Osnovy teorii Galua.

172

V. Ypae-nenun C 3ada~nblmu epynnamu

Boo6Ije He MO) wT 61b1Tb HMnpHMHTI4BH~tl, a nOTOMY B STOM cjiyqae HeT
Haxto6H-ocT CTaBHTb yCJIOBHem o6JaaaaHHe iwHKJom (n - 1)-ro nopHixua.
1110 KacaeTCH orpaHH'4teH IVA, H-anaraemoro Ha BetI'H~HI-y npo0CTbX MOAyJaetf, TO03Jtecb emy nloJV1e>KHT TOJnbKO To ripocroe 414C110, KOTOpoe COOTBeTCTByeT TpaHCUIO3HLLHH. B 3TOM cnyqae LLHKneHHEb~ft TIHf HmeeT, BHAL
I Zs=(2) 1,1,..., 1),
flHpCme eAIHHLaa BXOAIHT n - 2 pa3a. 14T06bW CyL1eCTBOBaJ1O He MeHee ii - 2
pa3J1HqHbix no MOAJlfhI0 p JIHHeklHbIX HOJIHHOMOB, J1OCTaTOqHO orpaffl4Hlm p
YCROBHem
5. 1 pHi M e P. llOCTpOHTh ypaBHCeuie 6-fl CTenHtH1 6e3 aq4M)eKTa. ALa14 IHKJneHHorO, T1Hna~ (6) BO3bmem p, 2, ~Ani -2 (55 I) - p'2 =3, AW = (2, 1, 1, 1, 1)-.
HenpHB0AHMblHi no, N10,aymo 2 nomiHHom 6-f CTefleHH Xoao)KeHl 6bJTb aeaH~eJem
HOJIHIoma x26 -x = x (X68 - 1). 3TOT HoIIHHOM coaepWHT ejite JIHeii"hie, iBwaapa1Hbe
H Ky6HqiecK~e AeJIHTeJIHw JlHHefiHbie HOJIHHOMb] Bxoqq-r1 TaKH{C B x2 - X - X (x-1)
Ksa.IapaTwe-B X 2 - x =x(x3 -11), Ky6w~eC~ne -B x2 =X X (X7 - 1). I1OJJHHOM
K 1 _ + 1 6yaiytrn A!VJUHTCeem HOJIHIoma x(X63-) B TO WeC Bpem$1
B3aIMIIO HPOCT C HOJiliHomamH X (x -),X (X3 - 1 4X (X7 - 1), a HOTOMY mo}KeT cojepAaBTb TOJ~bKO HeI1pHBOAHmbeI Ho moayuJo 2 Ae.4IHTAH 6-fl cenemu. CTaJIo 6I1Tb, OH
neHpHBOAIIM 110 M0Ly.1o 2. IlOBTOMy mu mo}1(em noOAOHTb
1(x)=X6'+ x3 + 1(mod 2).
[lo m~o~ayao 3 noaIIHOM 1 (x) aoJI)xeH pa.3jara.TbCq Ha JnwiefiHbilf HOJIHHOM H1 HeBPHBOJAIM5f HO f0HHOM 5-fl CreiieH. OqHUH 143 Ta]KHX IIOJIHOMOB 6b1.n HaMH ywe iiaflAeH B3 n. 3. H4CXOJqHI13 Hero, mbI mo)*Cm HOJIO}KHTb
f1(X) =X (XI, + X4-x ~XSX- 1) (mod 3).
HafcoHell, Ho V0JoayIo 5 nOJIIHHM f (X) 40Ixoien paanarambC$ Hia 4 JIHflefilbJX HO.51H-!IQma H OALHH HenpHBOJLHMb~fl KBaJapaTHb1Hl IIOJIIOM. flociieAHHrnAoi OJH(eH 6blTb xReall-8 1 4 + I. 3TTBJH OM eH'T npoCTO pa~aaaraeTcH Ha
KB2ApaT~bIe MHO>KHTeAlH:
X I= X~1 x- 4 -(X2 + 2) (X2 -2) (mod 5),
KOTOpbie B CHJIY Cfcaianoro aouKj1(1w 6WlTb HenpHiBOAHl~bl. IIO3TOMY MWi mo)1em no-.I1O)KHTb
f (x) ==(x4- )(x-+ 2>=x6+ 2x4- X2 - 2(mod 5).
ConOCTaBAsR1 Bce Hakae~HHble YCJIOBHISI, MbM HlOJ~y'IIM cjie~yioutyio CHCTeMy cpaBHeiHfl AJll 1{oq4kpiLUieHT0B Ai noxio~mHa f/(x):,
Al 0, A2~ 2, A~s 0, A,& - 1, A 5 o, As - 2(m od 5).
Pewzaff 3TH CpaBHeHHq, noay'uHM:
Al ==10, A2 =-2, As_- -5, A4 -6, Ar, ---lo,' A6 ==3 (mod 30).
TaKHm o6pa3oM BC5IKoe ypaBHeHHe
X's +(1O +30a)x + (2 + 30P)X4+(-5 + 3OT)x8+(- 6+ 306)xs +
+ ( — 1O+3OE) X+ (3~+30%) = O
we a, P# T, 6, s, )% JpoH3BOJlbHbwe iueane qrnc.aa, HmeeT B xaqeCT~e rpynnu railya
cHtMMeTpH'lecxyjo rpynny.