ngu veil moi a, E X, hula phuong trinh ax = bed nghiem trong X, hoc pinning trinh bx = a c6 trong X ; thg thi :
67
X = X U I c
Ching mink. Gilt sit ab-1 E X . Ndu phttang trinh ax = b c6 nghiam trong X, nghiam d6 phiti c6 clang 6a -1 e X, vay - ab-1 E { c-1 / cEX}. Neu phuang trinh bx = a co nghiam trong X, nghiam phAi cd clang ab-1 E X. fly
X C X LO {c -1 I c E X}. .
Baa ham tilde ngttle Iai lit hien nhien. fly ta cd
X = X U {CI c E X}. n Dinh II 19. Dao lei, gilt Si:
= X U c-1 I c e X}.
Th6 till vol moi a, b E X, hoac phttang trinh ax = b c6 nghiam trong X, hoc phacing.trinh bx = a ad nghiam trong X.
Cluing mink. Gil sit a, 6 E X. X6t phuang trinh ax = b. Phuang trinh nay cti nghiam trong X , dd la x = ba-1 . Theo gilt thief ye X, bat EX hoac 6m-1 Et c1 / cEn. Neu ba E X, Mu d6 c6 nghia phttang trinh c6 nghiam trong X ; nen ba-1 e / c E X 3 , to c6 ba-1 =c 1, c E X, hay ab-1 = c, hay phitang trinh bx = a co nghiam trong X. n
Ti: hai dinh tren, ta ditty rang : "Vai moi a, b E N, hoac phdang trinh a + x = b co nghiam trong N hoc b + x = a c6 nghiam trong N". Cho nen
Z = N U (-1, -2, -3, ...I.
Trong khi do, vai 'mg a, b E N. , kitting phAi ban gib ta rang co ax = b c6 nghiam trong N* hay bx = a co nghiam trong
Chang han a = 2, 6 = 3, el 2x = 3 vA 3x = 2 deu kheng c6 nghiam trong W.
BAI TAP
1. Lap Mc bang town cho cac tap hop gem hai phAn tit, ba phan tit d6 duct nhang nhain
2. Tha rem Se tap hop eau day vol phep than da cho cci 14p thinh mat nhdm :
68
1) Tap hop cac s6 nguyen voi phdp cOng. 2) Tap hop cac s6 h8u ti yea phdp Ong. 3) Tap hop cac s6 thoe voi phdp Ong. 4) Tap hop cac so phtic voi phdp Ong: 5) Tap hop Se s6 nguygn la boi eda mot s6 nguyen m voi
phdp Ong. 6) Tap hop cac s6 thoc duong voi phep nhan. 7) Tap hop the s6 thoc khan 0 voi phep nhan. 8) Tap hop cac 56 pink c6 modun bang 1 vol. phdp nhan. 9) Tap hop cac so' phdc khan 0 voi phep nhan. 10) Tap hop Se 56 hdu ti cd dang 2", it E Z, voi phdp than. 11) Tap hop the can philc bac n dm 1 voi phdp nhan. 12) M = (1, -1} voi phdp nhan. 13) Tap hop Se s6 th1e duong voi phep toot T xac dinh
nhu sau : aTb = a2b2 voi m9i s6 thoc a, b > 0. 14) Tap hop the s6 thtic c6 clang a + ba (a, b E veil
phep Ong. 15) Tap hop can s6 thoc ed dang a + ba (a, b E Q va
a2 b2 t 0) voi phep nhan. 16) Tap hap can s6 phtle S gang a + bi (a, b E Z) vbi phdp Ong.
17) Tap hop cat vecto n chigu cart khOng gian 1.9fd phdp Ong veeto.
18) Tap hop cac ma tram vuOng cap n voi phdp Sag ma tram
19) Tap hop the ma tram vuOng cap n khOng suy bign voi phdp nhan ma ban.
20) Tap hop can ma trait vuOng cap 71 S. dinh thttc bang 1 voi phdp nhan ma tran.
21) Tap hop cat ma tram vuong cap 71 c6 dinh thdc bang t1 voi phep nhan ma tran.
22) Tap hop cac da thdc (c6 he so time) voi phep c9ng cac da
23) Tap hop gdm da thtic 0 va cac da than c6 bac khOng qua it (n la mat s6 nguyen, it 0, oho trutoc) veil phep Ong cac da thdc.
69 ,
3. Chung minh rang moi nua nhdm khac reng hau han X let mot nhdmneu va chi ngu luat gifin uoc that hien dale veil moi phan to coa X.
4. Cho X lit mOt•tap hap toy Y. Ki hieu Horn (X, X) la tap hap cac anh xa to X den X Vol phep Man anh xa Horn (X, X) cd lap thanh met nhdm hay kheng ? Chting minh rang be phan 8(X) Ma Horn (X, X) gem the song anh tit X dgn X la met nhdm voi phep nhan anh xa. Hay tim cap cda S(X) trong trMing hap X co n phan tit
5. Cho X la met nhdm voi don vi la e. Cluing minh rang thu veil moi a E X ta cd a2 = e, thi X la aben.
6. Cho met ho Mang nhdm (.2g, e / ma cac phep tain deu Id hieu bang den nhan. elating minh rang tap hop tich de cac
voi phep town xac dinh nhu sau • a E
(Xtx)a E / • (Ya)a E f = (XaYa)a e /
la met nhdm (goi la tich cac nhdm 2c).
7. Cho X la met tap hap khac rang cimg via met phep than hal ngei kgt hop trong X a la mot phan to ctia X. XI hieu
aX=faxixEXI,
Xa = xa I x EX}.
Chfing minh X la met nhdm khi VA chi khi voi moi a e X to cd aX = Xa X.
8. Chung minh rang•moi be phan khac r6ng 6n dinh cila mot nhdm hau han X la met nhdm con cua X.
9. Trong the nhdm 6 2) nhdm nao Ca nhdm con °Oa nhdm nao ? 10. Chung minh rang trong nhdm tong cac se nguyen Z,
mot be phan A coa Z la. met nhdm con can Z Mu va chi Mn A cd clang mZ, m E Z.
11. Trong nhdm cac phep the S 4, cluing minh rang cac phep the' sau : e, it = (12) (34), b = (1 3) (2 4) va c = (I 4) (2 3) thanh lap met nhdm con dm S4 . Maim con dd cd aben khong ?
12. Cho Y lit met be plan dm met tap hap X. Chung minh rang be phan S(X, Y) oda &X) gem cac song anh f • X —a•
70
sao the f(Y) = Y la mot nhdm con cart S(X) (bai tap 4). Tim
s6 phan to tha $(X, Y) trong truting hop X ad tt phan tit va
Y cd mot phials tit.
13. Cho A va B la hai 1)0 phan tha mot nhdm X. 1h dinh
nghia
AB=fablaeA,bEB)
A' ={¢-1 1 aeA}
Ch'ing minh cac dAng thlic sau day :
a) (AB) C = A(BC).
b) (A 1 ) 1 = A.
c) (AB) -1 = B-1 A -1
d) Mau A la melt nhdm con dm X thi A-1 = A.
14. Cho X la milt nhdm va A la mat be phan khac rung tha X.
Chting minh A la nhdm con cila X khi va chi khi AA 1 = A.
15. Cho A la mot nhdm con dm nhdm X irk a E X. Chting
minh aA la nhdm con ciut X khi va chi khi a E A.
16. Trong mot nhdm ,X cluing minh rang nhdm con sinh ra bai 60 phan 0 lit nhdm con tam thtiOng {el, e la plain to trung lap cfm X.
17. Gia sit S la Wit b0 phan khac rang dm mot nhdm X. Chung minh rang cac pilau tit cilia nhdm con sinh ra bai S la cac phan to cd dang x/x2 ...xn vol cac xl , xn thufie S hoar
3 1 . Tim nhdm con tha nhdm nhan the s6 huu ti duong sinh ra bed b0 phan cac so nguyan to'.
18. Cluing minh rang moi nhdm con clia mot nhdm xyclic la mot nhdm xyclic.
19. Cho X la mot nhdm vat phAn tit don vi la e,aeX cd
cap la rt. Chung minh rang ak = e khi Ara chi khi k chia hat cho n.
20. Cho a, b la hai phan tit thy 31 cila mot nhdm. Chttng minh ab va ba cd thug cap.
71
21. GiEt sit X lit mot nhdm xyclic cap n va a E X la into phan tat sinh aim no. Xet phan tit b = ak . Chung minh rang :
a) Cap cua b bang nld, 6 day d lit UCLN ctla k va n. b) b la phan to sinh dm X khi va chi khi k nguyen t6 vdt
rt (tit dd suy ra s6 phan ta: sinh dm X).
22. Gia sit a, b la hal phan tit cila met nhdm, va gia sit ta cd cap ciut a bang r, cap cita b bang a, vat r, s nguyen t6 ding nhau va them nita ab = ba. Chang minh cap cua ab bang rs.
23. Chung minh rang mai nhdm cap v6 han d@u cd v9 han nhdm con.
24. Cho X va Y IA nhfing nhdm xyclic cd cap IA In va n.' Chung minh rang X x 1( IA mOt nhdm xyclic khi va chi khi va n nguyen t6 cang nhau.
25. Cho A la mOt nhdm con cunt mOt nhdm X. Gia sit tap hop thuang X/A cd hai phan tit. Cluing minh A IA chugn tic.
26. Thong nluirn eke phep thg Sn, be phan An gam cac phdp
thg chin la nhdm con chutin tic cua S.
27. Gilt sit' X IA mOt nhdm xyclic va n han, va a E X la met plight to sinh Goi A la nhdm con dm X sinh ra bat a3. Chang minh rang cac lop trai cila A bang cac lOp phai ctla A va s6 cac lop do bang 3.
28. GM sit X la mOt nhdm, ta goi la tam oda X 69 phitn C(X) =(a E XI ax = xa vii m9i x E X).
Chitng minh rang C(X) la mOt nhdm con giao loan ciut X va mci nhcim con cita C(X) la inert nhdm con chugn tat dm X.
29. Tim tat ca ale nhdm con va nhdm con chugn tic ctla nhdm cac phep the' S3 .
30. Gift sit X la mar nhdm, x va y la hai phtin to caa X. Th goi la. /wan at aim x Ira y phtin t4 xyx-ly -I . Chung minh rang nhdm con A sinh ra bdi tap hop cac ham to dm tat ca the cap phtin to x, y cita X la mat nhdm con chugn tax caa X goi la nhdm cac loan tit, va nhdm tinning XIA la aben.
72
31. Cluing minh•rang muein cho met nhdm timing XIH cart mot nhdm X la aben, at ed va du la nhdm con ehuAn Sc H chtla nhdm the hoan tit caa X.
32. Tim nhdm the hoan tit' cua 83 .
33. Cluing minh Ring moi nhdm ed cap be han hoac bang 5 la aben.
34. Hay tim the nhdm thdang cua
a) Nhdm tong caz s5 nguyen la bei eim 3 tren nhdm con the s5 nguyen la boi oh' 15.
b) Nhdm Ong the s5 nguyen la bQi caa 4 tren nhdm con the s6 nguyen la bgi east 24
c) Nhdm nhan the so" thtte khan Q tren nhdm con the só Rule ddang.
35. Cho D a tap hop the duOng thong A trong mat phAng ed phdang trinh la y = ax + 6 (a t 0, 6 a nIning s6 Uwe). Anh xa
DxD—, D
(A1, A2) A3
trong dd A 1 , A2 , A3 tan brut th the phydng trinh la
y = aix + 6 1 , y = a2x + 62 , y = a1a2x + (b 1 + 62) the dinh
mgt phep toan hai ng6i trong D.
a) Cluing minh D la met nhdm vOi phep toan An.
b Anh xa
co : D R."
A 1-0 a
trong dd R" la nhdm nhan the s5 thy° khae 0 va A la dttang thAng cd phdang trinh y = ax + b la met dong eau.
c) Xae dinh Karp.
36. Cho G I , G2 la nhiing nhdm vai don vi theo thtl tp la e t. e2 va G la nhdm tich G I x G2 (hal tap 6), A va B la the bg
phan G 1 x {e2 },. (e l ) x 02 dm G. Xet the anh xa
P1
(x i x2) X I
P2 " G G2
(xI x2) 12
ql GI -> G
(x e2)
q2 G2 -3. G
x2 7. (e) , x2)
a) Chung minh p 1 , p 2 la nhung toan cau Xac Binh Ker p1 va Ker p 2 .
b) Chung minh q 1 , q2 la nhang dun Mu. Xac Binh Img 1 va Imq2 . Tit dd suy ra G 1 clang Mu vat A, va G 2 long eau voi B.
c) Chang minh A va B la nhUng nhdm con chug]) tac va AB = BA = G.
37. Chung minh rang m9i nh6m xyclic hitu han cap n 1:1411 Bang Mu vial nhau (clang Mu vUi nhdm ceng citc s8 nguyen mod n).
38. Chung minh rang MOI nhdm xyclic ve han dau dang cau vai nhau (clang cau vtli nhdm Ong cac s6 nguyen Z).
39. Gia sat X la met nhdm va Y la met tap hop duoc trang bi met phen town. Gia su cc; met song anh
f: X Y
tiler) man tinh chat f(ab) = f(a)f(b) vOi MO a, b E X ChAing minh Y.cang voi pile]) than da cho trong Y la met nhdm ; them nfia la aben nau X aben, la xyclic nau X xyclic.
40. Cho X va Y la hai nhdm xyclic co cdc phan td sinh theo 'Hatt to la x vet y va cd cap la s va t.
a) Chang minh rang guy tac co, cho Wong Ung vai phan to xa E X phan to Ofif E Y, trong do k la met sa to nhien Hide 0 cho truer; la met ct8ng can khi va chi khi sk la INN cna t.
b) Ngu sk = nit va lh (tang cau thi (s, tn) = 1,
74
41. Cho X lb mat nhdm giao hoan. Chung minh rang anh xa
so:x—>x a ak
vet k la mat s6 nguyen cho three, la mat dang eau.
Xac (firth Ker so.
42. Cho X la met nhdm. Anh xa
co : X X
a p—t a -
la mat ttt ding eau coa nhdm X khi va chi khi X la aben.
43. Cho X la mot nhdm. Chang minh rang tap hop the tp
clang eau caa X cling vet phep nhan anh xa la mat nhdm.
44. Gia sit X, G 1, G2 la nhang nhdm, G =Gi x G2 pa
f : X —> G 1 g : X G2 la nhung anh xg. Xdt anh .
h X —+G
x h(x) = (f(x), g(x)).
Chang minh rang h la mat &Mg eau khi Ira chi khi f va g 14 flitting &Mg eau.
45. Prong tap hqp X = Z3 , vdi Z la tap hop cac s6 nguyOrt, -ta xcic dinh mat phep Man hai ngoi nhu sau :
(k 1 , k2, k3)(11, 12 , 13) = (k + (-1)k313, k2 1 2 , k3 1 3)
a) Chang minh rang X cling vet phep than d6 la mat nhdm.
b) Chang minh rang nhdm con A sinh ra bei plain to (1, 0, 0) la chudn tic.
c) Chung minh rang nhdm thuong X/A ding eau vdi nhdm Ong cac s6 phdc cc; dang a + (di vdi a, b E Z.
46. Chung minh rang :
a) Nhdm Ong cac s6 time ding eau vdi nhdm nhan cac s6 time (Wong.
75
b) Nhdm Ong the ad phdc co clang a + bi Arai a, b nguyen Bang eau WO nhdm tich Z x Z trong do Z la nhdm cang cac ad nguyan.
47. Cho X la mot nhdm. veri mai phtin tit a E X to xet anh xa fa : X —> X
x 1—* a -Ixa.
a) Chdng minh fa la mot tit clang cau cda X, goi la to ddng edit trong xae Binh bdi phth tit a.
b) Cluing minh cac tor clang eau trong lap thanh mat nhdm con caa nhdm the to clang eau cda X (bai tap 43).
c) Cluing minh mat nhdm con H cda X la chuan Sc va chi nth fa(H) = H vbi moi to dang eau trong fa cda X. V1 11 do chi, cac nhdm con chuan tile cling can goi la the thorn con bdt hien.
d) Chung minh anh xa a fa la mat ddng cau tit nhdm X clan nhcim the to clang cat, trong dm X via hat nhan the ddng eau do la tam G(X) (bai tap 28) cda X.
c) Chdng minh X/C(X) dthg eau vol nhdm cac tor ding eau trong cim X
48. Cluing minh rang nth f : X Y la mat ddng cau td mat nhdm hbu han X den mat nhdm Y thi
a) Cap tha a E X chia Mt cho cap etia [(a)
b) C4 caa. X chia Mt cho cap caa [(X)
49. Cluing minh rang nhdm Y la anh ddng cau ail mat nhdm xyclic hitu han X khi va chi khi Y la nhdm xyclic utt cdp dm no chia hat cdp cha X.
50. Hay tim tat ca cac clang eau tit
a) Mat nhdm xyclic cap n clan chinh
b) Mot nham xyclic cap 6 dean mat nhdm xyclic cap 18 .
c) Mat nhdm xyclic cap 18 clan mat nhdm xyclic cap 6.
76
51. Chung minh rang ngoai clang cau tam thvOng ra thi khang th mat &Mg eau nth tit nhdm thong cac s6 hvu ti clan nhdm rang the th nguyen.
52. Gilt sit C" lit nhdm nhan the s6 phtic khde 0, H la tap hop the th phuc eim C' nam tren true thuc va true do. Chung minh rang H la mat nhdm con dm C' va nhdm thtiong C'/H Bang cau add nhdm nhan U cac s6 phdc th moduli bang 1.
53. Cluing minh rang nhdm thvang R/Z (R. la nhdm cling
the s6 thuc, Z la nhdm Ong the s6 nguyen) cang cau vai nhdm
U (bid tap 52).
54. Goi X la nhdm nhAn the ma tram vuong cap n kh6ng suy Bien ma the phan tit la thtle. Hay chting minh
a) Nhdm finking tha X tren nhdm con cac ma tram cd dinh thde bang 1 clang eau vai nhcim nhAn the s6 Hive khdc 0.
b) Nhdm thuang mitt X tren nhdm con the ma trail eel dinh thtic bang ±1 clang cau yea nhdm nhan cac s6 thuc dating.
c) Nhdm thvang cue X tract nhdm con the ma tran th dinh thlic dvang la mat nhdm xyclic cap had
55. Gib. sit X la mat vi nhdm (nhan) sink bai mat phan tit a th cap v6 han, nghia la :
X = { a" I n
ChUng minh vi nhdm X (Dinh 11 17) la nhdm ve Cie phan
to cua k la ... ct -3 , (1 -2 , cr i , a° , a l , a2 , a3 nghla la
X = X U{ Cl, C2, a-3, }
77
CHUONG III
VANH VA TRUtING
§1. VANH VA MIEN NGUYEN
1. Venh
Dinh nghlit I. Ta gel la earth met tap help X cling voi hai phep town hai ngOi, da cho trong X ki hiau theo Old tut bang ode dau + va . (ngubi ta thubng ki hieu nho vay) va pi la phep Ong va phep nhan sao cho the diet, hen sau them man :
1) X cling vai phep ging la met nhdm aben.
2) X cling voi phep nhan la met Mat nhdm.
3) Phep nhan &An ph6i dot voi phep Ong v6i cac phtin to lily y x, y, z E X ta c6
x(y + z) = xy + xz,
(y + z)x = yx + zx.
Phan tit trung lap dm phep Ong thi ki hieu la 0 WI gel la phan to khifing. Phan to dal xting (dal veil phep Ong) eta met phalli tit x thi Id hieu la -x Ara goi la ddi cua x Neu phep nhan la giao hoan thi ta bau tianh X la giao hoon. Neu phep nhan c6 phan to trung lap thi phan tit d6 gel la phan tit don vi cunt X va thubng ki hieu la e hay 1 (nau kheng cci ski nhdm Ian).
Vi dg. 1) Tap hdp Z the se nguyen ciing v6i phep °Ong va phep nhan thong thuelng la met vanh giao hoan c6 don vi goi
78
la vanh cac s6 nguyen. Ta cling cd vanh the so hfiu ti, cat s6 time, cac s6 phtic (cac phep toan van la phep Ong va phep nhan thong. thuong).
2) Tap hyp Z/nZ cac s6 nguyen mod n Ming Nth phep Ong va phep nhan ode s6 nguyen mod it (ch II, §1, bai tap 2) va ch II, §2, 3, vi du) la mat Wirth giao hotin co don VI, got la vanh sac s6 nguyen mod n.
3) Tap hop cac ma trait vuOng cap rt,n, >1, (vita cac phan tit lk thud °hang han) cling vet phep Ong va phep nhan ma tran la mgt vanh cd chin vi. Vanh nay kitting giao hodn.
4) Tap lutp cat s6 nguyen la bat ova mat s6 nguyen n > 1 cho fru& la mat vanh vai phep gang va phep nhan thOng fiking. Vanh nay la giao hodn, nhung khOng cd don vi.
5) Tap hop cac ma tram yang cap n > 1 co dang
a l a2 an 0 0 ... 0
0 0 0
trong dd the ai la nhang s6 thuc, ding vat phdp Ong va phep nhan ma tram la mat yank Vanh nay khOng giao hoan, khOng cd don vi.
6) Gia sir X Id mat nhdm giao hoan ma phep toan ki hiau bang da'u Ong +. Tap hyp E cac to :long cau tit X Mu X dityc trang hi hai phep than kat hop : mat mat, ph6p toan Ong (t, f + g((f + g)(x) = f(x) + g(x) vet moi x G X) khian cho E la mat nhdm Ong giao hoan (phein tit khOng la anh xa 0 xac dinh b61 0(x) = 0 vet moi x E. X, phan tit dt3i cua f la anh xa -f xac dinh bed (-f)(x) = -(f(x)) vet mot x E X) ; mat khac, phep than nhan (f.g).—n fg. Th co (g + h)f = gf + hf va fig + h) = = fg + fh That vey, ((g + h)f)(x) = (g + h)(f(x)) = err)) + + h(f(x)) = gf(x) + hf(x) (gf + hf)(x), vb. (f(g + h))(x) = + + h)(x)) = f(g(x) + h(x)) =- f(g(x)) + f(h(x)) = fg(x) + fh(x) = = (fg + fh)(x). Vay cling vat -hai phep toan dd la mat vanh, got la vanh the ty Ang cdu clic" nhim X. Vanh nay cd don vi,
79
do la anh va ddng nhat cim X, nhung khang giao hoan Mu X c6 qua 2 phtin tit.
Ngoai the tinh chat lit mat nhdm tong giao hoan vit mat naa nhdm nhan, mat Nth con cd m¢t sd tinh chat suy ra tit 1u4t phan !Mai.
Dinh li 1. Cho X lez mOt vtinh. Vbi mot x, y, z e X ta cb :
(0 x(y - z) = xy - xz, (y - z)x = yx - zx.
(ii) Ox = x0 = 0.
(iii) x(-y) = (-x)y = -xy, (-x)(-y) = xy.
Cluing minh. (i) Theo Mat phan ph6i ta cd xy = x((y - z) +z) = = x(y - z) + xz. Ta suy ra x(y - z) = xy - xz. Gang thac thu hai chang minh ciing trong tu.
(ii) Theo (i) ta cd Ox = (y - y)x = yx - yx = 0 = xy - xy = x(y - y) = x0.
