X goila idean nguyen t6 ngu• va chi lieu veil u, u E tich
18. Gia si tX 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 + n 2 x1 , nt n2) (x1 , n1) (x2 , n2) = (x i x2 + n1 x2 + n 2 x1 , nt n2) a) Chung minh rang X x Z la mot vanh cd don vi. b) Anh xa
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
h a : 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) = h a
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
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
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
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)
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
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
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
CHUONG IV