VI xI x, nen xI ab = x Theo b) xIa hoac xI b Ned
Q la trttling s6 hdu tithi van d4 phdc tap him nhieu D
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
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 - 75) 2 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. + a ix + + a mx m 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 , b e ..., b s_ i va p Elblag ehia Mt as va b s. 11k xet h@ s6 c s+e cua da thile tich f(x)g(x).
crvs = (... + ar_ ib.E1 ) + a rb s + (a rkibri + ...) trong do p chia
Mt the tdng trong cad dgu ngoac, nhvng Wing chia Mt fiat
a rb s vi p la nguyen to. Do chi p khOng ehia Mt c m . •
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
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)