(Hi) Tit (i) va (ii) ta dttoc x(-y) = x(0 - y) = x0 - xy = = 0 - xy = -xy = Oy - xy = (0 - x)y = (-x)y ; to suy ra (-x)(-y) = xy. Das bi0t, yid moi s6 nguyen n > 0, ta duoc (-xi" = xn thu n than, va (-xi' = -xn mkt n le. n
Tit (ii) ta suy ra Mu vanh X ed don vi va c6 nhieu hon mat phan to thi e # 0.
2. Win can thong. Mien nguyen.
es day ta hay t6ng plat hda khai ni9m vire Ira bai b trong vanh cac s6 nguyen.
Dinh nghia 2. Gia sit X la mat Nth giao hoan. Ta ban mat pilau to a E X la b¢i cim mat phan to b E X hay a'chia cho b, ki hiSu a i b, n6u ed c E X sao cho a = be ; ta can ndi rang b la ube dm a hay b chia hat a, 10 hiau b I a.
Nhtt vay theo Binh li 1 (ii), moi phtin to x e X la Mc aim 0 ; nhung do lam dung ngOn ngit, ngabi ta dials nghia :
Dinh nghia 3. Ta goi la ueic aza 0 moi phan to a # 0 sao cho c6 6 , # 0 thea man quan 110 ab = 0.
80
Ta suy ra ngay tit dinh nghla rang align to 0 va cetc vac Goa 0 khang plied la chinh quy. Trong mat vanh khang c6 Ude Mut 0, 1119i phan to khac 0 deu la chinh quy. That fly, quan he ab = ac Wong duong cal quan he a(b - c) = 0.
Dinh nghia 4. Ta gai la mien nguyen mat vanh ea nhieu Min mat phan tu, giao hoan, cci den vi, khang cci dem cfia. 0.
Vi do : Vanh cac s6 nguyen Z la mat mien nguyen.
3. Vanh con
Dinh nghia 5. Gia xu X la mat vanh, A la mat ho phan cim X an dinh d§ii vai hai phep than trong X nghia la x + y EA va xy E A vei moi x, y E A. A la mat Wink con Mut vanh X nail A ming voi hai phep town earn sinh tren A la mat vanh.
Dinh 11 2. Gid szi A itt mat be phan kb& ring cem mot 'Ranh X Cac dieu kien sau day llz Wong duong :
a) A Its mot yank con ala X
b) Vai nua x, y E A, x + y E A, xy E A, - x EA.
c)Vol moi x, y EA, x - y EA, xy E A.
Chung mink. a) kao then b). Vi A la mot vanh con cho nen ta c6 ngay x + y va xy thuac A vial moi x,y E A. Mat khac vi A la mat vanh nen n6 la mat nhdm veil phep Ong, ta suy ra -x E A eel moi x E A (ch II, §2,2, dinh li 7).
b) keo theo c) theo (ch II, §2, 2, he qua mia dinh II 7).
c) kelp theo a) Trade hat ta chit 3, rang cau phep than cam sinh tren A Ming c6 firth chat kat hop va phan ph6i. Do de Ming vol he qua dm dinh II 7 (eh II, §2, 2) ta c6 A la mat earth con dm X, n
Vi do. 1) BO phan {0} chi gam ea phan to khang va 60 phan. X la hai vanh con °ea vanh X.
2) BO phan mZ g6m the s6 nguyen la bai dm inert s6 nguyen m cho tnidc la met vanh con aim vanh caz s6 nguyen Z.
81
Dinh Ii 3. Giao cua mat hy Mt ki nhung oanh con cua mat vanh X la mat ultnh con am X
Cluing mink Wang to nhu trong nhdm.
Gilt sit U la mot ba phan eta mat vanh X. Th6 thi U chtia trong it nhgt mat vanh con eta X, co th6 X. Theo dinh If 3, giao A cua tat cis the vanh con eta X chda U la mot vanh con eta X chilli U, Wash con nay got la vanh con eta X sinh ra bdi U.
4. Mean va vAnh thircing
Dinh nghia 6. Ta gal la idean trai (idean phdi) eta mot vanh X mat Nth con A eta X filth man diet' kiOn xa E A (ax E A) vai mpi a E A va moi x e X. Mat vanh con A eta mot vanh X goi la mot idean eta X ngu va. chi Mu A vita la idean trai vita la idean phiti eta X.
Tit dinh nghla to suy ra ngay We kink
Dinh Ii 4. M5t 1)5 phan A khac rang eta mot vanh X la mat iciaan cua X ngu va chi nth CAC dial' Men sau them man :
Da - beAvaimpia,bE A.
2) xa e A va ax e A MO a E A va moi x e X.
Vi dg. 1) BO plait' {0} va. IDO plan X la hai 'dean cua vanh X.
2) BO phan mZ gam the s6 nguyen la bet eta mat s6 nguyen m cho truth la mat Wean eta vanh the s6 nguyen Z.
Dinh II 5. Ciao caa mat by bat ki nhang idean cua mat (Ranh X 16 mat idean cua X.
Chang mink Wong to nhu dinh ii 3.
GM sit U is mot ba phan cua mat vanh X. TM thi U Ostia trong no it nhat mat idean eta X, co the' X. Theo 'huh 11 5, giao A cua tat ca cac idean eta X Chita U la mot idean eta X chtla U, idean nay got la idean sinh ra bbi U ; ngu U = = a2 , an } thi A goi la idean sinh ra bat the phgn tit
a2, ..., an . Idaan sinh ra bat mot phtin tit goi lit idean chinh.
82
Dinh li 8. Gid ad X la met udnit giao hoan c6 don uf va al , a2 , an E X Be phan A caa X gam the phan to co long
% Intl + x2a2 + + xnan voi x i , x2 ,..., xn E X la Wean caa X
Unit ra bdi al , a2, ..., an.
Chting mink. Gia sit a = xiai +... + xa, b = v ia l . +... +ynan la
hai phan tit thy y thuoc A va x la mat phan to thy g thutic X. Th ed
a — b = (x iai + + xnan) — (vial + + ynan) =
(x iai — yiai) + + (xnan — ynan) =
(xi — + + Oen — ynian E A
xa = ax= x(xicti + + xnan) = xxisti + + xxnan E A.
fly A la mat idean clia X. A chda the ai vdi i
vi a1 = Oai -F + lai + + Oan , i = 1,2, ...,n.
Cu& ding m9i idean chila an thi ming dada
sla t ,..., xnan vdi x,„ E X va do d6 chda xiai + + xnan .
Kdt Man A la giao cua tat. ca cac idean chda 1a 1 an I tdc
la idean sinh ra beil ct i , a2,..., an. n Tit dinh nghia ta ming suy ra ngay
Dinh li 7. Ngu X let mot vanh co don Ed ua ngu A la mot Wean cart X chda don vi elm X Mi ta co A =
Bay gid ta hay xet mat idean A thy y cim mat vanh da cho X. VI A la mat nhdm con cim nhdm aben cOng &la X, nhdm thitang XIA la mat nhdm aben hoan toan xac dinh then (ch II, §2, 3). Cite phan tit dm XIA la cac Idp khac nhau x + A dm A trong X. Ta hay trang bi cho XIA mat phep Man nhan dd nd trO thanh that vanh.
Dinh li 8. NM A la mat idean cita MO( X, - thi :
(i) Lep xy + A chi phu thwic veto the lap x + A wa y + A ma khong phu thuoc vim sa ltia alum caa the phan to x, y nY the lap do.
83
X/A cling vai hai phep Man
(x + A, y + A) x y + A
(x + A, y + A) n—• xy + A
lit mat vanh got vanh (Mang Leta X tren A.
Chang mink. (i) Gia sit x' +A = x +A va y' + A = y A. Vayx'-x =a EAva y' - y=bEA, hayx'=x+a
va y' = y + b.
Do d6 x'y' (x + a)(y + b) = xy + ay + xb + ab. Vi A lit met Wean, nen tit a, b E A ta stor ra ay, xb, ab E A fly x'y' + A = xy + A. in ki hieu (x + A)(y + A) = xy + A. Nhu vay ngoai phep Ong (x + A) + (y + A) = x + y + A trong X/A, ta cd Olen nhan xae dinh boi (x + A)(y + A) = xy + A.
(ii) Phep nhan trong X/A re rang la Ice% hap do tinh kat km caa Olen nhan trong X. Ngoai ra to cling cd hat phan ph6i. Vay X/A la mat vanh. Neu X la giao holm thi X/A cling giao hogn. Neu X c6 don vi 1 thi 1 + A la dun vi cua X/A. n
Vi du. Vanh (hung cua Z tren idean nZ gyi la vanh cac s6 nguyen mod n. Phep Ong va phep Ethan trong Z/nZ xae dinh boi
+ nZ) + (y + nZ) = x + y + nZ
(x + nZ)(y + nZ) = xy + nZ
KI hieu x + nZ bang x vet lay n = 4 ta cd bang Ong va bang nhan cua Z/4Z nhu sau
5
5 0 I
I T. 2 a 5
2 a 5 I
a a 5
I
5 5 0
a 0
o a 2. I
84
5. DEng cau
Dinh nghia 7. Mat clang can (vanh) la mat anh xa t,i mat vanh X dgn mat vanh Y sao cho
f(a + b) = f(a) + f(a)
gab) = f(a)f(b)
trig mot a, 6 E X. Nan X = Y thi cling eau f got la milt to dung eau ena X
'Pa acing dinh nghia don cgu than eau, clang clu Wang tv nhu da dinh nghia trong nhdm.
Vi du. 1) Gia sit A lit mat vanh con dm mat vanh X. Dan anh chinh tan
A --a X
a a
la mat ddng eau goi la don cgu chinh the.
2) Anh xa eking nit& dm mat vanh X la mat dung eau, gal la to clang cau ding nhgt cim X.
3) Gia sit A la mat idean ciut mat vanh X. Anh xa h : X XIA
x - • x + A la milt (long eau tit vanh X den Nth thvang X/A. Dong eau nay can la toan cdu goi la toan eau ch{nh tac
4) Gia sit X va Y la hai vanh, anh xa
Y x 1—n 0
WA 0 la ph/in to kheng cua Y la mat clOng egu goi la clang cdu
Duren day chting to hay dtia ra cac (huh Ii tuang to tthu trong nhdm ma vigc cluing minh, hoac Wang to hoc gip dung the lc& qua trong nhdm, xin nhudng cho CrOC gia.
Dinh li 9. Gig au X, Y, Z la nhang vanh, f : X —. Y va g : Z l¢ nluing rthng can. Thd thi tich Guth xa
85
gf : X -> Z
cling la mot deng eau Dac bit tich 'caa hai clang cdu la mot ddng cdu.
Dinh li 10. Gid sit f : X -• Y Ca mot (Ong cdu tit met ultn th X din met Minh Y. Thd thi :
(i) f(0) = 0
(ii) f(-x) =-f(x) udi moi x e X.
Dinh 1i 11. GM sit f : X -I. Y la mot deng cdu tit mot uanh X den mot uttnh Y, A la mot uanh con cud X dix B ld mid idean elm Y. Thi thi :
(0 flit) la mot uanh con edit Y.
(ii) ri(R) ld mot idean dm X.
HO quit. GM set f: X -A, Y ld mot deng cdu tit met uanh X din mot uanh Y The. thi Imf la mot uanh con ce a Y ua Kerf /e/ mot idean cud X.
Dinh Ii 12. Gid sit f :X Y ltz mat tang cdu tit mot ultnh X den mot uanh Y. Thu thi
(i) f lit mot toan dnh nezt vet chi nia Imf = Y.
(ii) f la mot don dnh next uo, chi mitt Kerf = {0}.
Dinh li 13. GM 80 f : X --•Y let mot clang cdu to mat uanh X din mot ain't Y, p :X -* X/Kerf M town aid chink Mc (ui du ,3) tit uanh X din uanh thuang cad X teen Kerf
Thi thi :
(i) CO mot deng nen day nhdt : X/Kerf Y sao cho tam gide
f X Y
X/Kerf
Id giao hod?).
(ii) Deng cdu f la mot don cdu yet Im 7 = f(X).
86
He qui. Vol mot clang eau f : X —n Y tic mot Minh X din mot vanh Y, to to
f(X) X/Kerf
BAT TAP
1. Ch'ing minh cac tap hop sau day voi phep Ong va phep nhan cac se lap thanh mat vanh
a) Tap hop the s6 c6 clang a + b 4 vat a, b E Z.
b) Tap hop the sti pink c6 clang a + bi voi a, b E Z.
Chting minh cac vanh et6 giao hoan c6 don vi.
2. Chang minh tap imp the ma tram vuOng cap n vdi cac phan tit la nhfing s6 nguyen lam thanh mat vanh voi phep Ong va phep nhan ma tram.
3. Ching minh tap hop cac da thdc cua x \di he so nguyen lam thanh mat vanh voi phep Ong va phep nhan da that.
4. Gia sU da cho trong mat tap hop X hai phep toan Ong va nhan sao cho : 1) X Ming vei phep Ong la mat nhdm ; 2) X Ming veil Olen nhan la mat vi nh6m ; 3) phep nhan pit ph6i d6i voi phep Ong. Chting minh X la mat vanh
5. Tim cac udc oda khOng trong vanh Z/6Z.
6. Chang minh Z/nZ (it x 0) la mat mien nguyen khi va chi khi n la nguyen td.
7. Chang minh tap hop X = Z x Z ding vdi hai phep toan
(a t, b i ) + (a2 , 6 2 ) = (a1 + a2, b i + b2)
b 1 )(a2, b 2) = (a tat, b 162 )
la mat vanh giao hoan, c6 don id.. Hay Um tat ca the vac dm khOng vanh nay.
8. Gilt au X la mat vanh ea tinh chat sau day :
x2 = xvdi mm x E X.
87
Cluing minh rang a) x = -x yea mai x E X . b) X la vanh giao ham. c) Neu. X la vanh kh6ng cd tnic dm 0, cd nhieu hon met
phan thi X la mien nguyen .
9. Clic vanh b Mc bid tap 1), 2), 3) la tug vanh con cha nhirng vanh nao ?
10. Cho X lit met vanh thy j, A vb. B la hai idean elk X Chung minh rang be phan
A +B= {a+b I aEA,bE B}
la met idean dm X.
11. Cho X la met vanh thy y , n la mat s6 nguyen cho tru0e. Chang minh be phan
A=IxEXI nx = 01 la met idean cha X.
12. Cho X la mat vanh thy j, a e X. Chung minh rang 136 phan oX=faxIxEX1
la met idean phai cha X, va be phan Xa = {xaIxEX}
la mat idean trai cha X.
13. GM sit X la met mien nguyen va n la cap (oh II, § 2, 2, dinh nghia 5) dm phan iv don vi e. Chung minh
a) n la met s6 nguyen te, b) Mei phtin tv khac khang x e X cd cap n. c) Be phan
mX = { mx I x E X}, vbi m la met se nguyen cho trite; la mat idean dm X.
d) X/mX = X nen in la b6i eta it,
XlinX = {0} nen in kheing phai la 136i cim n
88
14. Gig. sit X le mot vanh giao hoan, th don vi. MOt idean A m X dm X g9i la Wean t6i dui va chi nth the idean caa X elute A chinh la X va ban than A. Mot idean P # X mia X goi la idean nguyen t6 ngu• va chi lieu veil u, u E X tich nu E P thi u . E P ha4c u E P, ChUng minh rang :
a) X/P la mien nguyen khi va chi khi P la idean nguyen t6.
b) XIA la trueing (V) khi va chi khi A la t6i dal.
15. Gig!. sir A la mot vanh, B 1a rat tap hop c6 hai phep toan Ong va nhan, va f : A B la melt song anh thiiit man
f(a + b) = [(a) + f(b)
f(ab) = f(a)f(b),
yea moi u , b E A.
Chung minh : a) B la mot vanh.
b) Nth A la vanh giao hoan thi B cung la vanh glad hoan.
c) Ngu A la vanh th don vi thi B cung la vanh th don
d) Ngu A la mien nguyen thi B cung la mien nguyen.
16. Hay tim tat °A cac to clang eau tha vanh the s6 nguyen.
17. GM sit f : -+ X la mot to clang eau cart vanh X. Chfing minh rang tap hop
A= {x E X I f(x) = x)
la met vanh eon ella X.
18. Gia sit X la.mot vanh thy y, Z la yanh the s6 nguyen. Xet tOp hop tich X x Z. Trong X x Z to dinh nghia the phep toan nhu sau :
n i ) + (x2 , n2) = (x 1 + x2 , n i + n 2)
(x1 , n1) (x2 , n2) = (x i x2 + n1 x2 + n2 x 1 , nt n2)
a) Chung minh rang X x Z la mot vanh cd don vi.
b) Anh xa
89
f : X X x Z
x (x, 0)
la mat don Su.
19. Cho A va B fa hai vanh tity g. Xet tap hop tfch d4 cat X = A x B. Trong X to dinh nghia cat phep teem
(a, b) + (c, d) = (a + c, b +
(a, b)(c, d) = (ac, bd)
Chung minh :
a) X IA mat vanh.
b) Cac be PbatCA = {(a, 0) la EA.1 va B = f(0, b )1 b e B} la nheng vanh con aim X clang cau theo Cul to val. A va B.
C) A Ira Ft la hai idean aim x sao cho A fl B = 110, 0)} va X = A + B (bat tap 10).
d) GIA sit A va. B IA nhang vanh S don vi, hay tim cat don vi caa X A va B.
20. Gia sit X IA mat vanh va a E X. Chung .minh rang :
a) Anh xa
ha : X X
x ax
la mat Hong cau (nhdm) tit nhdm Ong aben X dan nhdm Ong aben X.
b) Anh sa
h : X E
a h(a) = ha
M mat Hong Su tit Nth X clan vanh E the to d6ng eau aim nhdm ()Ong aben X (1, of chi 6).
c) Tim Kerh. Chang minh rang h la don cau khi X cci don vi.
21. Gia sn't X va Y IA hai vanh, f : X Y la mat d6ng Su to vanh X d6n vanh Y, A va. B theo thd to la hai idean mkt
90
X Ira Y sao cho f(A) C B, p . X —> X/A va p' : Y Y/B la the
Man egu chinh tat (5, vi du 3). Chung minh rang t6n tai m6t
dung cgu duy nhgt f tb X/A dgn Y/B sao cho hinh vuOng
I X Y
p P'
i X/A Y/B
la giao hot', We la 7 p = p'f
Ngu f la mot toan cgu thi f cd phai la mgt than cgu hay
khOng ?
§2. TRITON°
1. Trubng
Dinh nghia 1. Ta pi la &Wong mgt mien nguyen X trong
dd 1119i phan tit Ithac khOng deu cd mgt nghich dao trong vi
nhdm nhan X. fly mOt vanh X giao hodn, cd don vi, cd nhigu
hem mgt phgn to la mOt trubng ngu vitt chi ngu X - {0} la mOt
nhdm &Si vOi ober) nhan cim X.
Vi dy. Tap hop Q cat s6 hito ti ding vtli phep cling va phep nhan cat s6 la mOt trubng. Ta cling cd trubng sd that R va
trubng s6 phtte C.
2. Tnatng con
Dinh lights,. 2. Gilt sit X la mOt trubng, A IA mOt bo phdn
dm X do dinh d61 volhai phdp than trong X. A la mgt trubng
con eta trubng X ngu A ding volhai phdp than cam sinh tit
A la mgt trubng.
Tit dinh nghia 2 vb. lip dung hg qua cna dinh li 7 trong (ch II, §2) cling vesi dinh li 2 (§1, 3) to dusc
91
Dinh U 1. Gid sit A ha mat 60 ghat' c6 nhigu han mat plain t* cga mid trubng X. Cat diet< kiln sau day la tuang duang :
a) A la mOt truang con cart X.
b) V6i moi y E A, x + y E A, xy E A, -x E A, x 1 EA ntiu x # 0.
c) Vol moi x, y E A, x.- y E A, xy-1 E A ndu y x. 0. Vi du . 1) X la trubng con cim trubng X. BO phan {0} khOng
phiti la mOt trubng con cim X, vi then dinh nghia mOt trotting cd it nhgt hai phgn ti.
2) Trubng s6 hilu, ti Q la trailing con cita trubng sti thitc R, ban than R lai lit trubng con aim trotting se phile C.
Coi mOt trubng X nhu mot vanh, ta cd the" at van de xet cac idean cim X. Mac du mOt trubng X cd thd cd nhieu trubng con, nd lai chi ad hai idean tarn thubng : X vit {0}. Thuc vay, gia stl A la mOt idean dm X va A A {0}. Vgy ad Wit phan to x m 0 thuOc A. Vi A la mOt idean nen x 1x = 1 E A. Do do A = X (§1, 4, dinh If 7). \ray hat nhan ola moi dgng eau
f : X Y
tit IS trubng X den mOt Dating Y chi cd thd hoot 11 {0} hot1c la X. Trong truitmg hop Kerf = {0} thi f la mot (km eau (§1, 5, dinh If 12), Kerf = X thi f la deng can khOng.
3. Tnsbng cac thuang
Gia sit X la mOt mien nguyen. Theo dinh nghia, the phtin tit khac 0 am X deu la chinh quy, nhung dieu dd churl dam bac thong cd nghich dao trong X, chang han trong vanh cac se nguyen Z chi trit 1 va -1 cd nghich ciao trong Z con the s6 khac thi khOng. Bay gib ta dat van de nhting X vao mOt trubng X 'sao cho moi phgn tit khac 0 cim X c6 nghich dab trong X . X la mOt mien nguyen nen X la mot vi nhdm nhan giao hoan. Theo (ch H, §2, 5 dinh li 17) ta cd the' nhung X vac, mOt vi nhdm nhan giao hoan X sao cho mid phan td chinh quy dm X cd nghich ciao trong X. Nh'u ta cd thg trang bi cho X them
92
phep toan cong de X In met trurong thi van de doge giai guy& xong. Cu the' ta co
Dinh Ii 2. GM sa X- la mat mien nguyen, ba phan Sc phan to khac 0 tha X. CO mat truong Tc viz mat don eau (vanh) f tit X can X co cat tinh eh& scut :
1 °) Can phan tEl cat' X co clang f(a)f(b) 1 vOi a e X, b e
29) Cap ( , p Ca duy nhdt sal kOm mat thing edit, nghia la neu co cap (Y, g) the* man cti6u kin 1 ° thi 60 mat thing ctiu so : Y sao cho tam slew
f X X
8\c/ W
Y
giao holm.
Chting mink Vi X la met mien nguyen nen X la mot vi nhdm nhan giao hoan. Ta xay chIng vi nhdm nhan giao hoan X va
anh xa f nhur trong dinh 11 17 aim (ch II, e2, 5). Bay gig ta hay xet hai phgn tit x = (a, b) va y = , d) thy y cud X .
Phan tit (ad + bc, bd) chi plan thuoc vao x va y ; that vay
gig. sit x = (a', b') , fly ab' = a 'b, ta suy ra (ad + bc)b'd =
= (a'd b'c)bd, Bie la (ad + be, bd) = (a'd + b'c , b'd) . Thong
tp clgi yea y. 'Pa thay ngay rang X ding yea phep cong . (x, y) x + y = (ad + bc , bd) lit met nhdm giao hoan ; phgn to
kheing la (0 , 1) ; phtin tit c161 mia x = (a, b) lit -x = (-a, b).
