VI xI x, nen xI ab = x Theo b) xIa hoac xI b Ned
la nguyen ban, the nen cite khOng cd utdc chung nito khan ngoa
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 + a ix + + 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 p 2 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 b ix + + b E Z, 0 r < h(x) = co + c rx + + csx' c, E Z, 0 < s c n
ta cd a = b c 0 0 a t = b lco + b oc I
ak = b kCo b k-IC I + " +
a n = b r e
Theo gilt thigt p chia Mt a. = b ee ; vay vi p lit nguyen t6, nen Mac p chia Mt b o hoac p chia hgt c_ Gia sit p chia ha
b 0, the' thi p khOng chia c o, 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 = b re , 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 _ICI + bock,
trong do a k, b k_ 1 , ..., 6 0 deu chia het cho p. Vay b kc o 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 x 4 + 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 + 3x 2 — 42x — 48
2. Gi/t sii - vdi p, q q, E Z nguyen t6 citng nhau, nghiam cua da thdc
an.xn + + + ao
vol ha s6 nguyen. Chung minh rang
a) p I a q 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.
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
Chuang - NUA NHOM VA NHOM
§1. Nfra /them