Osnovy teorii Galua.

~ 7. Pa3pewubiue epynnlb 4

49

7. HlOCTaBBHM B camom O6fleM BHJLe BOHlP0C o6 aHaAHTHqecKom
rH p e A C T a B JI e H H H nOatCTaHOBOK CTeneHH P. 3Tro0 3HaqaeT, tITO JAaR
KawK2oft 3aJaaHHofl nOICTaHOBK"l
( 0, 1, 2,...,p-1'
a0,, a2.a_, )
MbI 6yanem HCKaTb TaKOtR [IO.THOM f1(X) C LuejibimH paLIHOHaJlbHMLMH i<0o(4 -4~HixvueHTamti, lOTOpbifl 6bi yJLOBj1eTBOP5UI YCJIOBI1HM
(7.2) 1(0) ==do f (1) ', f (2) == a, J(P - 1) o (mod p).
He TpyXWO y6eaiuTbcS] (B r.TaBe 1I, ~ 1.5 Mbl OCTaHOBHMC5I Ha 9TOM
no~iipo6Hiee), 'ITO aTHiM YCJIOBHH1M YRIOBq eTB0p51eT nIOJIMOM (P -I)-OR (HiqHi
HHNKe) CTeHeHH
AX) F~~~x) + J p+ -+ 4(mod P),
ruae F(x) == x(x- 1)(x - 2).. (-p+1). CTOR'ijiue B 3HMamHaTeJsix 4HClcla
F'(k) (k = 0, 1 p p-i1) paBlibI
F'(k) =k(k-1)... 2.1.( — 1).( -2)... (k - p+ 1)=
(lPk k! (p —k+1)!
3TH.3HameHaTeJH eJil~J~OT Ko09t~i4j~lHeHTbi I1OJHHoma f(x) upo6HbzMH; HO
Bce OHH B3aHMHO HPOCTbl c /3, a HOTOMV cpaBHeHH5I
XROI1YCKaI0T peLLeHHl B 1aejlbix '-iiciiax. EcniH Mbl, nOJlb3YfCb.3THMH pewneHHHMH, BO3bMeM B KaB4eCTBe f(x) IOJUiHHOM C iienbimi Koq4p4JHiuueHTaMH
F(X _oX __+ alXl ot1xp-1l
x x - 1 ~ x-p-t-
TO MTT HOJIMHOM T0oKe 6yueT YAOBJIeTBOPR1Tb ycJ1oBH5Im (7.2).
CTatiesi 3aHncl4IBaTb T1OJCTaH0BKH 0,1 2 J. -P 1 B TaKom ii3He:
(x ->f x). E cuiu 3aaaHbl B aHajiHTH'IeCKOM BHJLe RBe nOaLCTaHoBKHi: (X — >f(x))
H (X-o>(P(X)), TO MiX npHBa~i (-fx)x-> ())mo)KT 6bl~b rpextCTMBJ1HO, TaK: (x - cp (f(x>)). B camom aenre, ecuii (x ->1(x)) nepeBOJLHT
a~ B P3, TO f(c() --- (mod p), H eCnrm (x -o p(x)) flepeBOJJ2T [3 B y, TO
(9 ([3) == (mod p). OTCioaa cuieayeT, WIO (P (f(ci)) cpE (P3) _- (mod p), a nOTOMY
nOJLCTaHOBi%, (x, y (f((x))) [iepeBOA1HT (X B T. 3To noua3bIBaeT, 'ITO flOACTaHOBKa (x — > c (f(x))) eCTb npOH3BeJLeHtH nOJ1CTaiHOBO1( (f(X-X)) H
MoxKiO, HOHLI3HTb CTCIIeHb HOJIHBoma y (f&)), pn3JLnll4B ero Ha F(X) H
B351B OCTaTOK( CT 3ToFo JteJIeHiSI. B camom aejie, F10,ILIHOM F(X) HplHHHmaeT
flpH 3Ha'IeHimIx x =O0 1, 2,.... I P-1 Hy~neBoe 3atieHfle. EcinH Q(x) 6yaeT
'IaCTHoe H I?(x) OCTaTOK OT aeqeH-iRn c(f((X)) Hia F(x), TO mbi 6yaem
HmeTb:
(9(fx)) = F(x). Q(x) + I?(x).
4 '1e6oTapCB.