Mat khac X Ming yeti phdp nhan (x, +Y =(a, b) (c, d) = (ac, bd) , thy tada bigt, la met vi nhcim giao hoan. Them nua ta cd thd thit dg thgy phep nhan phan ph6i clOi yea Thep cong. Vey X la met vanh giao hoar( cd don vi. Mai
phan to x = (a, b) khiic phan tit (0, I) , do do a m 0, cd nghich dao la x I = (b, a) . Vey X la met batting. Anh xa
f : X cho toting -Ong voi mei phan t9 a E X plain tii (a, 1) E X th tinh chat
f(a + b) = f(a) + f(b)
93
ngoiti the tinh chat do Met trong dinh li 17 dm (eh H, §2, 5). Do do f la mot ddn eau (vanh) tit X den X . Cap (X, f) xay dung nhu vay cd tinh chat 1°. Cuoi thing gib. sit (Y, g) la cap cling thea man tinh chat 1 ° nazi bai toan..Qnh xa
: Y
x = f(a)f(b) 1 g(a)g(b) 1
la met song anh ban ten phep nhan : so(xy) = w(x)(o(y), nhu ta da bia Mat khan ta cling a clang thit rang
co(x + y) = so(x) + co(y) yea m0 x, y E X .
' Tit d6 p la met clang cau va hien nhien ta th g = pi
Vi f la mot don eau (vanh) nen ta (tong nhat the phan to a E X yeti the Odin to f(a) E too. Ltie de mei phan tit xe.k co the vial &MK clang- a ex va beg*. Ta eon Id hieu phan tit ab-1 bang alb (ch II, §2, 1). Vay a = all = ablb voi 1110i a E X va b E X. Hai phan Di a/b 'Tack! la bang nhau khi va chi khi ad = bc. Ciie quy the cOng va nit la
a/b + cld = (ad + bc)/bd
(a/b)(e/d) = ac/bd.
Doi ella a/b la -a/b = a/-6. Nghich dim eim a/b vii a m 0 la b/a. Ta thay lai cat guy Sc efia se Mtn ti.•
Dinh nghia 3. Gia sit X la mot mien nguyen, rc la mot truing: X goi la mot truing cac thuong mitt mien nguyen X Mu co 'Mt ddn eau (vanh) f : X -> X sao chi cap ( X , f) them man tinh chat 1 °) caa dinh li 2.
Tit dinh nghia 3 va dinh li 2 ta suy ra trading the thuang cast moi mien nguyen ten Da WI xac dinh duy nhat (ly lai met deng eau).
Vi du. Truing cac thuang cim vanh cat so nguyen . Z la tinning cac se huu ti Q.
• BAT TAP
I. Chang mink mien nguyen hitu han lit mot truing.
94
2. Chung minh vanh the ea nguyan mod n IA mot trading khi va chi khi r7 la nguyen t6.
3. ChUng minh rang trifling cac so him ti khang c6 trtiOng con nao khac ngoai ban than nd.
4. Chung minh rang 14 than
A= ja +6 tff I a , 6 E mot trifling eon aim trilling s6 that It.
5. ChUng minh rang bO phan
A= {a + b + c I/41 a, b,ceQ}
la mot truang con dm trtiang s6 tittle Ft
6. Gia sit X la mot trading, e is phgn to dun vi dm X. Xet 136 phan
A= {me I n E Z}
a) Chung minh A IA mat vanh eon etia vanh X, A ca phai la.mot mien nguyen khang ?
b) Chting mink A dang eau cal vanh cac s6 nguyen Z khi e ea cap vo han va clang eau yeti vanh min s6 nguyen mod p khi e ad cap p
c) Trong tniang hop e al cap p, hay cluing minh A IA mot trifling.
7. Gia sit X IA mat trifling, Y la mat tap hop da cho ding vlii hat phep twin tong va than trong Y, I : X Y la mat song anh td X dgn Y them man
f(a + b) = f(a) + f(b) f(ab) = f(a)f(b)
vol moi a, 6 E X.
Chang minh 'Y1a mot trudng va X = Y
8. Hay tim :
a) Cae to tong eau dm truling cac sd Mtn ti.
b) Cac tp d6ng eau ciut trifling cac s6 phtic gift nguyen cac s6 thuc.
c) Cac to clang tau cua truong the sa thug
95
9. Chung minh rang tap hop the ma trail ed clang
vai
a b {-6
a, b IA !lilting se thuc la mat truxig d6i yid phep Ong NT/ phep nhan ma tran, trueing nay dang thu trtaing the sd pink.
10. Chung minh rang tap hap the ma trail c6 clang
[ a el 26 a
vie a, b E Q IA mat trtiang the phep Wan van la the phep eang va nhan the ma tran) clang eau WO trtiong A 0 bid tap 4.
11. Tim tttiong the tithing eta mien nguyen A trong bai tap 6.
12. Chang minh rang mai trithng deli co trutng con be nhat ((man he thd to la quan he bao ham) clang thu hoc yeti trubng se htiu ti, hone yid trtiOng the s6 nguyen mod p vii p la mat se nguyen t6 (bai tap 11).
13. Gia sit X la mat &tieing, A la mat vanh con eta vanh X.
a) Chang minh rang nail A co nhieu hon mat phan t8 va A c6 don vi, thi phiin td don vi eta A triing via phan to don vi eta X, va lue do A la mat mien nguyen.
b) GM sit A lit mien nguyen. Ching minh rang ba phan
P = (6-1 I a, e b O )
lit mat tniiing con eta X vb. P la trueng the thuong eta A.
c) Chang minh rang P la trifling con be nhat trong the throng con eta X chda A.
14. Gia sit p la mat 56 nguyen to. Ching minh rang tap hop cite s6 hitt ti ed clang mkt, trong do n nguyen to via p, la mat mien nguyen. Tim trueing the thuong eta mien nguyen thy.
96
CHUONG IV
VANH DA THUG
§1. VANH DA THHC MOT AN
1. Vann da then mOt 6n
nal niem da flak la khai niam ma ta da lam quen it nhiau a ph6 thong. 111 got la da Oak, met tdng c6 (bang
a, + a ix + + am?"
trong d6 cat a, i = 0 n, la nbitng s6 that va x la met chlt.
Phe-p Ong va phep nhan da thdc la
(a, + a ix + + amx") + (b0 + b ix + +b/) =
a, + b o + + (am + em).?" + in +1 + + bnxn .
(a, + a ix + ...+ anin)(60 + b iz + ...+ bnx") =
(Lob, + (aob i + aib0)x + + (aobk + + +I
+ akb„)xk +... + amen?" trong d6 ta gift sit n Tn.
es day thong ta hay dinh nghia da thdc met each tdng quat Jadn va chinh xac hon. Gilt ad A la met vanh giao hoan, cd don vi ki hieu la 1. Goi P la tap help the day
(a,, ai ,.., ,...)
trong do cac al e A vai moi i E N va bang 0 tat ca trit met s6 hem han. Mau vey P la mot be phan via lay thaa de` cam 441.!
97
(ch I, §1, 15). al dinh nghia phep ceng va phep nhan trong P nhu sau
(1) (an, air -, + (be, =
(a0 + be, ai + an + be ,.
(2) (a0 , , ae,..)(be, b p ..., =
(co) eV - c —)
vet
akbo = E abi, k = 0, 1, 2,... ek = aobk i+j=k
VI cdc al va bi bang 0 tat ca trit met s6 huu han nen cdc a1 + bi va ci ming bang 0 tat ca trit met se huu han, cho nen
(1) va (2) cho to hat phep toim trong P. Ta hay chdng minh P la met vanh giao hoan cc) don vi. Trttec Mt hi4n nhien pile)) Ong la giao holm M kat hop Phan tit khong la day
(0, 0, ..., 0, ...),
Agri tit dal cila day (a0 , a,,,..) la day (— a0, — ar,...,
— an,..). Vey P la met nh6m Ong giao hoan. VI A la giao hoan, nen
i+j=k i+j=k
do do pile)) Men la giao hoan. Do phdp Man trong A c6 tinh chat kat hop va phan phtli dal vai phep Ong, nen vol mei m = 0, 1, 2, ... -ta cd the viat
E ah E bit/ ) = G ah(bfj) = h+k=m +j=kh+i+j=m
E (alibi) = E ahbi )
h +i+j=m j+1=m ‘1.1-4-i=1
tit dd to al Agri nhan trong P la kat hop. Day
(1, 0,..., 0,...)
98
la !Than tis don vi cim P. Vay P la 1111# vi nhdm nhan giao hoan t Cu6i cling lug phan phdi trong A cho phen ta vigt
E cti(bi + = E alb,/ + E aPj i+j=k
vol moi k = 0, 1, 2, ..., trong - P.
Bay gib ta hay Kat day
x =
at co then quy tac nhan
x2 = k3 = • •
=
i+j=k i+j=k
ta suy ra
(0, 1, 0
(2)
(0, 0, 1, Q...
(0, 0, 0, 1,..
(0, 0,..., 0, 1,
tit do lust phan phdi
0,
n
lh quy valc vi&t
x° = (1, 0,
Mat khac ta xet anh xa
A v P
a (a, 0„.., 0, ..)
Xnh xa nay hign nhien la mat don eau (vanh) Do do tit gib ta clang nhat plign to a E A veil day (a, 0, ..., 0, ...) E P, tit vi vay A la mat vanh con coo. vanh P Vi mai phan to cim P la mat day
(%,ate , ...)
trong do the ai bang 0 tat ca trit mat so` hitu hen, cho nen mai
phalli to caa P co clang
(cto, a lp ..., ac , 0,...)
trong cid a0 , E A kh6ng nhat thigt khac 0. \riga (long nhat
a vOi (a, 0, 0, ...) va viac dua vao day x cho phop ta vigt
99
(a., %, 0, ) (a., 0, ) + (0, a1, 0, ..) + + (0, a., 0, ...) = (a., 0, ) + (a i , 0, ) (0, 1, 0, ...) +
+ + (an, 0, ) (0, . 0, 1, 0, ) = a. + aix + + ax" =
= anx° + st lx + + an x". Nob( to thubng ki hieu cac phan td elm P vier dual clang
+ aix + + an?
bang f(x),
Dinh nghia 1. Vinh P goi la heath da thhc ctia tin x idly he hi trong A, hay van tat vInh da thac cua an x tren A, va hiOu la A /X/ Cite Orin tit cua vanh dd goi la da thee oda
an x lay he trong A. Trong mat da thdc
f(x) = ox + aix + + an?.
cac i = 0, 1, n gal la cac he hi caa da thdc. Cac as goi la cac hang hi mist da thitc, dim biOt ace = o goi la hang hi hi do.
2. Bac ctla m01 da thac
Xet mat day (an, a. . , an, )
thuoc vanh P Vi cac ar bang 0 tat ca trb mot s6 hitu hart nen nau
thi bao gib cUng cd mat chi s6 n sao cho a.x 0 ye a, = 0. I > n. Theo nhu tren, to vita
(an , ..., an, 0, .) = + aix + an?
Dinh nghin 2. Elbe caa da thud khac 0
f(x) = aox° + +an _ +
vbi a. * 0, n > 0, la n. He to a. goi is Ile to cao nlilit cila fix)
100
NMI vhy ta chi dinh nghia bac coa mat da flit khac 0. D61 vii da thdc 0 ta bao no khOng c6 bhc.
Dinh li 1. GU/ sit f(x) ua g(x) Ia hai da Mac kheic 0.
(i) Neu bac f(x) khac bac g(x), thi ta co
f(x) + g(x) m 0 ua bac (f(x) + g(x)) = max (bac f(x), bac g(x)).
Neu be f(x) = bac g(x), on neu them nha
f(x) + g(x) # 0, thi ta co
bac (f(x) +g(x)) s max (bat f(x), t4c g(x)) .
(ii) Neu f(x) g(x) x 0, thi ta co
bac (f(x) g(x)) Lc. bac f(x) + bac g(x).
Viac chung minh khOng c6 gi kh6 khan, xin nhubng cho ban doe
Dinh li 2. Neu A la mot mien nguyen f(x) tia g(x) la hai da Mac kluic 0 cart vanh A[x], thi f(x) g(x) # 0 Mr
bac (f(x) g(x)) = bac f(x) + bac g(x).
Chang mink. Gia sd f(x), g(x) E A[x] la hai da thdc khac 0
f(x) = ac + + amain (am # 0)
g(x) = bo + + (bn # 0)
Theo quy tac nhan da thdc ta
f(x)g(x)= aobo + ...+(a.bk + +akbo)xk + + amboxn+m .
am va b. Winn 0, nen amb # 0 (A khOng Tide cda kliOng), do
d6 f(x) g(x) # 0 va bac (f(x) g(x)) = m + n = bac f(x) + bac g(x)1
HO qua. Neu A la mien nguyen, thi A [x] cling In mien nguyen.
3. Phep chia vii du
Trong muc 2 to da thdy ngu A la mat mien nguyen thi A[x] clang la mat mien nguyen. Ta td dat eau hei : ngu A la mat tutting thi A [x] co phai la mat tuning khOng ? Cau hOi duqc
101
tit lbi ngay hie khac, A[x] khong phai la met truing vi da three_ x chang han khong co nghich &to. Thy fly trong truing hop nay A[x] la met mien nguyen dew Met, no la met vanh oclit (eh V, §2) nghla la met vanh trong di( cc( phep chic voi du.
Dinh 11 3. Gid sit A la mot trilling, f(x) va g(x) x 0 la hai da three ezia vanh A[x] ; thd thz bao gib wing co hai da three day nhat q(x) va r(x) thitec A silo cho
f(x) = g(x) q(x) + r(x), noi bdc r(x) < bdc g(x)
nen r(x) # O.
Cluing minh. TrUtIc hdt ta hay chling minh tinh duy nha
fix) = g(x) q'(x) + bac r'(x) < be g(x)
neu r'(x) # 0
Ta suy ra
0 = g(x) (q(x) - ex)) + r(x) - r'(x). IsIdu r(x) = r'(x), ta co g(x) (q(x) - q'(x)) = 0, vi g(x) # 0 va A[x] la met mien nguyen, nen , suy m q(x) - q'(x) = 0 We la q(x) = q'(x). Gia sii r(x) # r'(x), vay
bac (r(x) - r'(x)) = her (g(x)(q(x) - qrx)) = bac g(x) + bac (q(x) - &fix)) (dinh If 2).
Mat khac theo gia thiet va dinh II 1
bac (r(x) - r'(x)) S max (bac r(x), bac r'(x)) < bac g(x) bac g(x) + bac (q(x) - qlx)),
&du nay mau thuan voi clang thile tren. Chu y : nau met trong hai da thdc r(x) va r'(x) bang 0 thi ta khong thd nit deli bac
nhung then do khong anh hureng Uri viec chdng minh, vi 10c do bac (r(x) - r'(x))
, bang W.c r(x) nen r'(x) = 0 va bang
bac r'(x) ndu r(x) = 0.
Con sit ton tai cua q(x) va r(x) thi suy ra tii thuat Wan dud! day. Tim q(x) va r(x) goi la three hien phep chia f(x) cho g(x). Da Utak q(x) goi la thuong, da thttc r(x) la du cua f(x) cho g(x). Viec tim thuong va du la tic Mac ndu b4c f(x) < bac g(x). Ta
set
102
chi can dat q(x) = 0, r(x) = f(x). Trong twang hap trai lai ta dung nhan xet eau day :
Neu ta biet mot da attic h(x) sao cho
fi(x) = f(x) - g(x) h(x)
cc) bac tittle stir be hen bac cua f(x) thi bai Man tra thanh don gian tam : tim thadng va du cha fi (x) cho g(x). That vay, nee fi (x) = g(x) qt (x) + r i (x), ta guy ra
f(x) = g(x) (h(x) + q i(x)) + r i (x)
tit do
q(x) = h(x) + q i(x), r(x) = r i (x)
Trong tittle tihn, vat
f(x) = + + + ao
g(x)=bne +bn _ixn 1 + + be , br, m 0 va n 5 m to nhan xet
am rang, My h(x) = thi da thtic fi (x) =
ce bac tittle sit be him bac cfm f(x), hoac fi (x)
tniang hop f (x) = 0, dv r(x) = 0 va thitang q
fi (x) x 0 ta tip tut vai fi (x), ta dude f2(x)... Day da tilde fi(x), f2(x)... ce bac giam dam Khi ta di den mot da tittle ce bac flute sit be ban bac mitt g(x) thi da thIc d6 chinh la du r(x.). Ngu mot da thtic cfm day bang 0 thi dv r(x) = 0. De nhin thay 1.6 hon ta hay vast ra the bade ma ta da thuc hien d6 duqc day fi (x),
fi (x) = f(x) - g(x) h(x)
f2 (x) = ft(x) - g(x) hi (x)
fk(x) = fk_ 1 (x) - g(x) hk_ k (x)
vai fk(x) = 0 hoac bac fk(X) < be g(x). Ding va yea ve cite clang
thtic de 1ai, ta duqc
f(x) - g(x) h(x)
bang 0. Trong
(x) = h(x). Ngu
103
f(x) = g(x)(h(x) + h i (x) + + hk _ 1(x)) + fk(x),
tit do
q(x) = h(x) + hi(x) + + hk _ 1 (x), r(x) = fk(x). n Vi du. Trong time flan cla time hien chic fix) cho g(x),
no.tbi to sap did nhu sau da lap day fi(x), f2(x)...
1) A la tntbng 56 him tl.
—x3 — 7x2 + 2x — 4 — 2x 2 + 2r — 1 1
—x3 + x2 — —2 x
1 x + 4
—8x2 + —2 x — 4
— 8x2 +8r —4
11 2 x
Tv d6
—x3 — 7x2 + 2x —4 =
(-2x2 + 2x — 1) x + 4 ) — x.
2) A la trubng cac s6 nguyan mod 11
—1x3 — 7x2 + ix — 4 - aX2 aX - - Tx3 + 1x2 - 5x 6x + 4
- 8X2 aX - g
- 8X2 gX - g
0 Vay
—1 — 7x2 2x — 4 = (-2x2 + 2x — 1)(6x + 4)
104
• Tit (hob nghia 2 (ch III, §1, 2) va djnh 11 . 3 ta co the khAc ha qua.
H. qua. f(x) chia hit cho g(x) khi va chi *hi du trong phep chin f(x) cho g(x) bang 0.
4. NghiOm cga mOt da thtic
Dinh nghia 3. GM si c la mOt phan tit tag g eim vanh A, /la/ = an 4- apc + + ani la mOt da thlic thy 3, cim vanh Aix] ; phiin tit
f(c) = an + aic + + ane e A dttqc bang each thay x bai c goi la gin tri caa f(x) tai c. Ngu f(c) = 0 thi c goi la nghigm caa f(x) Tim nghiem ciut f(x) trong A goi la gidi phuong trinh dui s6 b4c tt
ant + + an L-- 0 (an *
trong A.
Dinh H 4. Gid sit A la mdt &yang, c E A, f(x) E Aix]. The can pile)) chia f(x) cho x - c In f(c).
Ching mink. Ngu ta chia f(x) cho x - c, du hoac bang 0 hoac la mat da tilde bac 0 vi bac (x - c) bang 1. Slay du la mat phan tii r E A. Ta cd
f(x) = (x - c) q(x) + r Thay x bang c, to dupe
f(c) = 0 q(c) + r, vfly r = f(c). n
HO quA. c la nghiqm caa f(x) khi vet chi khi f(x) chin hit cho x - c.
Thqc Man phep chia
f(x) = anx" + 1 + + an
cho c, ta dude cac he td cua da tilde thing
q(x) = b oxn i + b izji 2 + + n _
105
cho belt cac cOng flute
bo = ao , bi = + cbi - 1
va du
r = an + _ 1 .
Vi r = f(c), ta say ra mat pluming phdp (phuong plulp Hoocne) dg tinh f(c) bang so dc" eau day :
a a, a
chb 0 n -1
trong do rani phan to taa clang thd nhi doge bang each cOng vao phan tit taring ang mla ding thil nit& tich cita c voi phan to dilng trot& clang tht? nhi.
Dinh nghia 4. Gia sit A la mat tniang, c E A, fix) E Aix] va m la mat ea to nhien 1, c IA nghiem bei cap m nau va chi nal' f(x) chia hat cho (x - cfn va f(x) khOng chia hgt cho (x - Trong trUang hop m = 1 noted ta can goi c IA nghiem don, In = 2 thi c la nghiem Jeep.
Ngtted ta coi mat da thttc ea mat nghiem bed cap m nhu mat da thole c6 m nghigm trimg vai nhau.
5. Pit W. dai sd ve phiin td• sieu vi5t
WA sit A la mat truong con cim mat truitmg K, c E K va f(x) = ao + aix + + ane
la mat da tittle dm. vanh AM. Lac do, vi ao , ..., an E A nen
ao, a1, ..., an E K, do de ta co thg coi f(x) la mat da thim lay he tit trong K va f(c) la mat phan to thuOc K. Ngu f(c) = 0 thi c goi la nghiem cka met da this° lay he to trong A.
Dinh nghia 5. Gial sit A IA mat truting con caa mat truting K. Mat phan to c E K goi la dai so tren A nau c la nghiem dm mat da Glue khan 0 lay lag to trong A ; c goi la sieu Met tren A trong trttang hop trai
106
/qui vay bito c lit dai ad trait A cd nghia la ton tai nhiing pilau tit ai e A (0 s i f ti) khang bang khang tat cat, sao cho
ao aic + + anon = O.
Vi du. 1) Cac phgn td aim truang A clau dai ad trail A.
2) Trong truaing sd thtte R, la dal sd trail tniang ad him ti Q, n la kali vial trait Q. Trong truting s6 pinic C, mgi sd plata la dai s6 tren truaing sd thue It. Thvc fly, MO ad pink
z = a + bi la nghiem efut da thaw x2 — tax + a2 b2 .
(Na air A la mat truaing con ciut mat truong K, c E K Ira f(x) E A[x] . f(c) goi la mat, da thac ciao phan to c lay he td trong A. BO phan cua K gam the phan td co dang f(c) ki hiau la Ate]. D6 dang that), A[c] la mat vanh con dim vanh K, goi la oath da Mac can phan to c lay he tit trong A. 'Pa cd th6 eh:Ling minh A[c] la vanh con be nhgt cim K chda A va c.
Dinh li 5. Gid sit A la mat truOng con cart mat non& K viz c la mot phan to that K. Neu c la sten oiN tren A thi A[c] ddng cdu tun yard] da Mac A[x] clic] do x. Neu c l¢ did sd tren A thi. AEC] dang edit Si mat uanh thuong cats uanh
Chang mink. X6t anti xa
: A[x] —+ K f(x)I—n f(c)
Hign nhien so la mat ding eau (vanh) va Imp = A[c].
Theo he qua tin Binh li 13 trong (ch III, §1, 5) to cd
A[c] = A[x] / Ker p
Trong trtiOng hap c lit sidu viat tren A, f(c) = 0 khi va chi khi f(x) = 0. fly Ker p = (0). Do do
A[c] = Mx] {0} = Aix]. n
BAI TAP
1. Trong vanh da tilde A[x], (A la trureing Z/3Z the s6 nguyen mod 3), hay tim tat ea eau da tilde cd bac la :
a) 2 ; b) 3 ; c) n.
107
2. Trong Nth da fink Z/5Z[x] hay tilde hign the !Map nhan
+ 4x OX2 + Ix + 4 ( _It2 + ix +3)2.
3. Trong vanh da fink Z/6Z[x] hay thuc hian phap nhan
(2x3 + + lx) (5X2 + 3x +
Vanh nay c6 tide coa 0 hay khong ?
