Osnovy teorii Galua.

209

20* ~~~~~~~Teopuai quceA

B11LIHTaRJ I1o4JeHHO 143 (4.7) (4.6), n1oJyii~M
[tp (i)-(1)j + [fp(dj)-iddDI + [yp(d2)- ~(ds)] +.+.
+c f(dhl- (dki) + [p (p -1)- 4(p- 1)] 0.
3itec~b K~aKoe Bbipam~eHe B l(BaJapTHb1X CKO6K~aX B cHJiy (4.5) He MO)KeT
t6blm OTpH1uaTeffbHh1M; HO TaK iKK cymm a 39TIIX Bblpa~KeHHI0 paB~a HyJIlO, TO
OHH He moryT 6bITb 14 nwOAOKHTeJlbHbIMH, a 11OTOMY
14 B qaCTHOCTH
XXVI. CpaBHeHHe
(4.8) xP -1-0 (mod p)
HimeeT POBHO T (p - ) rnepBOO6pa3HbIX KOpHefl.
C.1e C T B He. Cyiu.eCTBYIOT TaKHe Lmc~aa a, 'ITO CTefleHH
1,a, a2,..., aP-2 HctiepflblBaIOT Bce B3aHmHo-n~pOCTbJe c p iKJIaccb
cpaBHeHAHOl11 MOJYIJO P.
3. rlepeftrtem K cjnyqaio, Korjaa m = pV, r~e P HeqeT~oe npoc~oe LifciCo.
IlYCTb a 6ymteT rnepBOO6pa3Hblff KO~eHb CpaBHeHH51 (4.8). Hcciieatyem, npip
ICaKHX YCJIOBH11X OH 6yi~eT Taiwe nepBOO6paa~bIM KOPHem cpaBHeHHR
(4.9) x'" (P __1 1 — 0 (mod pV)
[IyCTb a P1 - + kp. Torxrn eCvm k He AtejrHTCR Ha p, To ap-' #
(mod P 2), a nOTOmy noi~a3aTeJnh, K KOTopomy nH~aH3JlJ)KT a no MOyJIJ1O p2,
He pa~eH P- 1. Ho, C.Rpyroil CTOPOHbl, OH eJLIJHTCH Ha p - 1, a nlOTOMY OH
pa~eH TO4HO cp (P') = P (P-1) T. e. a 51BfigTC51 nepBoo6paaiibIM KOPHem
110 moayni1o p'.
'-ITo6bl BblSICHHTb xtenro flPH v= 3, B03Bearem a B P (P - 1)-.io CTeneHb.
TIOfiy'IHM:
ap(-1 =l+ p) -1 +~k +P(P-l1) k2p2~~
2
+ kPpPz 1- + kp2 (mod p3),
r~ae k He eJI~HTCS1 Ha p. IlO9TOMy noKa3aTeJab, K KOTOPOMY n1pHHaJLJ1)KHT a
no0 MOayaJIO p3, He sBiusieTC1 JAenHTeaem qHHcna p (p -1,a nOTOMy, aLeJJIcb
Ha1 P - 1, AORT)KeH COBiI1aCTb C p2 (p - 1) = cp (p3).
i7poJ.IoJDIaaH paccyy~aeH~e xis 6ojiee BIICOKHX 3Ha.1eHHA1 V, I1pHXOXaHM
IC Teopeme:
XXVII. UW111 Toro, HTo6bJ nepBoO6pa3Hb1A KOpeHb CpaBHeHHS1 (4.8)
6wi1. T ajoKe nepBoo6pa3HbIM KOPHem Cpa1BHeHH1 (4.9), Heo6xoaltHmo H.LoCTa1TO'HO, 'ITO6bi ixejioe 'IHCJIo k = -~ He JaeJHJIOCb Ha p.
flwnb3yfiCb STHM, XLOKaweKm Cy[LLCTOB0BaHHe nepBoo6pa3HbIx KOpHeft cpaB4ieHHS1 (4.9). UtiJ11 SToro paCCMOTPHM KcaKofl-HH6yxb nepBOo6pa3Hb1fk KopeHb