4. Trong vanh Z/57,[x] hay time Man phep Chia
f(x) = + 1%2 ir T cho
g(x) = —2x2 + 2s —
5. Trong vanh Z/7Z(4 hay xac dinh p de du cim Olen Chia
+ px + 5 cho 122 + bang 0
6. Trong Walla Q/4 cluing minh rang da thtic
(x +1)24 —x2° —2x - 1
Chia hgt cho
a) 2x + 1 ; b) x + 1 ; c) x.
7. Chang minh rang da fink x2 + 14 C Z/15Z[x] c6.4 nghiem trong Z/15Z.
8. Gia sit A la mat truemg va f(x) la mat da thdc bac n ciut vanh Afxl. Chdng minh f(x) ed khong qua n nghiam 'Man Het trong A.
9. Dinh li 3 cri can citing nila khong ngu A la mat vanh va g(x) la mat da tilde co ho tit cao nhat bang dun vi ?
10. Chang minh rang dinh li 4 va he qua ciut nd con dting vol A la mat vanh.
11. Chting minh kat qua trong bai tap 8) can dung yid A la mat mien nguyan.
12. Xel clang eau p trong dinh II 5. Chttng minh Ker W la mat idean nguyen to (ch III, §1, bat tap 14)).
108
§2. VANH DA THUC NHI U AN
1. Wirth da thdc nhidu an
'Prong §1 ta da xay dung vanh da thdc mOt In A[x] 14 he tit trong mOt vanh A. DO la vanh ma cac phan tit la dm day (a., a t, ..., trong d6 cac ai E A bang 0 tat al frit mOt s6 him Ilan. Day (0, 1, 0,...) &lac hi hiOu la x. Tat nhien ngu ta kG hiOu day do la y thi ta lai M hiOu vanh da xay dung la A[y] va g9i la vanh da tilde caa In y. Bay gib ta hay dinh nghia vanh da thdc ona it an My 110 td trong mOt vanh A bang quy nap nhu sau.
Dinh nghia 1. GiA sit A la mOt vanh giao hoan cci don vi. Ta dal
A l = A[x 1 ] A2 = A1 [X21 A3 = A2 [x3]
An = An_dxn]
vanh Aa = An_ i [xn] ki hiOu la A[xp x2, ..., xi.,] va goi la vanh da thdc cita n In x 1 , x2, xn ldy h6 tit trong vanh A. MOt phan tit cAa An g9i la mOt da thdc cita it In x 1 , x2,..., ; 16y 14 td trong vanh A, nguili ta Id hiOu no bang f(x l, x2,..., xn) hay g(x l, x2,..., xn)...
Tit dinh nghia 1 ta c6 day vanh A. = A C C A2 C C A.
trong do A1_, la vanh con cast A, i = 1, 2,..., rt.
Bay gib ta hay xet vanh A 1 [x2] =A[x1 , x 2]. DO la vanh da tilde eta In x2 lay hO tit trong A l = at]. Vay mai phan tit cda A[x 1, x2] cci thg vigt duel clang
(1) f(x1 ,x2) = an(x t) + al(x1 )x2 + + an(x 1)x2 vfri ale ai(xi) E
(2) a,(x 1) = Ai° + bit x l + .. +bem ir. i = 0, n
vl A[x 1, x2] la mOt vanh non ta c6 phep nhan phan ph6i d6i veil 014 Ong, do do f(xi , x2) con co thg vigt
109
(3) Art , x2) = c 1411412 +c2421 422 + + einxion 402
vOi cac c i E A, the an , a i2 la flitting s6 tut nhien va (a, 1 , a,2 )
(ail, ai2 ) khi i x j. Cac ei goi lit the ha to va the ciefief goi
la the hqng to cda da this f(x 1 ,. x2). Chang han x6t. da thde As p x2) E Z[X , x2] vbi Z lit vanh cac s6 nguyan
f(x 1 , x2) = x3i — 1 + + 2).x2 + (4 +a, - 1)x22 =
= x3i — 1 + x1x2 + 212 + :434 + ICA —
Da this f(x 1 x2) = 0 khi va chi khi the ha t8 ci dm no bang
0 St ca. That Sy nelt the e i = 0 thi 1.6 rang f(x1 , x2) = O. Deo lai gia f(x 1 , x2) = 0. Via f(x 1 , x2) dual dang (1), ta dugc cac da this (2) bang 0 tat ca, We la
bio = = bin = ° =
Dining cac c 1 ,. , cm trong (3) chinh la the
= 7I, do chi
c i = C2 = ••• = cm = 0.
Bang guy nap ta chang minh mOi da BSc f(x 1 , x2, ..., xn) cda vanh x2, ..., :r t.] co th6 via dual clang
(4) f(x i, x2,..., xn) = c ixain ++ cm .491 .../enno vbi cac
c i E A, can a i1 , am, i = m, la. nhang so t9 nhian va (air ..., am) x (n p ..., khi i x j. Cac c 1 pi la the ha td,
cat gra la cac hang td curt da thtie f(x 1 , x2,...,
ES than f(x1 , x2,..., xn) = 0 khi S. chi khi cac ha ti dm no bang 0 tat ca.
Cho hai da thtic f(x 1 ,..., xn) va exp. xn) bao gib ta sung co th6 via chi-mg olden clang sau day
AS"—, xn) = Xfi:t in + + cnixai mi _49. (6)
g(xi xn) = + + xna.
trong de (alt , an) . # (a 1 , aj d khi i x j.
1 10
Chang han, vol fix,, x2) =+ x ix2 va ,g(xi, x2) =
= ax1 2 x — x 1+2 + x2' ta viat 2
N I , x2) =xi + 0xi4 +xix2 +0x2
g(xi , x2) = 0+21 + 2X IX22 — XiX2 +x2
ta c6 tang, hien, tich mitt fix, xn) va xn) la Do cac tinh chat mitt cac phep than trong vanh Afx„, , ni
fix„„ xn) g(x i , xn) = daxai n ...x0„m
txx i , xn) ex t , xn) = E cedie,ii+ap +a,, i,j
i = M ; m
Tit mat da Doi° bang 0 khi vb. chi khi the 110 ti tha ttd bang 0, ta suy ra hai da thdc xn) xn) Wing nhau khi va chi khi thong c6 the hang to y nhu nhau. That vay set hai da thtic fix„, xn), xn) thy y (5). Hien coa chang la
. xn) - xn) = E (c1 — i=
Do d6 xn) - ex = 0 khi va chi kill c - di = 0, i =1 tire la f(x,,..., x n) = g(x i ,..., x n) khi va chi khi c i i = M.
2. BO
Dinh nghia 2. Gia sit /Try x,) E A [xi , xn] la mot da thole klitte 0
f(x„, , = ..yanm + + c,n.X1.0 1 .••4 0,1
vol the c # 0, i = 1, ..., M vb. (au , (a p, khi i # j. of la bdc caa da thdc trap xn) dot veri an xi s6 mu th cao nhat ma x i c6 dome trong e hang tit cim da thug.
111
Ngu trong da thde xn) an xi khang cd mat thi bac ciia xn) dal yea nd la 0.
Ta g9i la bac cars hang tee ...4n tong cac s6 mG aii + + (ti eda cac dn.
Bat cda da Mac (dal bed than thg the an) lit s6 Ion nhgt trong the bac elm the hang tit dm nd.
Da fink 0 la da tilde khang co bac.
Ngu the hang td eita fix 1 , xn) cd ding bac k thi f(xp ..., xn) g9i la mat da three thing cap bite k hay mat eking bt).e k. Dac biOt mat clang bac nhat gia la clang tuyin tinh, mat clang bac hai gel la dang than phttong, mat dung bac ba gel la clang lap phuong.
Vi du. Da thdc
Az, , x2 , x3) = 2x1244 — 4x2 + 34 +4 -5x344-44x3 +6
s b6+ IA 9, nhung &Ai veil x 1 nd cd bac la 2.
DA sap xgp cac hang to elm mat da tilde f(xp xn) khae 0 ta cd thg sap x0p nd theo the lay thita tang hay giam dal vai mat tin flan dd. Chitng han, trong vi du tren ta co thg sap xgp Pap x2, x3) thee cac lay thita Iui mitt 4.
f(x 1 , x2 , x3) = 4(1 — x2) + x i(2443 — 5x324) + 34 41-52x3 + 6
hay theo the lay thila lid dm x2
f(x t , x2 , x3) = — 44x3 + x3(2.x ix53 — —4x2 + + 34 + 6
hay theo the lay thita lui dm x3
f(4, x2 , x3) = 34 +2% 14%35 — Ax i4x32 — 445 —x2ix2 +x2i + 6.
•Ngoai cac each sap x6p dd, nguai ta can cd mat each sap xgp gal la each sap xep theo !di tit dien (giting each sap xgp the chit trong td then). Cad) sap xgp nay data tren quan he Qui to toan phan da xac dinh trong tich dd bac (n a 1) yea N la tap hap cae s6 to nhien (ch 1, §2, bai tap 10). Ta co, theo quan he that to cig,
(a 1 , a2,..., an) > (b 1 , h2,..., bn)
112
nee va chi ne'u cd met chi se i = 1, 2,..., rt sao cho
al = ai = 6 1 _ 1 ,ai > bi
Quay lai da thtic f(xi , x2, 13) trong vi du tren, the so` mu trong mai hang tit cho ta mOt phan tit thuec N 3, cu the ta cci 7 phan t8 thuoc N3 : (1, 3, 5), (2, 1, 0), (0, 0, 9), (2, 0, 0), (1, 3, 2), (0, 5, 1), (0, 0, 0) ma theo quan hO thti ta dang xet - cheng stip thu tut nhu sau
(2, 1,0) > (2, 0, 0) > (1, 3, 5) > (1, 3, 2) > (0, 5, 1) > (0, 0, 9) > (0, 0, 0)
vol (2, 1, 0) la phan tit Ion nhtt. Viet cac hang to throng ang caa f(xl , x2, 13) theo thd tren
fixt x2 x3) = xix2 + 4 +20‘144- 6x1x32t3 4r52x3 + 34 + 6
ggi la sap xtp f(x i , 12, x3 ) theo lei ta dign.Hang tit -4x2
tudng ling vei phan to len 'that (2, 1, 0) gat la hang td cao nhdt coa. f(xi, 12, x3).
Dinh li 1. Gid sit f(xi , mOt da the& udi hang to
cao nit& g(xi , xn) Id mOt da thtic vat hang tit
cao nhdt la dxbi ...xbn. a& gid sU > (bi,..., b,,). Thl thi
hang tri cao nhdt diet da the& tang f(xi , xn) + g(xi xn)
la all _4,
Chiang mink. Ta hay vitt f(x l , va x) duel dang
xn) = cliqu + + crnxIsi
= d1a1;11 „, xan= dmx7ml xnamn
voi c ix it it = = 0
i ,•••, a,_ 1, > (all , aid i = In .
Ta xn) + xn) =
(c 1 + di).x7n + + (cm "+ d
OAIs6...-on 113
Do do hang tit cao nhgt aim da thdc tang lit
(e l + 4.ta = ann. n Ha qua. Old sdft (x1,xn), fk(xl , la nhang da
them to hang hi cao ?theft theo the tzt la
celki oh gid sit
(an, ..., am) > (a21, a2n) > aim)
The thi c ixin la hang tit cao nhdt cart da thee tong
xn) + + fk(xl, xd.
Tap hop sap the Di IV con la mot vi nhdm giao hoan din via phdp Ong (ch II, §1, hal tap 5)
(a l , an) + (b p ..., b,) = (a 1 + b i , ., an + b.)
Phgp Ong nay va quan he the 41 trong cd tinh chat B6 de 1. Neu
..., air) > (bp ,..,
thi (05 + c1 , + c.) > (b 1 ..., bn + cn)
vet most'c3 , ..., en) E Nn .
Chung mink. VI
(ct i , ..., an) > (b i,•bn)
ngn cd mOt chi s6 i = 1, sao cho
a t b a. = b. a 1 > b. 1 1-1. , do dd
at + c i = b i + c t , + c1_ 1 =
a1 + ci > b +
Hg quit. Neu
, an) > , tin)
uct , cn) > (di, , de) thi (a1 + + cn) > (b +d1 , ..., bn + dn)
114
Clang mirth. Theo bd dg 1
(ai + cp ..., an + en) > (al + b„ + cn) >(1), + dt by, dii). n Tit 136 de 1 va he qua cua no, ta say ra
Eljnh H 2. Gin set f(xi, xn) vh g(xi, xn) lit hai da auk khan
0 caa vanh tit] c6 Mc hang to cao nhdt theo thd tit lit
c lxjii .. xnm oh dixin NM cid] x 0 Oa hang tit cao nhdt eta
da that tich •••, xn) x,,) la cidixlii +ba
Chang minh. Gia su
xn) = citin + + ettain „,x:e
g(x i ,..., xn) = rbn in ciprei rnt x:ron
da Ogee sap xgp theo 161 to dign. Digu do c6 nghia la
aln) > (an,..., aid veil mai i = 1
va (b 11 ,..., bin) > birit val moi j
Ta hay chang minh
d I . tam + b 1
la hang to cao nhgt cua da thee tich
xn) xn).
Nhan f(x j ,..., xn) vdi xn) ta &roc
xn) = di + ... +bjn
ij
i= 1, ..., 1
j = 1, m
Mai hang to cs xla .,. x:e +bln cho ta pagn tv
÷ bJ1r” ath + bird E Nn
Theo bd de 1 Ira he qua dm nd, ta c6 rat bgt Bang thin
115
+ b in) > (
b in) > (aa + b ii ,
abt b in) > (a11 + as, by) ,
Vay hang tit
C i d i aqii + b 11 + b
chinh la hang W cao nhgt e8a da thdc tich. n
HO qua. Ndu A la ntbt miM nguyen thi xn 1 cling bay.
3. Oa thtic d6i xOng
GU sit A la mot vanh giao haul cO don vi. Trude hdt ta hay xet vanh da thile 2 an , x2]. Trong vanh nay ta chu y tai
hai da thdc Tao biat sau day
f(xi , 12) =x l + x2 , g(xi , 12) = xi x2 .
Cac da thdc nay cO tlnh chat : nab ta thay x 1 bang x2 vii x2 bang x 1 thi da thdc along thay &di
f(x i , 12) =xl -I- x2 = x2 +x 1 = f(12 , xi)
ext , 12) = xi x2 = x2 xi , xi)
Tsang vanh Aix]. , x2i khong phai chi ed hai da dote do c6
dab chat nho vay, chang han da tilde
r(xi , 12) = + 4 - xi x2
cling cho ta
V(xi 2) p(X2 , xi)
MOt each tdng quid
116
Dinh nghia 3. Gia sit A la mot vanh giao hoan cd don vi, f(xi , . , x,) la mot da than aim vanh 21[tr 1 xn]. Ta bao
f(x i ,.., xn) la met da these dot xung can n do nau
f(x i , x2 , ., xn) = x1(2) , ., x1.00)
yea moi phep the
( 1 2 ...n =
v(1) v(2) ... r(n)
tegio) suy ra tit f(x i bang each thay trong
x,) x 1 bea bdi x1(n) .
Vi Trong Walla Z[x i , x2 , 13] da theta
f(x 1 , x2 , x3) = 4x2 + x2 x3 + x3 x 1 + x1 xi + x2 4 +
+ x3 1 x2 + 21 1 12 13
la d61 xdng. That vey ta they ngay rang
f(x i , x2 , x3) = f(x2 , 13 , xi) = f(13, xi , 12) =
= 1(12 , xi , x3) — f(x3 , x2 , x1) = f(x1, x3 , x2).
MuOn ed f(x3 , x2 , xi) clang hen, ta thay x 1 bang 13 x2 bang
x2 , ; bang x1 trong f(x 1 , x2 , x3) , ta duet
f(x3 , x2 , xi) 2 = x32 x2 + x2 x 1 + xi x3 + x3 x22 + x2 xi +
-Ex x2 + 2x3 x x 3 3 2 1
do de. f(x1, x2 , x3) = f(x3 , x2 , x1 ).
Dinh Ii 3. B3 ph3n gbm cdc da that d6i rang caa vanh A[xi ,.., xn] la mot yank con caa vanh .A.[x i ,., xn]
Ch'ing mirth. GU sit f(x 1 xn) Ira ex, , xn) la nhang da
thac d6i xang, nghia la
, xn) = AXT " T(n))
117
Va
g(zi xn) = gixtio
yid moi phdp thg t. Thg thi
x„) g(xt c.) xn) =
coo) — g(x.rio co)) ;
f(x i xn) =
xrdg(co) ,-, vii mot phep thg T. n
Cdc Agin to cart A la nhang da thtic dot tray mai phan td a E A co the yigt
a = ax? xn° .
Moi da tilde del ringf(x1 xn) E4 A nhgt thigt phai chira tat ca it an Nth phai cc; ding mat bac dai wit mai gn do : that fly nguf(x 1 xn) oft mat hang td trong d6 chaa mat an xi
vai s6 mu la k, thi no °Ting c6 hang tit suy ra tit hang tit ay bang each thay thg x, bai xj , tic lit hang tit ch/a vat ding se ma k.
Cac da thtic sau day
= + X2 + + Sr,
= X2 + Xi X3 + Xn — 1 xn
53 = X1 X2 X3 +x1 X2 X4 + xn _2 xn _Ixn
ern —1 = XI X2 ;" Xn — I n — 2
= x112 ... xn
ming la nhang da th/c clef ring, ggi la cac da theft d6i xang co bdn. Chung citing mat vai tre quan trong trong if thuygt cac da th/c dra ring do dinh li clitai day. Trude khi de cap Uri Binh
n))
ring dac biet. That
us\
Ii d6, to hay dea vao ki hieu sau day. Gia sit g(xi ,..., xn) la mat
da thee tha ,..., xis] , phtin to cim Aki co dude bang
each thay trong g(xi ,..., xn) : x 1 bed eri, x2 bee 62, ..., bai an
Ed la met da thee- the, the da thee 461 ring to ban, ki hieu
la g(81 ,..., an) . Chang han Arai n = 2 va g(x i , x2) =- 4 — x2
thi g(st , 62) = 6j — 62 . VI 6 1 , ..., an la nhiing da thtic dai
ring men on) Ming la met da thee elai ring theo dinh
11 3. Chang han
g(51 , 62) =6i — 62 =(x+ x2)2 — x l x2 = 4 +x i x2 +
la mot da thee d61 ring dm x 1 va x2 . Nhu vOy, MOI da thee
eaa the da thae dal rang co ban 61 , ..., an IA met da thee dai
ring tha n x 1 ,..., xi, . Dieu ciao lai Ming thing nci la nal dung
mkt dinh li 4 sau day, thta tren cite b6 de :
Bo dg 2. Gid sti f(x i ,..., xn) Ca mot da thdc ddi ming khac
0 at =71 ... xann la hang td cad nhdt act no The thi
a1 3 a 3- 3 an
Chung mink. Ta phai Chung minh ai yin moi i =2,...,n.
Vi xn) la 461 rung, nen f(x i in) phai chtia hang tit
suy ra tit hang tit well x:n
bang each thay xi _ I bai xi va xi bai xi _ . Nau ai > ai _ thi
(a1 ,..., ai _ 2 , ai , a1 _ 1 ,.., an) > (a1 ,..., a1 _ 2 , a1 _1, ai ,..., an)
do d6
axis
khong phai lit hang tit can nhat, mau than vai gia thigt. n dd 3. Gid sit al , .., an /4 nhung se to nhien sao cho
ai 3 az .. 3 an
119
The thi da thac
f(x l ,..., xn) = 671 - °2 lei - °3 anay - 0,,,ro
trong da 6t ,..., 6^ la cac da thac ddi xang co bad, eh hang tit cao nhdt la
...
Chang ainh. Cac hang to cao nhat Ma cr i , 62 , theo thd to la
Crn - 1 , Cn
X 1 , Xi x2 X I X2 ... Xn _ 1 , x i X2 ... xn .
.4 clang dinh li 2 to cd hang tit' cao nhat Ma fix i xn ) la
—22 (x i x2).2 a3 (x i x2 ... x^ _ 1 )a _ 1 (x, x2 ... xrdan
= x7 , x:a. n B6 de 4. Cid sit al la mat s6 to nhien. BO ;Man M caa !sr,
v6i N la tap hap cac s6 to nhien, gain the phan hi
(t1 , t2 ,..., tn)
sao cho al 3 tIL t2
la hau han.
Chting mink Goi L la tap hop hat han g6m cac s6 to nhien 0, 1, ..., a , . Hign nhien L^ hvu han vft M C L^ , do dd M
Witt han. •
BS d6 5. Gia sa g(51 dn) let mat da Mac caa cac da
the& ddi xang co ban
us) = di e21 11
trong d6 c; x 0 ; i = 1, m, vet , ain) m taw ..., i x . j. Thd thi an) = 0.
120
Chang mink. Thay 61 bang x1 + x2 + + xn , 6n bang
x1 x 2 ... xn trong g(op ..., on) ta dude met da thdc cua citc an x i X2 xn
g(xi + x2 + + xn , x i x2 xn) =f(x 1 xn
= E f, (xi x„) i =1
vat
n = ci (x, + x2 + (xi x2 .
= 1, 2, ..., m.
Hang tat cao nhat caa da thdc f (x 1 , xn ) theo dinh li 2 la
ci alc. (x i x2)12,2 ... (x 1 x2 Xja in = ei
vat ail + a2 + + am = ku
ai2 ain = ki2
ain = kin
Hang td cao nhat cUa mai da thee f. (x , xn ) cho ta phan
tit (kit ,k12 kin) E Ta co
k2 • kin) m (k11 k12 , kfr,) khii # j
vi ndu
(kit, ki2 ,
thi ail = k kit = kJ, — kj2 =
ail = kit — kin = ki2 — ki3 = ai2
= ajn
kin) = (k11 , k.2 , kin) vet i m j
121
vai i # j, matt thuan via gia thi(t. Vi N'1 sap the to than phan nen be phan hriu han Om the plain t' , km) i = 1,
2, m c6 phan tit km nhat, clang han (k 11 k12 , kin) la
phtin tit Ian nhat. Theo he qua elm dinh li 1, c i tets xji;I IPI la hang tit can nhift cua f(x t , xn) V2y
g(51 , ..., on) =f(x 1 xn) la khitc O. n
HO quit. Gid set
x) = cm 4o1
h'(xl , xn) = + + , a xn um m xI to xnum
itt hat da Mite trong d6 am) # am) khi i # j,
sao cho
on) = h'(op on);
thd thi
Cluing mirth. Gia sit # e l Did
g(op ..., on) = on) — ..., an) =
(c 1 — C/2 1n (cm — ern) 07.1 61ritmn •
Vi c # c' 1 nen — e' 1 # 0. Theo bd (IS 5
an) # 0 .
Nhung theo gia thiet g(op ..., on) = 0, mem thuan. n Dinh li 4. Gid srt f(xl , x2 , xn) E Alxi , x2 , ..., mot
da thee dot sung khac 0. Thd thi co mOt vet chi mot da lilac
h(xp x2 ,..., xn) E Afx l , x2 ,..., xn)
sao cho f(xi , x2 ,..., ;) = hasi, 62,
trong do an 62, ..., an la cdc da Mac d6i xang co ban.
122.
Cluing mink Ta hay sap xep f(xi , xn) thee 161 to 'dign, gig
sit
ax73 xa2z xn"
la hang tit cao nhat tha f(x i , xn) . Theo 66 de 2, ta th
a3 a an
B6 de 3 cho ta bids da fink
a 61;1- az lei - (13 6nn 1
ciing cd hang tit cao (that la
a tE t
'Cat hieu
fl (xi n) = f(x, , • — 622 a3 Cann
Ngu fl (x 1 , x,,) m 0, ta hay sap xep no theo 161 din
va gia sit
la hang tit cao nhat oda n6
Theo clink li 3 : fl (x1 , ..., cling la mot da thde del xiIng,
va do dd thee be de 2
6 1 6 2 bn .
Mat Made, to bigu Mule cua hieu hai da thac (m0c 1), ta cci
(al , a2 , an) > (b i , 62,",
do d6
Xet hi0u
12 (x 1 , . xn) = (x i ,
123
Ngu f2 (x 1 xn) # 0, ta hay sap xep n6 theo lei tit diet) Ira gia sit
y xit „. ynn
la hang tit cao nhgt cim nci ; Ming li luan nhtt del veil fi (x i , xn) ta ductc
c 3 e2 3 3 c
vie (b 1 , b 2 , bn) > (c 1 , c2 , ..., en)
Qua trinh d6 kheng the dai ra vO tan vi the Win tit (a l , an) , (b i , bn) , (e 1 , ., cn) ma to dude la nhung phan
tit cim tap him M nun han trong be de 4. Vay qua trinh Mtn phai them dill , We la sau mat se Mu han Jou& ta se phai dupe
0 = Tx (xi x„) A CI? 12 4 - 13
Ta suy ra tit dd rang
f(xi , x2 , " xn) = a a e22.
1 —1 / 1 Cnn 6.11 2 C22 3 ... Unn .
Vay da thee h(xi , x2 , ..„ xn) can tim la da fink
11.(2C i , .X2 , 2., x,) = et / a1 a 3 x,241, xb b 2 xbz —63
2 1 I I —1 1
Gia sit co mot da flute
le(xi , xn)
hit; , otn) = f(xi ,
an) = an ).
tip dung he qua dm b'O' de 5 ta co
h(xi xn ) = h'(xi , xn ). n
sao cho
TM thi
124
Phep chdng minh coa dinh Ii 4 !Mang nhitng cho ta bigt stt ton tai Na duy nhgt ciut da thtic h(x , xn) , nd don cho ta
mat phuong phtip thuan tin de thitnh lap h(xi ,.:., xn). Vide tim da thlic h(xi , xn) sao cho f(x t , xn) = h51 ,..,an) got la bidu thi da thug diti ring xn) qua cac da thile d61 ring co ban C1 , ..., dn .
Vi du. Trong vanh da tittle Z [xi , x2 , x3] trig ha s6 nguyen
hay bidu thi da thac ddi ring
x3 + x2 X3 + 2x 1 x2 + 2x x + 2x 2 3 1 3 2
qua the da tittle d61 ring co ban 6 1 , 62 , 43 .
Truoc hgt ta nhan xet rang
2x 1 x2 + ?xi x3 + 2x2 x3= 262
\ray chi can xet da tittle dai ring
f(xi , x2 , x3) = + .32 +x3.
Hang tit cao nhgt cua nd la xi x3 x° x3°- Theo dinh li 4
t 2 ta lap da thug
/3 (xi , x2 , x3) = f(x 1 , x2 , x3) — Oro - 0 a(3)
= + + x33 _ (x, + x2 + x3) 3
—(3x2i x2 + 3x2i x3 + 3x 1 x2,2 + 3x 1 x2 + 3x2 x3 + 3x2 xj + fix, x2 x3).
Van theo dinh li 4, ta lai sap xgp nd theo lot tit dign va tim hang tit cao nhgt elm 11 (x1 , x2 , x3) . Nhung trong trUting
hap Cu the' nay ta nhan xet rang
xj x2 + xi r3 + x l r22 + + x x3 + x2.3 =
= (x1 + x2 + x3) (x, x2 +x l x, + x2 x3) - 3x1 x2 x3
Do do fi (x i , x2, x3) = —361 62 + 3a3 .
25
Duel cling
x3 ) =61 —36162+363
2x 1 x2 + 2x 1 x3 ± X3 2x2 =
va
= 62 + 3153 + 252 •
Trong thgc tign ngdOi ta con (Ida ra mat s6 nhan xet de viac bigu din dirge nhanh clidng hon. Ta hay xet try& Mt mat da Vide dgi ximg clang cap xn) EA[x1 xnl cd hang
td cao MI& la
a a a X I
l X2
2
Vay bac dm /(x, xn) la + a2 + + an . VI cac da thilc
dal ring ca ban al , 62, ..., 6 la Bang cap cd bac theo thd ta la 1, 2, ..., n, nen da thdc tieh
a - 012 - 43
Ming clang cap va cd bac la
al - a2 + 2(a2 - a3) + + nan = a l + a2 + +
Do dd
11(X1 , X2 , ...; xn) = fix ]. 7 x2 , d - a bai l 42 012 -43
Ming (tang cap va ed bac la a 1 + a2 + + an ngu nd khac O.
Sap xap fl (x i ,..., xn) theo 161 td din va gia six /34 xinn la
hang tit cao nhat cua nd thg nil
6 1 + b2 + + bn = a l + a2 + an
va (a 1 , a2 ,.., an) > , 62 , .
Theo dinh li 4 ta cd mat day hdu han phan td cna lc"
(6) (a 1 ,..., an)• > ton) > (c 1 , c 2i) > .
126
thent man tinh chat
a 1 et. an
(7) b i bn
va trong truing hop 6 day
(8) a l + + an = 6 1 + + b„ =
Do cd them tinh chat (8) nen s6 phtin tit aim day (6) giam di nhieu. Tap hop can phetn tit aim day (6) la met be phttn cot% tap hop him han
M = , tin ) , (tmi , tmn)}
trong dci
>
va.
iii + ti2 + + tin = ct i + a2 + + an
vet i = 1 , 2, rn. Theo dinh 11 4 f(x i , xn) via dyed clang
Ef(Xi 3Cn ) = upr i -1
vet ti e A vet ti = 0 nen (tti , , tin) khOng cd mat trong day (6)
hay quay lai da thee
f(xi , x2 , x3) = 4 +
Tap hop M d day la
= {(3, 0, 0), (2, 1, 0), (1, 1, 1)1
Do dd
fix, , x2 , x3) = t1 61- 0 602 - 0 + r2 621 -1 621 - 0 6-03 +
r3 ' 621 = 6-; 22 61 62 + C3 ;
127
ti la lig tit eim hang tit cao nhat tha /Oil , x2 , x3), do dO gi 1. Mutin co r2 , r3 , ngubi to thay x 1 , x2 , x3 theo thd to
bang 1, 1, 0 ta Male
2 = 8 + 2g2
do do r2 = -3. Sau do thay x1 , x2 , x3 theo the to bang 1, 1, -1 ta co
1 = 1 + 3.- 13
hay a3=
3
Vay x2 , -x3) = - 36162 + 363
Pinning 'tip vita thing goi lit phoong phap vgri h6 td Mt dinh.
Chit 9 : Khiing phiti bao gib pinning phap nay clang co higu lye Mu A la mat vanh hay Mtn nila la mat min nguyen, nhung hitu han.
Trong truang Mp da that xn) khang phai la clang cap thi ta nhan xet rang the hang tit ding mot cap cos. no lap thanh mat da tilde d6i xting clang cap do do ta co f(x i , xn) la tang dm nhung da Hide d6i ring ding cap, chang him trong vi da tren
4 + YZ + x3 ÷ 2,1„2 2.1x3 2x2x3
la tang aim da thde dal xdng (tang cap bac 3
x3 + X3 X x3 1 2 3
va da th e dgi ming ding cap. be 2
2•T 1z2 2t1i3 + 2%2'3
ling dung. Tim the s6 nguygn a, p, y sao cho
a p y = 6,
a3 p3 + y3 = 36, • a + fi + y = 6.
Theo vi du tren ta co
128.
P3 + Y3 = + P + y)3 — 3(a + + (a,8 ay + PO+ 3a/3Y
'Pa suy ra co + ay + fly = 11 .
Mat khac xet da thdc f(x) E Z [x]
f(x) = afix fifix r).
GM sit a E Z, ta c6
f(a) = (a — a)(a — (3)(a — y).
Vi f(a) = 0 khi va chi khi mat trong the thus 86 a —a, a —p,a —y bang 0, cho nen the nghiam cua f(x) la a, p, y. Khai trign f(x) ta Mtge
fix) = x3 — + fi + 11)? + (a/3 + ay + fiY)x afiY
= x3 — 6x2 + th -6
vi a + /3 + y = 6, 4 + ay + fly = 11, apy = 6.
Da thtic f(x) co tang cite hO sd bang 0, do d6 f(x) co mat nghiem la 1. Theo he qua dm dinh li 4 trong (§1, 4), f(x) chia het cho x — 1. Chia f(x) cho x — 1 ta duce
f(x) = (x — 1)(x2 — 5x + 6).
Da thtic x2 — 5x + 6 cho ta hai nghiem la 2 va 3. Vay Cite s6 nguyen a, p y can tim la 1, 2, 3.
BAI TAP
1. Trong vanh Z ix1 , x2 , x3] biau din da thiic xi +4 3 qua the da thdc deli ming co ban.
2. Prong vanh Z/2Z [xl , x2 , x3] biau din da thdc
lxi + 1xZ + 1x3 qua the da thdc dgi rung CO ban
3. GUI h@ phuong trinh
x +y +z = —3
x3 + y3 + z3 = —27 xd yd z4 = 113
Old 09 129
cHUONG V
VANH CHINH VA VANH OCLIT
Al. VANH CHINH
1. linh chit sit hoc trong vAnh
Trong chttang nay cluing to nghien cau the tinh chat s6 hoc cda vanh, nghIa la cac tinh chat deli yeti quan he chia hat (ch III, §1, 2).
CAA ml A la mitt ratan nguyen ma phtin tit dan vi ki hiau la 1. Cac it& cda dcin vi con goi la cac phan td khd nghich, chdng ldp thanh mat nhdm nhan U ma 1 la phan tit don vi. Chang han trong vanh Z cae sit nguyen cac phan tit Milt nghich la 1 va -1 ; trong vanh da thtic K [x] vdi K la mitt Hitting, cac da thee be 0 nghia la cac phan tit khac 0 cda K la the phan tit khA nghich.
Sau day la mot s6 tinh chat ye ehia het trong mitt mien nguyen ma viac chting minh khang c6 gi khd khan.
B6 de 1. (i) a I a.
(0) e I b va b I a keo theo c I a.
(iii) u khd nghich, u I a yen mai a.
(iv) Nati b I u obi a khd nghich, thi 6 khd ?which.
(v) Quan h¢ S xac dish nhu sau : xS x' khi x' = ux obi u khd nghich, la mitt quan he Wang &tang ; x va goi a lien kit.
Vi du. 1) Hai phan tit cda Wham nhan U la lien kat.
130
2) Trong vanh the nguyen Z hai s6 nguyen a vb. -a IA
lien kat.
' 3) Trong vanh da thde Kix/ voi K IA met *tieing, hai da
tilde f(x) irk af(x), a E K va. a m 0, la lien kgt
B6 de 2. x ua x' Ca lien kit khi vet chi khi x I x' ua x' I x.
Gia sit a E A, tap hop cac bet xa, x E A, czia A la met idean
dm A kg hi0u Aa hay aA idean nay . la idean sinh ra bai a,
ngubi ta goi n6 la met idean chinh (oh III, §1, 4).
B6 de 3. a I b khi utt chi khi Aa D Ab.
140 quit. x utt x' itt lien kit khi va chi khi Ax = Ax'. Dac
bigt u la khci nghich khi utt chi khi Au = A.
Dinh nghia 1. Ctic phan tit lien kat vOi x va cac plain tit
khA nghich la Mc detc kheng thtic skt eta x can cac ubc khac
eta x 1a cac uetc thus sic eta x.
Vi du. ±2 ±3 la cac 1.10c tilts su eta 6, can ±1 va ±6
IA the the kheng thoc stt.
Dinh nghia 2. Gia sit x IA met phan tit khan 0 vA kheng
. kha nghich eta A ; x gel la met plan ta bat khd quy eta A
ngu x kheng e6 ude thus so.
• Vi du. 1) Gaz s6 nguyen to va cite se del Aim Sang la Se
phan tit Mt klia quy cua vanh Z.
2) Da tilde x2 + 1 IA da Niue bat kha quy cua vanh R [x],
R IP truhng so thou Nhttng trong' vanh C [x] v6i C IA trubng
s6 pink thi x2 + 1 kheng phai la bat khA quy vi nu.S ti6c
tharc sv chang han x + i va x — i.
Dinh nghla 3. Neu c I a va c I b thi c got udc chung
eta a va. b. Phan tit c goi la ubc chung !Ott nhtit cunt a v2t b
ngu c IA Se chung eta a va b va neu moi tide chung Sit a _va b la utic cua c
Tit b6 de 2 ta c6 hai 06c chung Ion nhat eta a va b la lien
kat, do d6 c6 thg coi la bang nhau neu kheng M nhan tit Ma.
131
nghich. Ta cling dish nghia Wong 01 vac chung lan nit& dm nhigu pit 0'1.
2. Vanh chinh-
Dinh nghia 4. Met mien nguyen A goi la yank chinh nau mot idean cua no la idean chinh.
Vz da . 1 Vanh Z cac s6 nguyen la vanh chinh. That vay, gia sit I la mat idean cim Z. Neu. I = {0} thi I la idean sinh bat 0. Wu I # (0), gia su a la s6 nguyen during be nhgt cda I va b la mat phgn to thy g clia. I. Ta c6 the gia sit 6 3 0, vi ngu 6 < 0 thi -b > 0 Ira -b Ming thuac I, do de ta My -.b. Lay 6 chia cho a, ta dugc
b = aq + r vat r la du, nen 0 r < a. Mat khac r = b - aq E L Nth) r 0 thi a kheng ph& la s6 nguyen during be nhgt cast I, mau thugn. Do do r = 0 va 6 = aq, tde la I = aZ la idean sinh ra bat a.
Mign nguyen A a day dude gia su la mat vanh Stith Cam phan to ma ta set la thuec A.
B6 d6 4. Lldc chung lon nhdt caa hat plain hi a va b bdt ky ton tat.
• Cluing mink. GO I la idean sinh ra bat a Ira b. Cac phan to / ed. clang ax + by vol x, y E A (ch III, 41, 4, dlnh II 6). Mat khac vi A la vanh chinh nen I sinh ra bai mat phgn tv d nao dd, phan to d cling thuec I nen d co clang
(1) d = ax + by, x, y E A
Ta hay &dug minh d la vac chung clia a va 6. Vi a,bEI=dA, nen
a = da', b = db' , a', 6' E A.
Do dd d la tide chung cast a va b. Them naa ngu c lb met talc chung cita a va 6, We la cd a", b" E A sao cho a = ca", b = cb", thg thi (1) tra thanh
d = c(a"x b"y)
132
Vity d la ubc chung Ion WI& caa a va b. n He qua. Neu e let mat ubc chung lein nha caa a at b, thi
co r, s E A sao cho
e = ar + bs
Cluing mink. Xet doe chung Ion nhat d rim M de 4. d va e la lien kit (muc 1), tile la ce mot phiin tit kha nghich u sao cho
e= du
Nhan hai vg rim (1) vdi u
e = du = alit + byu = ar + bs, r = xu, s = yu. n Dinh nghlit 5. a na b la nguyen t6 cling nhau melt cluing
than I lam lute chung lOn nhdt.
Tit he qua coa bei de 4 ta suy ra
B6 de 5. Allu a, b nguyen t6 cling nhau thi cd r, s E sao cho
I = ar + bs.
He qua. Neu c J ab vet c, a nguyen t6 cling nhau, thi ci 6.
Cluing mink. VI c, a nguyen t6 rung nhau nen theo b6 de 5 ce r, s E A sao cho
1 = ar + cs.
Nhan hai vg rim ding tilde voi b
b = abr + Ms.
Vi c I ab nen c6 q E A sae cho ab = cq. Do de,
b = e(qr + bs)
tdc la c I b. •
B6 de 6. aid sit x Ca mat plain tit Mt khd quy viz a la mat plan tit bit ki. The thi Mac x I a hoac x ua a la nguyen to geng nhau.
Cluing minh. VI x la bat kha quy nen the tide cua x la cac plait tit lien kgt vai x va the phgn to kha nghich, do de met
133
t ube chung kin nhat coa x va a chi S thg la m0 pit til lien kat via x hoac met phan tit UM nghich. Trong trubng hop thit oh& ta cd x I a, trong truing hop thit hai x va a la nguyen US Ming nhau. n
Be de 7. Gici sg x let mat phan tit kheic 0 ea khOng khd nghich. Cdc menet de sau day , Id twang duang :
a) x Id bat khd quy
b) x I ab thi x I a hock x I b.
Cluing mink a) keo theo b). Theo log de 6 ta co hoac x I a hoac x va a nguyen to cimg nhau. Neu x va a nguyen to Ming nhau theo he quit eta b6 de 5 ta cci x 1 6.
b) keo theo a). Gia sit a la mot itac eta x, thg thi cd b E A sao cho
x = ab.
VI x I x, nen x I ab = x. Theo b) x I a hoac x I b. Ned x I a thi kat hop vbi a I x ta cd x va a lien kat. Neu x I b thi kat hop yin b I x ta eo x = ub, u la kith nghich. Do do
x = ab = ub.
Nhung x z 0, nen b m 0, do do ta suy ra a = is vi A la mien nguyen. Cho nen mot thin a eta x chi S thg halm la lien kat veil x hoac lit kita nghich, vay x la bat kha. quy. n
Be de 8. Trong mOt ho Ix/Ong rang bdt ky F rduing Wean caa A sap the ta theo quan h¢ bao ham, co mOt idean M caa hp F la tot dai trong F (ch I, §2, 3, dinh nghia 7).
Cluing mink. Gia sit /0 lit melt idean eta F. Hoak lo 11 tei
dai trong F va nhu vay la xong, hoac cd mOt idean ;eta F
sao cho 71 z /0 vii. It D I. Nati /1 la t6i dal trong F thi thg
la xong, neu khong ta lai cd mbt idean /2 eta F sao cho /2 z h va /2 D Il . Tigp lac qua trinh nay, hoac la ta &rye mot idean M eta F t60 dai trong F, hoac lit ta duoc mOt day ve han nhang idean phan. bigt trong F :
Ia C; C I» C. 1 C
134 ..
Ta gia sit tinting hop sau thy ra. Gqi I la hop
I = U neN
De dang thgy I la met idean caa A. VI A la met vanh chinh nen 'dean I chtoc sinh ra bai met phan tit x e I. Theo dinh nghia dm hop mi met s6 tit nhien n sao cho x E In . DMu nay
keo theo I CI, va do do 1 = In + p matt thuan vol gia thiet
the Mean cita day la phan Wet. n Dinh li 1. Giti sit x liz mOt ',han tEl khac 0 oh khOng khd
nghich. Thu thi x co the viet docii Ong
(2) x = p ip2 P„
uoi cdc p t , i = 1, ..., n, C¢ nhitng gilt tit bdt khd quy.
chthig mirth. Goi Fla tap hop the plain tit Hieing kha nghich x x 0 sao cho x kh6ng viet chicle duet clang (2). 'Pa hay chting minh F = 0. GM sU F x pj. Ta H hieu bang j he the 'dean Ax vai x e F. Theo bd de O, F c6 met plaint to In sao cho Am la t6i dai trong Trude het m kheng bgt MIA quy, vi nen m Mt kha quy thi m co clang (2). in kitting Mt kill', quy thi in ed tide thue su, °hang han a la met tide thoc sti cim m, dieu
chi cci nghia la ed b E A sao cho
m = ab
Nhu fly b cling la met ttdc cua m, b kheng thd la lthii
nghich vi se keep theo a lien ket vat m, 6 kheng the lien Mt
vai m vi se keo theo a kha nghich, do d6 6 plied la tide title
so eim m Vi a va 6 la nhfing Ude thoe su ena m nen theo bd de 3 va he qua caa ng to co
Am C Aa, Am x Ad
va
Am C Ab , Am mAb
Do Am la tai dai trong T nen Aa va Ab kheng thuee 1 do
do a va b kitting thuec F ; a va b den khac 0, khac kha nghich va khOng thuee F, nen a va b phai viet throe dud' clang (2)
135
PC
b = P1+1
diet' nay keo theo
m = ab = p1 Ppi+l P
mkt thuan vdi m E F. n Dinh li 2. Gid ad
X = P1P2 Pm = qiq2 qn
cat p i , pm , qi , a nhiing phan hi bat khd quy. The thi m = n , rid wit niqt at dank ad thich hop ta ca qi = ups , i = 1, ..., m.
Chang -mink. Theo bd de 7 nhan to Mt kha quy p i cfia x phai IA ude mia mot q, nit() dd. VI A la giao haat nen ta the gia thigt rang p1 la ubc ciut Nhung q 1 la Mt kha quy no khOng c6 vac thuc su, do do p1 lA ude khOng thuc su cua qr Them nila pi khong kha nghich, oho nen phai c6 p 1 va lien kgt the IA qi = voi ul kha nghich. Nhu vAy ta dude
P1P2 Pm = b iPlq2 qn
Vip 1 m 0 ta suy ra
P2 " Pm = 11 02 — qn .
Theo bd de 7,192 IA WIC Caa mot q, nap do vOi'i 3 2. lh cci the gia thigt rang p2 la tide cua q2. Dodd ta dirge q2 = u2p2
voi u2 kha nghich. Nhu vay ta thole
P2P3 Pm = b ib2P2q3 tin* VI p2 # 0 ta suy ra
p3 4.. pni = u 1u2q3
Sau khi da lap la! qua trinh dd m Ian, ta dojo m n va 1 = u1u2 . umqm
ne=
khang nghich nen ta phai co' m = n. n
Nhv fly dinh li 1 cho ta bit moi phan tit x # 0 kheng kha nghich aim met vanh chinh d@u cd the phan tich thanh tich nhUng Olen to bat khit quy va dinh If 2 cho ta bier sit plan tich lit duy nhgt aeu kitting ke den cac nhan to kha nghich.
Vi du. Trong vanh cac sit nguyen Z ta cd
18 = 2 . 3 . 3 = (-2)3 (-3) =
Goi K la trubng cite thticing (ch III, §2, 3) cim vanh chinh A, the thi mai philn thaEK deu cd the vial chub dang a = a/b viri a , b E A nguyen t6 thug nhau. Thgt vey giit sir a = a76', vid a', b' E A. Goi d E A la uec Chung lirn nhgt eim a' va b' ; ta c6 a' = ad va b' = bd, via a, b e A. Hien nhien a, b nguyen td Ming nhau va a = a/b.
Bay gib ta hay chitng minh met tinh chat se-hoc quan trong khac aim Nth chinh ngoid tinh chat da cho d dinh 11 1.
Dinh li 3. Gia sd K la trztang cac thuang caa vanh chinh A, a E K la my nghiem eint da fink.
f(x) = x" + ark?' I + aix + (cti E A) .
The thi a E A.
Cluing mink. Theo Whit tren ta ea the via a =a/b, vol a, 6 EA nguyen to cling nhau. Vi f(a) = 0 nen ta Buy ra, sau khi thay
a bhng a/b va nhan yeti :
d' + tqctrian-1 + + aice 2 + ctobr = O.
Nhu vgy b chia het ; vi b nguyen to veil a nen itp dung lien tip he qua cim b6 de 5 ta chive b chia ha a. Do d6 6 la met phan tit kha nghich Ma A, tle la 6 1 E A, digit nay key
theo a = a6 -1 E A. n tinh chat dm vanh chinh cho beg dinh It 3 ta dinh nghia :
Dinh nghia 6. GUI sit K la trueng cac thltang eita met mien nguyen A. Neu moi a E K lit nghiem dm met da tittle c6 dsng
137 ,
f(x) = x + an _ixn-1 + .. + a ix + a (ai E A)
thuOc A, till. A gal la bane clang nguyeh.
Khai niem vanh ddng nguyen thing vai tre quan trong trong • if thuyat s6.
Chung ta mbi chi dva ra met vi du ve vanh chinh, do la vanh the s6 nguyen Z, trong khi cac tinh chat the vanh chinh rat phong phti va dude sit dung nhieu trong than ; kh6 khan cho vi du ye vanh chinh narn b ch6 cluing ta chi mai nam doge nhung khai niem ban dau eim men Dai sa. Bay gib ta hay c6 gang (Iva ra met vi du Hui hai va vanh chinh.
Vi du 2. Xet vanh the s6 nguyen Gaoxa :
A= ZVI = a + bi j a, b E Z)
Chung ta hay cluing minh A la vanh chinh. Muan fly, chting ta hay nhite lai khai niem m6 dun &la mot s6 pink z = a + pi, a, /3 € R., ki hien I zl : I z I = 4a 2 + p2 , (14 =0 as = fl = 0) ma trong trttbng hop z E A, thi I zj 2 E N nghia lit 14 2 lit met s6 to nhien. Them rata, ta they ngay
Q[i] = + Oil a, fi la trubng can thaw:1g tha A. Then frac s6, ta ming they ngay eho met s6 him ti a, bao gib ta Ming tim thew met s6 nguyen a sao
cho a - al 4 2. Dod6 to th th6 khang dinh dttoc rang :
(*) Yx E Q DI, 3 z E A, Ix - z12 1< 1.
Bay gib ta hay lay met idean I khang tam thttang tha A Tap hop
X= 114 2 0 #zEll
IA met be pia 'Mac rang caa N va khOng theta O. Vi N sap thu ta tat, nen X co Mien tit be nhat, goi u lit s6 phitc thuec I sao eho 11(1 2 la so ttt nhien be nhat the. X. Xet mot phtin tit thy
138
y o e I. Hign nhien — e Q M. Theo khang dinh (*) tan tai
z E A sac cho 2
z 1
hay Iv - zuI2 < 1u1 2 .
Nhung u, v e z e A, vay v - zu El. Mat kluge I u 1 2 la phgn to be nhgt mitt X, nen v - zu = 0, hay u = zu, hay
I = Au. Vay A la vanh chinh.
BAI TAP
1. Trong vanh da than R [x] vii It la truong se that, chting
minh cac da Onto axe +bx + c vii b2 — 4ac < 0 la nhang da thde bgt kha quy. Dieu do ca con dung naa khang ngu coi can da thdc do thuOc vanh C [x] Nal C la trudng s6 fait ?
2. Xet vanh da thite K [x] vii K la mot Gutting.
a) Chting minh rang moi da fink bac nit ciut deu la bat
Mgt quy. Ngu K la mot mien nguyen thi dieu ck5 can dung khang ?
b) Cluing minh rang cac da Unto bac hai va bac ba eita K[x] la Mt kha quy khi Ira chi khi chting khOng co nghiam trong K.
3. Trong Nth Z[x] vii Z la Wirth can s6 nguyen, xet xem cac da thdc sau day co phai la Mt kha quy hay Milting
f(x) = 2x + 8,
g(x) = x2 + 1,
'h(x) = z2 + 2x - 2 ?
4. Gift sit' a, b la hai phan tit' cda met vanh chinh A, nguyen to ding nhau. Chting minh idean sinh ra bat a va 6 chinh la A.
5. Gift sit p la mat phan tit khan 0 cua mot vanh chinh A. Chang mink p da bat kha quy khi va chi khi Ap la idean
dal (eh III, §1, Mi. tap 14).
39
6. 'Prong mat vanh chinh chting minh Sc idean nguyen (eh III, §1, bai tap 14) khac WI la cac idean tai dai
7. Chung minh met trdeng la met vanh chinh.
8. Vanh thuung dm mat vanh chinh cd phai la met vanh chinh ltheng ?
9. Venh con east mot vanh chinh cd phai la mat vanh chinh kOng ?
10. Vanh Z[x] cd phai la mat vanh chinh kheng ?
11. GU sit A la tap hap the sd phtic cd dang a + bC-73 vol a, b E Z.
a) Chang minh rang A ding ved phap Ong va phep nhan cac se phdc la mot mien nguyen.
b) Chting minh rang 2, 1 + t , 1 - f 3 , la nhang phan tit bat kha quy dui A. Tit (Id suy ra A killing pile( la vanh chinh.
12. Chdng minh vanh
A = a + bi / a, b Z1
la chinh
13. Gia sit a va b la hai Olen td dm met vanh chinh cd dang phan tich nhu sau :
a =pZi pp Pe
b = 141 p1:2
trong dd the pi la nhang phgn tit bat kha quy, the a . pi la nhang se td nhien, i = 1 ..., rt. Chfing minh phgn to
d = prntr(c),, Pi) pinisetz . (32) pnmi n(an fin)
la mat thee chung Idn nhgt etla a va b, trong do min( / 3,) la phgn tit be nhat cua atp PO, i 1,...,
§2. VANH °CUT (EUCLIDE)
1. Vanh Oclit
Dinh nghia 1. Gia sit A la mat mien nguyen, A t la tap hop cac phan tO khac 0 dm A. Mien nguyen A cling vet mat anh xa (goi la filth xi) Oclit)
: A* N
tit A* den tap hop cac s6 tv nhien N thea man cac tinh chat :
1 ° Neu bPa va a # 0 thl 6(b) 5 6(a) ;
2° Voi hai phan to a va b thy 3.1 dm A, b x 0, co q Ira r thubc A sao cho a = bq + r va 6(r) < 5(0 ndu r m 0 ; goi la mat vanh Oclit.
Phan tit r goi la du. Neu r = 0 thl b chia Mt a va theo 1 ° ta co 6(b) d(a). Nhu vay dieu kiOn can de mat phan to b
la vim ctia mat Sian t* a * 0 la 6(b) s 5(a),
Ngodi ta bao mat vanh Oclit la mat vanh trong d6 c6 phap chia vat du.
Vi du. 1) With cac s6 nguyen Z cling vdl anh xa
Z* —0 N
n •—n
la mat vanh Oclit.
2) Vanh da thdc K [x], via K la mat trudng, la mat vanh cant. Anh xa d a day la anh xa cho Wong Ong voi mat da dole f(x) x 0 bac dm no.
3) Vanh cac se nguyen Gaoxo la vanh Oclit vOi 6(z) = I zi 2 .
Tinh chat 1 ° cim Dinh nghla 1 nhin that' a clang con tinh chat 2° c6 the dua vao cluing minh trong vi do 2 ctla (§1, 2).
Dinh li 1. Net A Id mat 'iamb Oclit thi .A la mat vanh chink.
Ch'ing minh. Ta hay ching minh mpi idean cim A la chlnh. Gia sit I la mat idean cue A. Ndu I. = {0} thi I la idean sinh ra hot 0. GM sit I m 101. Gal a la phan to Mule 0 cAa I sao cho 6(a) la be nhat trong tap hop 6(C), I. la tap hop cac plain
141
td klatic 0 Ma 1. Gia sit x la met phan t8 tuy y cua I. Theo tinh chat 2° ta co q, r E A sao cho
x= aq + r
Vi a, x C I, nen r = x - aq C I. Neu r m 0 ta co 6(r) < 6(a), mart than vdi gia thin 6(a) la be nhat trong 6(/*). Vey r = 0 vb. = Aa. n
Ta da biet trong met vanh chinh, chung Ian nhgt cua hai Win ter a va b Mt ky ten tai (o, 2, b6 de 4).
Neu A la met Nth Oclit thi khOng nhfing tide chung 16n nhat cim hai phan to ton tai, ma ta con co the thus hien nhung phep chia (vdi du) lien UM de co udc chung Ion nhat. Dieu de clua tren b6 de sau day :
B6 de 1. Gid sit A la mot blintz chinh, a, b, q, r la nhzing 'than tit cart A Wei man quern h4
a = bq + r
The" thi ark chung tan nil& czia a ua b lei talc chung Ian !theft cern b viz r.
Cluing mink. Gci I IA idean sinh ra bdi a, b va J la Wean sinh ra bai b, r. Tn a = bq + r, ta suy ra a e J, do de I C J. Td r = a - bg, ta suy ra r I, do do J C I. fly I = J. Nhttng A la met vanh chinh, nen ton tai d E I sao cho Ad = I. Theo be de 4 Ma (§l, 2), d la uhc chung 16n nhgt cila a va b. Nhttng I = J nen d clang la udc chung nhgt cua b va r. n
By gib gia sit A la met vanh Oclit va ta dat van tie tim uoc chung Ilan nhtit cua hai phan tit a, b E A. Neu a = 0 thi r6 rang tide chung ldn nil& Ma a vit b la b, vi very ta hay gia sit ca a l&n 6 dart khac 0. Thte hien phep chia a cho b ta dada
.a = hqo + ro vdi d(ro) < 5(b) nM ro # 0
Neu ro # 0 ta lai chia 6. cho
b roq i + r j vdi 6(r 1 ) < d(ro) neu r 1 # 0.
NM r1 # 0 ta lai chia ro cho 1. 1 :
ro = riq2 + r2 vdi 6(r2) < 6(r1) neu r2 # 0.
142
Qua trinh chia nhu vay phAi cham Mit sau mot se heal han bit& vi day' cac so' tur nhien
5(b) > 5(ro) > 6(r 1 ) >6(r2 )...
khong thd giam vii han, tile Ia sau met s6 len chia, ta phhi di tdi mat phep chia ma du bang 0
rk - I = rkqk t 0. 4 dunk- bti di/ 1 ta cd r k = item chung len nhat cua r k va
0 = udc chung Ion nhat cua rk _ I va rk = utIc chung Ion nhat caa rk - 2 va rk _ I = = vac chung Idn nhat Oda rI va r2
= tide chung len nhat cua r va r = ttoc chung ldn What mla b vb. ro = ueic chung len nhat dm a va b .
Vi du. Tim toic chung lon nhat caa
f(x) = x6 - 22 + x4 - x2 + 2x - 1
va g(x) = x5 - 3x3 + x2 + 2x - 1.
Traffic khi thtte hien cac phep chia lien tiep ta hay nhan xet r&ng ttdc chung ldn nhat cfm a va b bang tide chung kin nhat cim a' va 0, voi a' lien ket dm a. Bay gib ta chia f(x) cho g(x)
X6 - 2x5 + x4 - x
2 + 2x - 1 x5 - 3x3 + X2 ± 2x - 1
X6 - 3X4 + x3 + 2x 2 - x x - 2
-2x5 + 4x4 - x3 - 3x2 + 3x - 1
-2x5 + 6x3 - 2x2 - 4x + 2
4x4 - 7x3 - x2 + 7x - 3
Ta nhan thay rang nett chia g(x) cho
ro(x) = 4x4 - 7x3 - x2 + 7x - 3 ta se bi he s6 phan, do de hp dung nhOn 'tat trern ta chia 4g(x) cho ro(x).
4x3 - 12x3 + 42 + 8.x - 4 7 - x2 + 7x - 3
7x4 x3 + 7x2
7x4 - 11x3 - + llx - 4
x
143
Thee be dg 1 item chung Ian nhgt rim 4g(x) va ro(x) bang
dim chung Ian nhat caa 7x4 - 11x3 - 3x2 + I lx -4 ye. ro(x). Nhung
neu this 7x4 - 11x3 - 3x2 + Ilx -4 cho ro(x) ta se bi he sg phan, do
do Elp dung nhan xet tren ta chia 4(7x4 -11/3 -3x2 +1a - 4) cho ro(x).
28a4 - 44x3 -12x2 + 44x -16 4x4 -7x3 -x2 +7x - 3
28x4 - 49e3 + 49x - 21
5x3 - 5x2 - 5.x + 5
Bang le chia ro (x) rho r2 (x) = 5x3 -5x2 - Si 5 , ta chia
ro(x) cho 1
r i (x) d4 tranh he sill phan
4x4 -7x3 - x2 + 7x - 3
4r4 4x3 4x2 4,
3 2 x- - X - + 1
4x -3
-3x3 +3x2 +3x -3
-3x3 + 3x2 + - 3
0
Vay uric chung Ian !Mgt cua f(x) va g(x) lb. x3 -x2 -x +1.
2. Ung dung
Bay gib ta hay rip dung cam kgt qua da thu chide ye vanh chfnh va vanh Oda dd nghien cdu vanh da tilde K[x] yid K la met triteing.
Trude Mt cam phan to kha nghtch cast K[x] la the pliant to khac 0 cua K. Car da thtic lien Mt caa da thdc f(x) la cat da thus ed clang af(x) vai 0 # a E K. Cac da thdc betc 'Mgt ax + 6
b lit Mt MIA quy, Chung ed met nghiem duy nhgt - -
a trong K.
7
144
Dd la the da thdc Mt kita quy duy nista ed nghiam trong K.
Cac da thdc Mt UM quy khgc nghia la the da flute Mt kha quy cd bac Ian han 1 kitting cd nghiem trong K. That vay gia
sit p(x) la mat da tilde bgt kha quy ad bac Ion hon 1 Ira cd
mat nghigm c E K. Theo (Chuang IV, §1, 4, ha qua dm dinh
11 4) p(x) chia Mt cho x - c
p(x) = (x - c) q(x).
Vi bac (p(x)) > 1 nen bac (q(x)) D 1, do do x - c la vac thtte sV
eim p(x), mku thugn vat gilt thigt p(x) bat kha. quy. Theo (§1, 2,
dinh li 1) moi da thde fix) ed bac > 1 cd thg vigt dval dang
sau sal khac mat nhan tit khit nghich.
f(x) = (aix + (akx + bk)mtp i (x)n i pi(x)el
trong dd the air + bi , i = 1, k, la nhang da thdc bac nhgt
khong lien ket, the pj(x), j = 1, 1, la nhting da thde bat khit
quy cd bac > 1 khOng lien kgt the mi va la Miring so 41
nhien. Ngu them dgu hang 0 ta rang f(x) kh6ng chda
nhan tit tuygn tinh. Viec chda hay khong chda nhan tit tuygn tinh khign cho f(x) ad nghiam hay khong trong K. That Sy gift
sit c E K, ta ad
f(c) = (a ic (akc bk)'"kp i(c)W I pkri.
Vi the pi(c) x 0, j = 1, ..., 1, nenf(c) = 0 khi va chi khi c
bi la mat trong the phgn tit - —a.
, i = 1, k. Them nita tip
dung dinh 11 2 (§1, 2) vg sv duy What dm dang phan tick ta ad : nen f(x) chda nhan tit tuygn tinh bi vai lily thita
bi thi fix) chia Mt cho (x + —)
m vat f(x) khong chia Mt elm
ai
bt bi (x + + . Do do - la nghiam beg cap nt i tha f(x). Dao
Lai gia sit c la mat nghiem bat cap in mitt f(x), the thi vi f(x)
146
Chia hat cho (x - khang Chia hat cho (x - * 1 nOn trong clang phan tich aim f(x) phdi co nhan tit tuygn tinh x.- c veil My thita la m. Kat Juan : the nghiem cua f(x) trong K eUng vat s6 bat cast chting dude cung cap bat cat nhan tit tuy(n tinh vit lay thus cua thong trong clang pilau tich f(x) thanh tich nhting da thtic bat kha quy va dho lair Vi
bac (f(x)) n m i + m2 + + mk
n4n ta 'co
Dinh li 2. Cid sit f(x) la mot da thac bac n > 1 can uanh Kfxl, wit K la mot truang. The thi f(x) co khang qua n nghiam. trong K, cite nghiqm co the' phan Net co tiding nhau.
Ngu clang phAn tich eda f(x) chi chtia nhang phAn tit tuygn tinh
f('x) = (a1x + (akx + bk)mk
thi Me do
= bac f(x) = /71k
s6 nghigm cart f(x) trong K bang so' bac caa f(x) Gia sit ta trong truing hap nay, got a l, a2, ..., an la 71 nghi9m aim f(x) trong K, age nghigm ea thg phan biet c6 the' tiding nhau, va gia sti
(1) f(x) = cce + eixn-1 + + ck _kx + en
f(x) phai ed clang phari tich lit
f(x) =co(x - ad(' - a2) (x - an).
sau khi nhan cite da thdc x - = 1, n, ta dude
(2) f(x) = co - 1 (a i + a2 + + an) +
+ 2 (aia2 + aia3 + + an _lak) + + (-1rala2 ak i
So sanh (1) vit (2) to chive cOng thac Vista
a i a l + a2 + + an = -
co
146:
C2 Co
cr ia2a4 +
+ an Co
C3
Co a la2 ... an = (-1)" c .
Ta
o
nhan thay rang the vg trai etla the thug than tren citing qua la cat da thdc &Si ring co ban dm the phan tit a l , a2 , an (ch IV. §2, 3).
Bay gib ta xet mgt da thdc bat kith quy p(x) e6 bac len hem 1 dm vanh Mx]. Theo nhu tit da nth, p(x) khOng c6 nghiam trong K. Ta hay dat van de thy dung met trubng E ehtla K nhu mot trubng con sao cho p(x) c6 nghiem trong E.
Dinh li 3. Old set
p(x) = ao + aix + + art > 1)
la mgt da thdc Mt Ma quy cita vanh KN. The thi co mgt tnamg E zeta ctftth duy nhdt (ly tai mOt ding cdu) no cho
1) K Ca mOt trubng con act E.
2) p(x) co mgt nghiem 0 trong E.
3) Mai phan tit a E E Met duy nit& du& Bang
a = bo +,b10 + + bo _ 10"-1 , bi E K, i = 0, ..., n - 1.
Chang mink. Goi I la Mean sinh ra bed p(x), I la tai dal (§1, bai tap 5). lihtt vay vanh thtiong E = Kfx.711 la met trttang (chili, §1, bal tap 14). T1 cling c6 the cluing minh E la met trubng mOt each true Hap nhu sau. E to rang la met vanh giao hoan cri don vi lit 1 + I khac ph&n to 0 + I ; ta hay cluing minh Mai phan tit' f(x) + I # 0 + I co righich deo trong E. VI f(x) + I # 0 + I, nen f(x) kheng phai la 1301 ola p(x), do d6 f(x) va p(x) nguyen t6 ming nhau (§1, 2, b6 d4 6). Theo (§1, 2, b6 de, 5), ten tai r(x), q(x) E K(x) sao cho
n§ts6... 10 147
f(x) r(x) + p(x) q(x) = 1.
Ta suy ra (fx) + I)(r(x) + I) = f(x) r(x) + L= I + I.
Bay gib to xat toan Su chinh Sc
K[x] --• K[x]/I
f(x) f(x) + I.
Thu hop cda h yea K la mOt clang eau
h : K —> K[x]/I
ma 4 hay chung min h n6 lit don cau That fly, gia sit a, b E K sao cho
a + I = 6 + I.
fly a — b E I, tdc la a -b la mot bOi oda p(x), dieu nay chi S thg thy ra khi a - b = 0. h la mOt don cau, elm nen ta hay don oh& cac phin tit a E K vol N(a) E Akin = E. Do dd K la mOt trubng con Ss. truang E. 1h ki hieu h(f(x)) = f(x) + 1 = Vtli ki hieu nay ta rb
Trij = + + . + En .? = 0.
Hitting ta da (tang nhat cac p hen tit a E K v6 i h(a) = h(a) = ¢ elm nen ta c6 the' vigt
an + a10 + ... + atiOn = 0
trong dd 0 = x. fly 0 E E la mOt nghiOm cda p(x). Cudi dung mOt phan thy y caa E c6 clang Jlij v6i fix) la mOt da thdc oda K[x]. Lay f(x) chia cho p(x), ta dole
f(x) = p(x) q(x) + r(x)
Vol r(x) = bo + bix + bo , e K, khong nhdt thigt khac 0 Lay anh aim f(x) bai h
fri)= p(x)q(x) + Frig = 0 + =
50 ÷ 51 ' • '÷ 5n-1?1 = bo 610
Hai p Ma tit
148
1 bo + 6 10 + +
vh = co + c 10 + + co_yr
bang nhau khi va chi khi = = 0, n - 1 That fly
gigt sit frx7 = Ta cci r .
(bo - co) +(b i - ei)0 + + (bo_i - 613-1 = 0
tdc la da thdc
(bo - co) +(6 i - cl)x + ... + (bo_i - crox" la bai eta p(x). Dieu do chi thy ra khi va chi khi the ha tit eta da tilde bang 0, nghla la = c = 0, , n - 1.
Nhu vay ta da ehting mink xong sat tan tai cua trtibng E. Gia sit bay gib cci mat trubng E' eiing cd the tinh chat 1 0, 2°, 3°. G9i 8' la mat nghiem eta p(x) trong E'. Ta hay xet (tang eau p xet trong (ch. W §1, 5, dinh It 5)
p : Kix] E'
f(x) 1(0) Ta cd p(x) e Kenp. Mat Mute K[x] lit mat vanh chinh nen
Kerp sinh ra bai mat da tilde p'(x). Vi p khang phgti la dong eau 0 nen Ker yo # Kix], do dti p'(x) kh6ng khgt nghich Kat Juan p(x) va p'(x) phai la lien kat, do d6 Kerr = I. Them nda do tinh chat 3°, E' C Imp = KW% fly E' = Imp, vit ta ed p la toan eau. 4 dung dinh It 13 eta (eh III, §1, 5) ta co mat don eau W : K[x] lI -> E' sao cho tam gitic
Mx] E'
P K[x]/I =
giao hoax'. Nhung a day vi c la mat town anh nen co- clang la town anh. do do Co IA mat dang thin Bang cau W duce the dinh bdi :
14,9
by + 6 10 + + tin _ 1 60 -1 n-• bo + 6 10' + . +
n6 gift nguyen cac pilau tv cim K va Men 0 thAnh 0'. Ta eta thong minh xong firth duy nhat caa truang E. n
4 dung dinh If 3 cho K la truang s6 time R, p(x) IA da th/c x2 + 1, ta duet E la tniang set pluic C trong dcl Mic Olen to c6 clang a + bi, vat a, b E R Ira i IA met nghiem cua x2 + 1. VI C = RixilI, I la Wean sink ra beii x2 + 1, nen phep Ong va phep nhan time hiOn nhu plat) cling vA p'hgp nhan da thuc vdi eh/ y IA i 2 = - 1.
To co
(a + + + di) = a + c + (6 + d)i ;
+ bi)
+ di) = ac + (ad + bc)i + bdi2
= ac - bd + (ad + bc)i.
Ta co the xay dung fruiting sd ph/c C bang cach My dell de clic R2 va xac dinh hai phep Man cling va nhAn trong R 2 nhu sau :
(a, b) + (c, d) = -lie, b + ;
(a, b) . (c, d) = (ac — bd, ad + bc).
D6 dAng cluing minh rang R2 Ming vdi hai pile)) toan trail la mOt trUnng. Sau d6, bang don eau
,_,, K2
a (a , 0)
ta clang nhat a We (a, 0) va Oat i = (0, 1), ta came
t2 = (-1, 0) = -1.
Vay i IA Wit nghiAm cfia da th/c x2 + 1 = 0. Cutii Ming ta tim thgy clang quen thulm dm ode s6 ph/c bang each via
• (a, b) = (a, 0) + (0, b) = (a, 0) + (b, 0) (0, 1) = a + bi.
114 qua. Old sr[ f(x) la ntOt da thac co bac n > 1 caa vanh da that Thd thi bap gib cling co mat &Jiang chi& K rilut
150
met trubng con aao cho f(x) c6 dung n nghiem trong do, cho nghiem c6 the phan biet hay kitbag.
Chang mink. Gui p(x) IA met nista' tit Mt kha quy Ma f(x) cd bac Ibn han 1 (nen the nhan tit Mt kha quy dm fix) dell Me 1 thI f(x) c6 cac nghiem trong K va van de titian giai guy& xong). Ta fly clang trotting E nho trong dinh II 3, da thile p(x) cd nghiem trong E, do do f(x) cang fly. Ngu f(x) co it nghiem trong E thi van de dune giai guy& xong, nalu MOO-1g f(x) phai. S clang
f(x) = (x — c)m1 ... (x — ti(x)
via e1 E E, i = 1, ..., k, va f1 (x) e E[x] vet 1 < bac f1(4 <
bac f(x). Ta lai me rang trubng E thAnh trubng E1 f1 (x) co-
nghiem trong E l . VI bac dm the da thile la hitu han nen eau
met sS bubo heti han me rang to phai dude met trubng E1 trong
do f(x) rb it nghiem. n Chu 9. Gia sit f(x) E K[x] la met da thdc ca bk rt Ibn bon
1 Ira E la met trubng mb rang So. trubng K (nghia la K la trtiting con eita E) sao cho f(x) cd n. nghiem, phan biet hay khOng, trong E. Cac nghiem a l , thueic E, nhung theo
oiling tilde Viet°, citc da thde dfii 'cling co ban dm eac Olen tit a 1 , an bao gib el-Mg thuee K. Vi My nen g(a1 ,..., an) la
met da that° &Si ming etla cite" Olin tit a l , . . , an , thi
g(a1 ,..., an) 11 met phan tit thuec K theo dinh 4 dm (oh
IV, §2, 3).
Vi du. da thfie f(x) E Q[x], Q la trubng so hal.' ti.
f(x) = x4 — x2 — 2 = (x2 + 1)(x2 — 2).
Ede da Unite x2 + 1 M. x2 -‘ 2 IA nhung da-thate Mt MIA quy. cua Q[x]. Ta hay ma rang Q theo phuong phap aim dinh 11 3
de x2 + 1 cd nghiem. Goi Q(i) la trubng ma Ong dd via i la
met nghiem eim x2 + 1. VI e Q (i), nen Q (i) chtla hai nghiem
151
aim x2 + 1. Bay gib ta lai ma rOng Q(i) de' x2 . — 2 cd nghiam.
Goi 4 la mOt nghiam ctla x2 — 2 Va ki hieu bang Q (i) (G)
tniting ma rang d6, ta ducic hai nghiam cim x2 — 2 trong Q (i) (a). Nhu vay trong trvong Q(i) (a), f(x) co ban nghiam la k —i, a, -a
BM TAP
1. ChUng minh rang vanh gam C£1C s6 plulc cd dang a + bid vai a, b E Z la mOt vanh Oclit.
2. Chung minh mOt trubng la mOt vim]) Oclit,
3. Gia sit A la mat vanh Oclit. Chang minh A la mOt trtlang khi Ira chi khi 6(x) la hang via moi x e A.
4. Gia sit A la mat vanh Oclit vai anh xa Oclit
: A` —n• N.
Chung minh tan tai mOt anh xa Oclit 5'
sao cho (5'(A) = 0, it) , n 0
hay d'(A) = N.
5. Gib. sit A la mat vanh Oclit via anh xa Oclit . Chang
minh a(u) plain to be nhat cua a(A) khi va chi khi u kha nghich trong A.
6. Gib. sit A la mOt mian nguyen. Chung minh dial) kian can da A Oclit la tan tai mOt phan t8 khOng kha nghich x E A sao cho mai lop dm A/(x) cd mOt dal dial) hoac kha nghich hoac bang 0 (dua anh /la Oclit d ciat A ye anh xa Oclit 5' trong bai tap 4 va kat hop vai bat tap 5).
7. Chung minh vanh
Z [1 + 1 +
a + b ( 2 ) I a, b E Z 2
'152
khong phai la vitnh dclit bang each 4 dung bid tap 6. (Ngadi 1 + iir9
] to c6 thg Chung minh Z [ 2 la vaaa chinh, nhangday
thong ta khOng lam vi thigu qua nhigu khan Main caa dal s6).
8. Gia sit fix) = x5 + x3 + x2 + x + 1
g(x) = x3 ÷ 2X2 + x ÷ 1
a) Tim tide chung Idn nhgt cost f(x) ud g(x) trong Q/x1
b) Tim udc Chung IOn nhat magi f(x) va g(x) trong Z/3Z[xJ trong do Q la truang so` hint ti Z/3Z lit trmang cat so' nguygn mod 3.
9 Gia sit
R(Ir3) = a bli-251 a, b E
Q(1=3-) ={a + b1,1 I rz,b e Q}
Q(a) = {a bm a,b E Q}
trong dd R la truZing cat s6 thac, Q la trotting Sc so hilu ti. Chung minh thug :
a) R(rS3), Q(,), Qom la nhang trudng voi (Map Ong ta phdp nhan thong thuong cat s6.
b) = R[x]/(x2 + 3) v61 (x2 + 3) la idean sinh ra bat
X2 + 3 trong KW.
Qom = Q[x]/(x2 - 2) vdi (x2 — 2) la idean sinh ra bat
x2 — 2 trong Q[x].
10. GM sit E la mot halting and rang dm tntOng K, u la
mat phgn to thuac E, f(x) mat da thtic bgt kha quy trong K[x] nhan u lam nghlam vit gia sit rang g(x) E K[x] tong nhan
it lam nghiam. Cluing minh rang trong K[x]
a) f(x) la da attic cd bac th6p nhgt nhan u lam nghiam ; b) g(x) chia hgt cho f(x)
11. Gia sit X = Q2 . Trong X ta xac dinh cac phap than Ong
vh nhan nhu sau •
153
(a, b) + (c, d) = (a + c, b + d)
(a, b) . (c , d) = (etc + 2bd, ad + bc) a) Chung minh rang X cang vol hai town do IA mat
truang.
b) Chung minh rang X'' Q(4-2-) c) Tim tat ca cac to dang cau cia X. lit do say ra rang
tap hop cac to dang c&u cua X Itt mat nhdm xyclic dui v6i phep nhan anh xa.
154
CHLIONG VI
DA THUG TREN TRUONG SO
§1. DA THUG VoI He 436 THUG VA PITOC
1. Tnstmg so phtic
Cho mot da thdc v6i hg s6 th0c thi chtia chAc da than do -
co nghigm trong trubng ad th0c, cu th6 da thdc x 2 + 1 kh6ng co nghigm trong trubng s6 thorn Dual day ta se thgy moi da tilde bac n vol hg s6 phac co dung n nghigm phdc. D4 chting minh ra hay dua vao cat 66 de sau day :
Be de 1. Moi da thdc yea 114 sd thgc co bac le co it nhdt mot nghient Sec.
Chong mirth. Gia ad
f(x) = a/ + an_r1 + + an an m 0, n 14.
Qua giao trinh giai doh ta bigt rang vol nhting gia tri dining va am mitt x kha 16n ye gia tri tuygt d6i, ham s6 f(x) co can dgu trai nhau. Vgy rb nhang gia tri thge So, x, a va b chAng han, sao cho
f(a) < 0, f(b) > 0.
Mat khan ham so f(x) la lien tuc, vi v&y co. mgt gift tri c cua x/nam gicia a va b, sao cho f(c) = 0. n
Ta chug rang phep cluing minh cfia b6 de 1 khong dai sd, no da 14 nhang kat qua dm giai tfch. Them naa ye ngen ngil ta da goi anh xa
155
R -• R
c la ham s6 f(x).
B6 de 2. Mpi da thetc bec hai ax2 + bx + c, ooi he s6 phdc, bao gia cling co hat nghiem phetc.
Cluing mink. Gni co l va co 2 la hai can bac hai cue. b2 - 4cic, -b + 1 -la + co 2 ta cd hai nghigm cda da th/c la va . n 2a 2a
B6 de 3. Mot da the& b4e ldn han 0 Si he a trate co it nhdt met nghiem 'plate. •
Cluing mink. Mof sd to nhien it > 0 dolt cd the via duel dang it = in la met se to nhien va n' la met s6 tp nhien le. Ta hay cluing minh bang quy nap theo m. V6i m = 0 khgng dinh la dung theo be de 1. Ta gia sit khgng dish IA dung cho m - 1 Ira chang minh nd dung cho m. Murin vgy, ta hay
xet met da th/c f(x) cd bac n = (m > 0) veil he s6 thoc. Coi f(x) nhu met da thdc cita vanh v6i C la trobng se ph/c, ta hay xet met trurbng m6 Ong E cua C sao cho f(x) cd dung it nghiem al , a2 ,..., trong E (ch V, §2, 2, ha qua cim (huh If 3). Gia sit c la met se time tuy j, at
(1) pi; = + e(ai cei), i s j.
Ta cd Cn2 phan tit thanh lap boi cac t6 hop chap 2 dm n phan tit a t , , an . Xet da th/c
(2) g(x) = fit) - fits). •
= + atxt 1 + int
Vi bac cast g(x) bang se Mc nhan ti x - nen ta cd
n(n - I) - 1) _ 1 / = Cn - 2 - - 2n1 2
vei q = n'(en' - 1) ; vi re' va. en' - 1 la nhung se Id nen q la met se le.
156
Ta hay cluing mink the he tit al , ...al tha g(x) deu la that
ca. Trude het the he W dd la ciic da thuc el& sung co ban cunt the Ai Vey theo (1) chung la nhting da thtic eua a i , ..., an vtli
he s6 Wye, vi c la met s6 thee. Han naa cluing con la nhting da thitc d61 icing cart a 1 , ..., a. That thy nett ta thay a i bang ai a i bang a i thi pl . van gift nguyen con the Phi va Ai; thi tin° del cho nhau (h i va j), dieu nay chat than khang lam thay del the he to at, ..., a 1 . Do dci theo (ch V, §2, 2, chid 3I) CAC he tit do la nhung se thne. Vey g(x) la met da thdc veii
he s6 thac ed ben 2v` -1q, vol q la 14. Theo gia thiet quy nap g(x) ed it nhat met nghiem phdc dang (1), We la cti It nhat met cap chi se (i, j) sao cho phan to
pii = cticej + c(ai + ai )
thuen truting se phde C.
Neu to gan cho c met gia tri them kluic thi ta se dude met da thde g(x) khan loaf (2), thata nhan met nghiem phdc dang (1), nhung n6i chung voi met cap chi se (i, j) khac. Tur
n6u ta gan cho clan luot Cn + 1 gia tri phan Met thi nhat
dinh phai chicle hai nghiem pink dang (1) vOl ding met cap chi s6
j) vi ta chi th td hop chap 2 tide n phan tit. Vay the nao
thug co hai se tilde c i va c2 khac nhau sao cho the phan to
a. a + cI (a. + a.)
an) + c (a. + a.)
ling 2
ved cling met cap chi s6 i, j deu la pink. Dat
a = anal + c i (ai +
b = a.a. + c,(a. + a)).
Ta cd a, b E C. Ta suy ra
a - b a. + a
j c — C2
a - b a Ia) t
ie I — C2
157
Vey a + a. era i dell la Ta suy re ai ai a nghiem cim da thdc hec hair
x2 — (ai adx +
voi he s6 phlic. Theo 14 de 2, a i va ai la Onto. Nhu fly to da cluing minh de thdc f(x) khong nhung S met, ma cd hei nghiem phtic.•
Dinh li 1. MQi da thdc bee tun hen 0 ()di he 86 phitc co it nit& met nghiem phdc.
Cluing mink. Dia sit f(x) la mot da thdc bez n > 0
veil he sti phdc. Dar
Fx7 = + +
vai cat ai la cat lien hop rim Mc a. i = 0, , n. X6t da thdc :
g(x) = f(x)f(x). Ta cd
g(x) = bo + b ix + . + b2nx2"
voi bk = E k 0,1, 2n i+j=k
vi Slc = I EPj = bk bv=k
nen the he ad bk la thuc. Theo b6 de 3, g(x) co it nhgt met nghiem phfic z = s + it
g(z) = f(z) ,(17 = 0.
Do d6 hoar f(z) = 0 hoer irjz = 0. NOu F27 = 0 = "do + z + = o -
thi ao + atz + + anzn = no + + + anr = 0
We la fq) 0
Nhu vey hoc z hoar a la nghiem cua f(x) n
158
114 quit 1. Cdc da thuc bdt khd quy cat' yank Chi), C la twang ad phdc, l¢ cac da thzic bac nhdt.
Cluing mink. Theo (Ch V §2, 2), cac da thee bac nhgt la Mt kha quy. Dia sit f(x) la met da th/c cfia C[i] co bac lon han 1. Theo dinh H 1, f(x) co met nghiem phdc c. fly f(x) ca met u6c thuc au x - c, do do f(x) khong bgt kha quy n
114 quit 2. Mot da Mac Mc n > 0 vat Mad phac co it nghilm
Cluing mink. Gilt sit f(x) la mat da th/c bac n > 0 yea M s6 ph/c. Theo M qua 1 dang phan tich caa f(x) thanh 'Lich nhang da th/c Mt MIA quy phAi la
f(x) = u(aix + b i)mt (ars + b2)mz + bkrk
trong A6 u la met ad phdc khac 0 vb.
7712 Ink = Th.
Theo (Ch. V, §2, 2), it nghiem phdc caa f(x) la m. 1 nghi@m b t bk
trong yea - —ai
, mk nghiem v6i n ak
Clui 9. Ngu fix) la met da th/c vdi he s6 there c6 met nghiem phdc z = a + it (t # 0), thi f(x) cling (than lien hap i cac z lam nghiem. That vay, gia sit
f(x) = ao + atx + +
vlt f(z) = ao + aiz + . +a/Z" = O.
• Ta suy ra
ac, + air + = ao + a + + an? = 0,
Mk la f(z) = 0
De dd to suy ra f(x) chia hgt cho da auk Mc hai
g(x) = (x - z) (x - 7 = x2 - (z + zix + zi,
z + I va ri la nhfing s6 dup.
159
116- qua 3. Citc da thac bdt khd quy cua Rfx.7, R la truane. so thuc, la cite da tin& bac nhdt cac da that bac hai axe + bx + c sot biat sd b 2 - 4ac < 0.
Chang mink. C fie da thee bac nhat va the da thee the hai ved biat 56 am re rang la nhung da thec bat kha quy ciut R/4. Gia sit p(x) mist da thee bat MR, quy cua R[x] v6i bac lon hen mat. Theo (oh V, §2, - 2), p(x) kheng c6 nghiam time. Theo dinh It 1, p(x) co mat nghiam phile z va theo chti y trail p(x) chia Mt cho da thee bac hai veri he s6 thoc
g(x) = x2 - (z + .1)x +
g(x) kh6ng ltha nghich va la It& cua nhan tit Mt kha quy p(x),• fly g(x) phiti la lien kat cna p(x), tee la
p(x) = ug(x), 0 x u C R. n Tit ha qua 3 ta suy ra mai da thee f(x) C Rix] ed bac
hen 0 deu co th6 vie% dual clang
(3) fix)= u(ct i x (ak x ak)'"t (a i x2 +13 i x+ydni
...(at x2 + fit s + y,fi ,
trong do u mat s6 thuc khac 0, cac a t x + b la nheng da thec
bac nhat, the a) x2 + /3j x + yi la nheng da thee bac hai vbi bit
s6 am, cac mt va ni la nheng s6 to nhien, i = 1, ..., k ; j = 1,
., 1. Cate nghiam thee cim f(x) duo° sung cap bed cac Mein td thy& tinh ai x + bt va the nghiam phut Mi the nhan tit bac hai
a. x2 + x + y. . nhan tit tuyan tinh x + 1)1 cho ta mat' 1 1
bt nghiem Rive - Tit , m6i nhan tit bac hai a1 x2 + f
i x + yi cho ta
hai nghiam phde lien hop
s.J + iti Ira s.
l - it. l
- # vei s = — vat. - " 2a I 2ci
160
Vay nen sj + iti lit nghiem bOi cap cite f(x) thi — iti Ming
Ia nghiem liOi cap ni caa fix).
Da thdc f(x) khong cd nghiem thac (nghiem pik) nau cac lay thita mi (cac lay thilit ni) trong dang phan deb (3) cila f(x)
clau bang 0.
2. Phuong trinh bac ba via b6n
Theo nhu trail mai da thdc vdi he s6 phtic va c6 bac Ion hcn 0 deli cd cac nghiam trong trotting s6 pink
Ddi voi cac da tilde bac hai ax2 + bx + c, ta cd cac nghiem IA
—6 + ru t —b + cu2 2a va
2a
trong d6 co l va io2 la hai can be hai cit'a b2 — tat Nhu vay ta da bigu thi cac nghiem cua phuong trinh
ax2 + bx + c = 0
qua cac he -so cfm phuong trinh bang cac phep tinh cong, nhan, chin, Ming lily thiM, khai can. Nguia ta ndi ngmed ta da gidi phuong trinh bang can th,Zc. Ta so thay duel day ta cd the giiti mat phuong trinh bac ba hay bac bon bang can thdc.
Phuong trinh bite ba. Ta hay net phuong trinh bac ba vai ha so phtic
x3 + ax2 + bx + c= 0.
Phuong trinh nay tr6 thanh
(4) y3 +py +47= 0
nau ta dat
a y=x+a.
DO giai phuong trinh (4) ta dat
161
(5) y = u + u,
ta doe sau khi thay vie (4)
(6) u3 3 + + (U + u)(3uv +p) + q = O. Van 66 caa chang ta bay gib . la tiro cite s6 phtic It Nth v theft
. man (6). Ta chti y Uri Sic s6 phuc u Ira v c6 tinh chat
(7) 3uv + p = 0, hay uv = —
Bao gib ming S nhcing s6 pink u va v them man (6) vi (7). That •vay nau u va v them man (6) thi u + v la mot nghiOm caa (4), vay ta S (5) voi y la mot nghiam phdc eta phuang trinh (4), do d6 kat hop vat (7) ta c6 u va v la hat nghiem oda phuang trinh bac hai
z2— yz — = 0
Nhu vay r6i tit (6) va (7), ta &Mc
3 , u 3 = —q, a3 U 3 = 3
Ta nhan thay rang u3 0, v3 lit Se nghiem tha phuang trinh bac hat
(8)
t2 + qt — = 0
u3= = —q
92 + p3
2 4 + 27
3— q 11 q2 p3
2 4 r7
q p3 2 p3 trong 66 —
4 + —27 la mot can bac hai
eta T la mot can bac ba caa
Gal al
2 3 1
2 + -
4 141
27
162
va u la mot can ba cua
2 4 27 • q q P
3 2
sao cho u 1 u 1 thda man (7), thi to cling e6 u2 u 2 va u33 this
man (7) veil
u2 = 43 = g2u1 va V2 = E2 v
a , a trong dd z = — + i —2—, = — i --2—
la hai can be ba philc cna don vi. Mini fly to dugc ba cap s6 phdc (u 1 , u 1), (u2 , 02), (u3 , u3) thda miin (6) va (7). 'Ih dat
yi = VI + V I 1 a
Y2 = EV I + E2 V -i = - (11 1 + V I) + h 2 (1st — lit)
1 in Y3 = E2 u1. + Evi = - 2 (ui + u1) - 2 (a 1 — u1) .
town va cha 3? rang u 1 , u thda man
(6) va (7), ta duct
YI +Y2 +Y3 =
Y1 Y2 + YI 313 + Y2Y3 = 3 3
Y1Y2Y3 ut = —q.
=p
Vay
Yi)(Y Y2)(Y Y3) = Y2 (y1 + y2 + y3) +
An Y2 Y1Y3 + Y2Y3) YiY2Y3 = + PY q .
Da thlic + py + q da duct phan tich thanh tich nhting nhan td tuyeh tinh, va ta dude cac nghiera cos. nd la
y2 , y3 (9). NgLidi ta throng vial tat (9) dual dang
(9)
Tit 9), sau khi tinh
163
y u + u =
2 ( 2) 2 + (3) 3
goi la rang thee Cac-da-n6 (Cardano). Nhu vay ta da giai duet phuong trinh bac ba (4) bang can than
Did D = -4p3 - 27q2, (8) va (9) cho ta
0'l - 3,2)2 (V1 - y3)2 (y2 y3)2 = _10804 _ vb2. = D.
Do dei (4) S nghiem bOi khi VA. chi khi D = 0.
Vi d4. Giai phuong trinh bac ba
1 01-3-
x3 - 3 +
\ 2 2 x + 2i = O.
Dat x = is + u, ta dune
u3 + u3 + 30z + -
Ta you eau u va u than. man
(10) uv = 2 +i 2 a
Do d6 u3 va u3 la hai nghiem tha phuong trinh bac hai
t2 + Zit - 1 = + z12 = 0 .
Phttong trinh nay S mat nghiem kap
3 3
mat can bac ba la i. Dat u l = i , ta any ra tit (10)
a a
_ I 2 22
( + ( 123) 3
2 )1 + 2i = O.
u - I u 1 2 2
164
Vay then (9) cac nghiem cua phuong trinh la
a x —
2 u 1 + u1 + —
2 '
1 a - (u i + u t) +
i 2 (u1 — u1) = —
1 ia i (u i u 1) — 2 (u1 u1)
trong do i
= x3 , vay + la nghiam kep coa phuong trinh.
Bay gib ta hay Kat phuong trinh (4) trong trubng hop p va q la nhang s6 thuc. Ta dot
3 2
A =P T vay D = —4p3 — 27q2 = —108A.
Ta hay nghien olio cac nghiam ctla (4) theo A.
a) A > 0. (8) cho ta bigt u 3 va u3 la thuc. Gal u l la can
bac ba thUc cua u 3 . The (7), u l la thuc va theo (8),
zi t m lit . Theo (8), phuong trinh (4) co mOt nghiam that yt va
hai nghiam pink lien hop y2 va y3 .
b) A = 0. Theo (8), ta co u3 =. u3 va IA thuc. Gig a t la can
bac ba thuc coa u 3 thl theo (7), u 1 tang plati la can bac ba
thuc °Oa u3 va ta c6 li t = u l . Vay theo (9), ta cd ba nghiOm
thuc y2 , y3 yeti y2 = y3 .
c) A < 0. Theo 136 de 1, phuong trinh (3) ad mOt nghiam
thvc, goi nghiam time do lit yt . Theo (8), u 3 va u3 la hai so
phdc kitting phAi lit thvc, do d6 cac can bac ba cua thong phai la pheic (o thvc). Mat khac theo (7) Ira (9), ta cd
165
fly u 1 va ui la hai nghiem pinning trinh bac hai
22
— Y1 2 — = °
vdi he s6 thoc, do dd it, va u l la phtic lien hop. Ta suy ra, trong (8),
ode nghiem y 1 , y2 , y3 M. that. Prong true:mg hop my vi A * 0
nen D * 0, do do 5/ 1 , y2 , y3 I& ba nghiem phan Met. Cac s6
Yt y2 , y3 la thoc, nhung muon tinh cluing theo cling ante
Cac-da-n6 thl lai phai lay can bac ha cna nhung so phtic. Ngubi ta da thong minh dude rang, trong trtiOng hap A < 0, khang thd bidu thi the nghiem cua phuong trinh (3) bang Mc can thUe vai luring that duoi can.
Phuong trinh bGc b6n. Phep giai. mot phuong trinh bac ban
x4 + ax3 + bx2 + ex +d = 0
vOi he s6 pink toy 3, se dtta ve phdp giai mat phuong trinh pho bac ba goi la phuong trinh giai bac ba. 1a flan hanh nhu sau :
a2 x2 Chuydn ba hang tit cudi sang vg phai rdi ceng
4 vao ca
hai vd, ta duoc 2
(x2 + atry ( — x2 — cx — d
Sau dd ta Ong vao hai vd cua phuong trinh nay tong
( x2 ± ax
+ 4
trong dd y la met an mth, ta dttoc
2 (x2 ax Y 2 a
(11) (X + + = ( — + y x2 + 4
ay y2 + - = x + - d
2 4 166
'Pa hay Itta chgn an phu y sao cho vg phai lh mat chinh phuong. Mien thg thi chi vide lam trial tieu bide sd dm tam thitc bac hai din voi x 6 vg phhi
a2 Y ( a-g- 2 — 4
(4 b Y) ( -4- - = °
hay
Y3 - by2 + (ac - 4d)y - [d(a2 - 4b) + c21 = 0.
DO la mat phuang trinh bac ba, gal la phuang trinh gidi cita phuang trinh da cho. Gia sit yo la mat nghiem cua phuong trinh
de. Dien yo van phuong trinh (10), ta dugc vg phai elm phuong
trinh la mat chinh phuang
n de
2 ax ( x + 2)
= (ax 0) 2
x2 + ax p,x2 + aX Yo
-2- -2- - 2
Hai phuong trinh bac hai d6 se cho tat ca bon nghidm elm phuong trinh bac ban. Vay phep giai mat phuong trinh bac bOn da dugc dua ye phep giai mat phuong trinh bac ba vit hai phuong trinh bac hai. Ta say ra tit do rang phuong trinh bac bon gihi dvtgc bang can thdc.
Vi dg. Dial phudng trinh bac bon
x4 _ 3x3 ax2 _ +2
Chuyan ba hang tit cute sang vg phai
_ 3x3 _3x2 + 3x _ 2
9x2 -
Sau de cang vito hai vg , ta dude
(x2 3x 2 3 2 (x — = — x- + 3x - 2.
167
Y2
Cuai cang Ong vito hai ve t ng + 7
(12) (x2 - 2 + )') 2 =
3
= 2 (y - -4) +3x ( 1 + =42 -2.
Ta flu eau y lam that flan biet s6
3 y2 9 ( 1 - 4 ( y ( - 2) = 0,
hay sau khi khai trien
.Y2 - 3) + 3' - 3 = 3)(Y2 ÷ 1) = O. Chon y = 3, phuang trinh (121 tra thanh
3x 3 2 3x 1 2 ( X2 T T • 2 3x 3 3x 1
2 2 - 2 2 '
3x 3 3x 1 2 2
hay x2 - 3x + 2 = 0 , x2 + 1 = - 0
Hai pinning trinh nay chip ta ben nghiem 1, 2, i, -1 . Su thuc ra trong &tieing hop cu the nay ta nhinthay ngay
nghiem 1 cart phuang trinh bac ben da cho vi tang s6 cac he s6 bang 0. Do &I ta co' the vier M lam n6i bat nghiem 1.
x4 - 3x3 + 3x2 - 3x + 2 = (x - 1)(x3 - 2x2 + x - 2) .
Vay van de bay gib la giai.phuang trinh
x3 - 2x2 + x - 2 =0
ma ta cei the yin
x (x - 2) + x -2 = (x - 2)(x2 + 1) =0 Ta d6,7 ta suy ra cac nghiem cfut phuang trinh.
Tir d6
168
Nguiri to chdng minh rang khong the giai bang can tilde cac pinning trinh tdng quest bac Ion hon bon (Aben, Galoa). Hon the nda, Galoa da tim ra &toe ties chufin dd. Nat mot phuong trinh da cho gial dugc bang can tilde hay khOng.
BAI TAP
1. Trong vanh Crxl (C la tniang s6 Mlle) hay plan tich cac
da thdc : x2 + x° — 1 — x2 — 4i + 3, x7 — 1 — i5 thanh met tich nhfing da thdc Mt kha quy.
2. Bidu din hinh hoc cat nghiftm cues da thdc
f(x) = xP — 1 ,
g(x) = (x — — b. (a x 0)
Tit dd suy ra rang f(x) va g(x) cd khong qua hai nghiOrn chung.
3. Tim cac nghi6m plate cua da thim
f(x) = (1 — x2)3 + 8r3 .
Phan tich da thdc f(x) thanh deb nhttng da thdc Mt UM quy vei M s6 thy°.
4. Trong vanh C[x] chting minh rang da thdc f(x) chia hat cho da thdc g(x) khi va chi khi moi nghiOm eta g(x) dau la nghiOm cna f(x) va mcd nghiOm bOi cap k eta g(x) cling lit nghiam boi cap Ion hon k eta f(x).
5. Trong vanh Q[x] (Q la trubing s6 hitu ti), chtIng- minh
rang da tilde f(x) = x31` + x3/+I X312+2 ehia hot cho da thdc
g(x) = x2 + x + 1 vdi k, 1, n la nhting s6 tu nhian thy y.
6. Trong vanh Q[x], chiing minh rang da thdc f(x) = x3 -
- 3n2 x + n3 , vdi n la mot ad to nhien khac 0, la mot da thdc bat MA. guy:
169
7. GM sit a = 1 + i.
a) Bigu din a dual clang Wang giac. Tim madun va ae-gu-men dm a" W. a'.
b) Vigt an va a-fi dual clang a + bi.
c) Bigu din hinh hoe can gia tri a" va vol n 8. d) Chting mirth rang MO so phtle z tuy yd@u c6 thg bigu
din (Wan mot each duy nhat dyed clang z = x + ya vat x, y la nhang so thgc.
e) Chung minh rang anh xa
f : C M
z=x+y •al—o[ x Y 1 —2y x + 2y
W vanh s6 phtie clan vanh the ma trap than vuOng cap 2 la mot Bang eau Tit do suy ra rang tap hap the ma trail, thge,vuOng
cap 2 clang [-2y x + 2y] la mat tniong.
f) TInh
[ 0 1 —2 21
8. Giai the phuang trinh bac ba sau day :
a) 4y3 — 36y2 + 84y — 20 = 0.
b) x3. — x — 6 = 0.
c) x3 + las + 15 = 0 .
d)x3 + 3x2 — 6x + 4 = 0.
9. Ch'ing mink (bang da thdc xang) :
(xt _ x2)2 (x2 _ x3) 2 (xi _ x3)2 = _4p
vai xt , x2 , x3 la the nghiam caa phgung trinh
X3 px +q = 0.
27q2
170
10. Giai the phuong trinh
a) x4 - 3x3 + x2 + 4x -6 = 0
b) x4 - 4x3 + 3x2 + 2x - 1 = 0
c) x4 + 2x3 + 8x2 + 2x + 7 = 0
4) x4 + 6x3 6x2 8 = 0
§2. DA THOC VI% HE SO HOU Ti
1. NghiOm hUu ti ctia mot da thtic ved h0 s6 hem ti
Trade St ta nhan xet rang Su
f(x) = an xn + + an (a n m 0)
la met da than vol he s6 hdu ti thi f(x) cd the vial (lured clang
f(x) = tri (an ? + + cto) = 6 1 g(x)
trong dd b la man s6 chung cua the pilau s6 ai va the ai la
nhung s6 nguyen. Vi f(x) va g(x) chi khan nhau met nhan to bac
0 nen cac nghiem Ma f(x) la Mc nghiem Mut g(x) Vay viec tim nghiem eila met da tilde vdi he s6 him ti dude cliia ve viec tim nghlOm mla met da thdc vdi he s6 nguyen. Mat khac ta tong nhan xet rang Su a la nghiem cim da thdc g(x)
a di + an ce1-1 +... + a a + a = 0
ta cd sau khi nhan hai ve yea a: 1
+ + a: -2 (an
(an at' + _ I (an - I a) + ao
7b. cd # = an a la nghiem thew da thdc
h(x) + an _ i x' + + ann 2 x + ao
171
veil he stS nguyen va he s6 cao nhat bang 1. Do dd mudn S the nghiem etag(x) ta chi vies tim eac nghiem eta h(x).
Ta dal van de a day lit tim cac nghiam hau ti etaeac da thac f(x) cri clang
f(x) = + an _ i xn I + + ai x + ao
vdi the ai nguyen. Gilt sit a la mat nghiem hitu ti eta f(x), the+hi theo ("nth II 3 eta (Ch V, §1, 2), a phki a nguyen. Matt !chive, to
+ an _ian + + aka + an = 0
ta co thg vi
a(an1 + an _ian-2 + + =
do dd al a.. Rini vey the nghiem nguyen aka f(x), neu cd, phai la nhang It& eta ao . Cho nen muen tim the nghiem nguyen etc fix) ta xet the tide eta s6 hang us do ao, va sau dd, thit xem the tuft de S phai la nghiem eta fix) hay khang. De hart the se lan th8. nguea ta chia the then xet sau day.
Gia sit a la mat nghiem nguyen aim f(x). The thi f(x) Chia het cho x - a
f(x) = (x - a)q(x)
va theo ad de Rode ne (Ch IV, §1, 4), q(x) la mat da thde vai he se nguyen. Do do q(1) va q(-1) la nhang s6 nguyen vb.
1
/(1) a
(1)
= q(1)' 1 + a Cr( 1) -
ngu a klthe 1 va -1.
Vi thy trade het ta tinh f(1) va f(-1) ctg xem 1 va -1 ed phai lit nghiem eta f(x), sau do ta xet the vac a ±-1 Sa a. sao cho
f( 1) 1 - a v- 1 + a
la nhang so nguyen de tha xem chUng ed phai la nghiem aka f(x), va do do se Ian thit eta to bat di mil Chung.
172
Vi du. Tim nghiem hitu ti Mut da thile
f(x) = x5 - 8x4 + 20x3 - 20x2 + 19x - 12
Trude het ta ed tong cac he se eiut f(x) bang 0 nen 1 la nghiem cda f(x). Bang so de Hodc ne (Ch IV, §1, 4), ta tinh eat he s6 cua da thtie thuong g(x) trong phep chia f(x) cho x - 1
1 -8 20 -20 19 -12
1 -7 13 -7 12 0
do dd eac nghiem con lai caa f(x) la cac nghiem cua g(x)
g(x) = x4 - 7x3 + 13x2 - 7x + 12.
'Pa nhan xet rang g(c) > 0 v6i m9i c c 0, nen g(x) khOng cd nghiem am. Vi vay ta chi xet cat u6c thing dm s6 hang tti do : 1, 2, 3, 4, 6, 12. Ta cd g(1) = 12, g(-1) = 40. VI eat s6
40 40 40 3' 7' 13
khOng phai la nguyen, nen 2, 6, 12 khOng Olaf la nghiem cua
g(x). Cite s6 3 ye 4 deulam cho
g(1) va g(-1) 1 -a 1 + a
nguyen, nen clung ed the la nghiem cult g(x). Mu6n their xem clung cd phai la nghiem cent g(x), ta chi viec tie') tau vao bang tie)) nhu sau :
-8 20 -20 19 -12
1 1 -7 13 -7 12 0
3 1 -4 1 -4 0
173
Vey ta co
f(x) = (x - /)(x - 3)(x - 4)(x2 + 1)
Cae nghiem nguyen cua f(x) la 1, 3, 4.
2. Da thee bat arid quy ads vanh Q[x]
D6i vdi trifling s6 thvc It va tntbng s6 phdc C, van de xet xern met da tilde da cho cfia vanh FtkJ hay C[x] co Mt kith quy hay khang rat don glint (01, 1) ; nhvng trong vanh Q[x]
Q la trttling s6 hdu ti thi van d4 phdc tap him nhieu. D6i vdi cac da thdc bee hai va ba cua 411[4, viec xet item S Mt kith quy hay kitting &roc dtta ve viec dm nghiem hUu ti gda cac da thdc do (eh V, §1, bill tap 2) : CAC da thdc bec hai va bee ba cim Q[x] la bat kha quy khi ya chi khi ehting kh6ng S nghiem hUu ti. D6i veri cac da tilde Mc Ian hon ba thi van dg phdc tap Min nhigu. Chang him da thdc x4 + 2x2 + 1 = (x2 + 1)2
ril rang )(hong co nghiem hitu ti new, nhting no co met WC thvc sit x2 + 1, vey khang phai la bat kha quy.
Trong mac 1 ta da nhen set ring mci da thde fix) vdi he s6 hdu ti deu S the vigt dudi clang
f(x) = 6-1 g(x)
trong dci 6 1Ft mot s6 nguyOn Mute 0, g(x) la met da thile veil he se nguyen. Trong vanh Q[x], f(x) va g(x) la lift Mt vity f(x) la Mt ad quy khi va chi khi g(x) lit bat kha. quy. Do dri tieu chum Aidenstaind ma ta dim m chafe day dg xet met da thdc cua Q[x] co Mt kha quy hay•khang la tieu chum cho cac da th1e veil he se nguyen.
De chuan bi cho viec chting minh tIeu chuifn ay, trout Mt ta gidi thieu khai niem da Dixie nguyen ban va ehtIng minh hai b6 de.
Dinh nghla 1. Gia sit f(x) la mot da auk vol he sec nguyen, f(x) gal IA nguyen thin ngu Se he se oda f(x) khdng co vdc Chung nail khdc ngoai ±1.
Cho mot da thdc vdi he s6 Nguyen f(x) C z[x], Id !lieu bang a vac chung Idn nhat Sit cdc he s6 cda f(x), ta cel
174
f(x) = ac(x)
vdi ((x) e Z[xl va the he se cda r(x) kh6ng ed Lida Chung nit° khac ngoal ±1, the la r(x) nguygn ban.
Ndu f(x) E Q[x] thi to ed thg vigt f(x) dttdi dang
f(x) = b COO
trong do fa(x) nguygn ban va a, b E Z nguyen t6 ding nhau.
2x2 15 1
Vi du. f(x) = 6x3 1
-2- + - = 17)(60x3 + 24x
2 - 75)
3
= 176 (20x3 8x2 28) '
Cac se 20, 8, -25 Itheng S doe chung nao kink ngthi 1 va. -1.
Be de 1. Tick cua hai da thdc nguyen ban la mat da thdc nguyen bdn.
Cluing mink. Gitt sit
f(x) = a. + aix + + amxm
g(x) = b. + b ix + + b rix"
la hai da axle nguygn ban. 1k chi can e.hting rainh rang cho met se nguyen td p thy 3,, p khang chia het the he s6 cita da thtic tich f(x)g(x). 136 rang p khong chia Mt the he se eita f(x) va g(x).
Gia sit p chia Mt cte,..., as_ i , be..., bs_ i va p Elblag ehia Mt as
va bs. 11k xet h@ s6 c s+e cua da thile tich f(x)g(x).
crvs = (... + ar_ ib.E1 ) + arb s + (arkibri + ...) trong do p chia Mt the tdng trong cad dgu ngoac, nhvng Wing chia Mt fiat arbs vi p la nguyen to. Do chi p khOng ehia Mt cm. •
116 de 2. Neu f(x) la mdt da tilde yea he se nguyen co bac Ian hon 0 utt f(x) khong bdt khd quy trong Q[xl, thi f(x) phdn tich duos thiznh mat tick nhung da Hale bac kluic 0 yeti he ad
nguyen.
Cluing mink Gih sit fix) khong bat kita quy trong Q[xl, thg
thl f(x) ed thg vigt
175
f(x) w(x)y(x) voi p(x) va 7;tax) la nhang talc Mac sa coa f(x) trong Q[x]. Theo nha tren da nhan xet, to cd thd viet
w(x)= b g(x), y(x) = h(x)
trong dd g(x), h(x) lit nhang da thac nguyen ban va a, b, c, d la nhang s6 nguyen. Do dci
f(x) = g(x)h(x)
P_ — = q bc1
— va p, q nguyen to ding nhau. Ta H hiau cache se
caa da thac tich g(x) h(x) bang the thi theo bd de 1, g(x) h(x) la nguyen ban, the nen cite khOng cd utdc chung nito khan ngoai
Pei ±1. Mat kluic vi f(x) E Z[x] nen the s6 — phal IA nguyen, do do
q chia het vi q nguyen vdi p. Ta suy m q = ±1, Mc la
f(x) = -±p g(x) h(x).
Vi p(x) va ydx) la nhang udc there set ona f(x) trong nen g(z) va h(x) la !Mang da tilde bac khan 0 dm Z[x]. n
Tien chutin Aidenstaina. GM sd
f(x) = ao + aix + + aorn (n > 1)
Id mOt da th,Zc obi M s6 nguyen, oh girl sit c6 mOt s6 nguyen t6 p sao cho p khang chic) hit he s6 cao nit& ate, nhung p chin hat Mc he ad con IM p2 khOng chia s6 hang ht do ao . Tha thi da th,2c f(x) l¢ bat kid quy trong Q[4.
Cluing mink. Gib s8 f(x) cd nhang ado thy° sa trong Q[x]. Theo bd de 2, f(x) od the vier
f(x) = g(x) h(x),
trong dd
g(x) = bo bix + + b E Z, 0 r <
h(x) = co + crx + + csx' c, E Z, 0 < s c n
176
ta cd a = b c 0 0 a t = b lco + b oc I
ak = b kCo bk-IC I + " +
an = b r e
Theo gilt thigt p chia Mt a. = bee ; vay vi p lit nguyen t6,
nen Mac p chia Mt bo hoac p chia hgt c_ Gia sit p chia ha
b 0, the' thi p khOng chia co, vi neu th0g thi p 2 se chia Mt
a. = b oco, trai vii gia thigt. p khOng thg chia. Mt myi he s6
coa g(x), vi ngu thg thi p se chia Mt a = bre , trai via gilt
thigt. Vay giA sii b k la he s6 dau tign cum- g(x) :hong chia hgt
cho p. Ta hay xet
a k = bkco +bk_ IC I + bock,
trong do ak, bk_ 1 , ..., 60 deu chia het cho p. Vay bkco phai chia
ha cho p. Vi p lit nguyen t6, ta suy ra hoac 6 k chia cho p,
hoac c. dila Mt cho p, may than Atli gia thigt ye b k va co .•
Vi du. 1) Da thdc x4 + 6x3 — 18x2 + 42x + 12 la bat kha quy trong Q[x]. That fly ta cd the ap dung tigu than Aidenstaind vdi p = 3.
2) Da thdc
xn + + + + p
yea p la met s6 nguygn to' thy yr, lit bat kha quy trong Q(x] .
BAI TAP
1. Tim nghigm hitu ti cart cac da thdc
a) x3 — 6x2 + 15x — 14
b) 2x3 + 3x2 + 6x — 4
c) x6 —6x5 +11x4 —x3 —18x3 + 20x —8 ..
d)
x5 + 2x4 + 6x3 + 3x2 — 42x — 48
177'
2. Gi/t sii - vdi p, q E Z nguyen t6 citng nhau, nghiam q ,
cua da thdc
an.xn + + + ao
vol ha s6 nguyen. Chung minh rang
a) p I aq Ira q I. an
b) p - mq la Mc cim f(m)
voi m nguyen ; dac Met p - q lit Mc cua f(1), p + q la Mc ctia f(- 1).
3. 4 dung bai tap 2) de tinh nghiem hvu ti cda da thdc
10x5 - 81x4 + 90x3 - 10212 + 80x - 21
4. Chung minh rang da tittle f(x) vet he sg'nguyen kheng cu nghiem nguyen ngu f(0) va f(1) la nhilng s6 le
5. Gia sit p(x) la mat da tilde vol he s6 nguyen va p(x) bat kha quy trong Zhcl. Chung minh rang, trong Z[x], Mu p(x)If(x)g(x), thl huge p(x)I f(x) hoc p(x)I g(4-
6. Trong vanh Z[x] clidng minh ring mai da thitc, Mule 0 va khan ±1, dgu cd the vigt dud) dung Mich acing da thdc Mt
quy .
7. Dung Heti chugn Aidenstaina dg cluing minh rang cac da thdc sau day la Mt kha quy trong QM.
a) x4 - 8x3 + 12+2 -fix +3
b) x4 -x3 + 2r + 1
c) + xP -2 + + x + 1 vati p nguyen tg.
FluOng don : dgit x = y + 1.
8. Tim digu kian can va du da da tilde x4 + px2 + q la Mt kha quy trong (Hal
9. Gia su fix) = (x - ct i)(x - a2)...(x - an) - 1, vat cac a1 la Mitng s6 nguyen plan blot. Chung minh tang fix) lit bgt kha
quy trong QDcl.
178,
MUC .LUG
Trang
Lai n6i
3
Chiterng I - TAP 1-10P VA QUAN Ht
§I- T4p htyp va Soh o
1. 1Chai them tap hop
2. BO phan caa met tap hop 5
3. lieu cua hai tap hop 6
4. T0p hop rang 7
5. Tap hop met, hai phin to 7
6. Tap hop cac N) phan ctla mot tap hop 8
7. Tich a cac caa hai tap hOp 8
8. Hop va giao caa hai Op hop 9
9. Anh xo
10. Anh va too anh 13
11. Don anh - Toan anh - Song anh 14
12. Tich anh o 15
13.Thu het:, va and Ong anh 17
14. TOp hop chi so 18
15. flOp, giao, Lich de cac caa met ho tap hOp 19
Bai tap 20
§2. Quan hf
I. Quan he hai ngdi 23
2. Quan he lilting &king 24
3. Quan he Oil III 25
MI tap 28
0. Scr cac nen de au If thuyit tap hqp
179
Chuang - NUA NHOM VA NHOM
§1. Nfra /them
1. ?hop Loan hai ng61 37 2. NOM rthem 39 Bai tGp 42
§2. alham
1. Nhom 43
2. Nh6m con 47 3. NhOm con chuOn lac va Dion, thIldng 52 4. DEng cdu
58 5. D6i 'ding NM 64 BM tap 68
Chuang Ip - VANTI VA TRUONC.
§1. Vanh va mien nguyen
1. Vanh 78 2. Doc cua khang. Mien nguyen 80 3. Vanh con. SI 4. Mean va vanh thtiong 82 5. D6ng 85 BM tap
87 §2. Tnr&ug
I. Traang 91 ' 2. TnIdng con 91
3. Trttong c&c thUong 92 Bai tap 94
Chuang IV - VANH DA Tale
§1. Vanh da that mat ho
I.' SIM da Mac met An 97
2. Bac caa mOt da thilc 100 3. Phep chid vii cht 101
4. Nghiem Na mOt da Mac 105
