Osnovy teorii Galua.

166

166 ~~~V. Ypa6enuef C 3afdaHHLmu zpynnamu

ITOm~~, i1OJb3yHCb c~OPMyJAOf UHemeMaHa, 6yJ~eM Hmemb:
fp (a.)]~ ( (a') == (p(a) (mod p),
OTKYJ~a mbi 3aKjuoqaem, WqTo( (ci) siBmieTC1 K0PHem cpaBHeHH115 xP' x (mod p).
Ho KOPHH SToro cpaBHeHHKI Hcqepni~hBa1OTCS1 95IAOM 0, 1, 2,..., p -i, a noTOMY BeJIHqtiHH (p (a.) 1)aBHa O1HOMY 113 9THX pa[LoHiaJlbHhX 'rncenl. TaKHM
o6pa3oM rioie BeJIlIqHH, Hie meH5111OHLHX flPH nl01CTaH0BKe (2.6) (H ee CTenleHRX), eCTb nose- pauLLHaOiHWhbhX Lmcenl, e CHAY qero rpynria ranya cpaBHeHH115 f(x) 0 (mod p) eCrb lIHKJwiHeCKasi rpyriua, O6pa3OBaHHasi CTenIeH5MH
rnoA.CTaHoBKH (2.6)). HMa<:
Te o pe ma 89. Epyrira ranya cpaBHeHHSI
(2/f) 1(x) ==O (mod p)
eCTb UHK.4KHqecKasi rpyrina, o6pa3oBaHHa% CTene~H5MH noXLCTaHoBKH, coCTo51ujeig 113 LLHKJIOB fI0PRUK0B nl, n2,... -, n,, ecaH f(i) pa3anaraeTCR
110 MOaiyJ1O p Ha pa33JH4qHb1 HeIIpHBOAImwbe MHO}KHTeu1H cTerreHeit
nl n2,. nk
9. Csius, me*KAy rpyinnaMH raJnya ypalBHllHH51 H cpaBHeKHji. PacCMOTPHM OA~HoBpemeHHO ypaBHeHHe
(2.8) f(x)=O
H cpaBeHner
(2.9) f(x) — 0 (m od p).
4T06bJ floCTPOHTb rpynny raaiya ypaBHeHHS1 (2.8), rnepe~ymepyem ero
KopHHw XI, x2,.. Xtl, H rIOCTPOHM ero, OCHOBHbwe MOAJlyH:
HOPHSIMII KOT0~bIX M1BJ1IOTCH COOTf1CTCTBeHHO XI1, X22..., X-n
H4To6bi HOCTPOHTb OCHOBHbIe moJLym1 an15 CpaBHeHH51 (2.9), elx6epem
B KaqeCTie rnep~oro moap~Ivi?p1(x) KaKoRt-HH6yab Ria Hel1HBO2IHMb1X rio MOJiy.71 p MHlO)HTeneff IOJ1HH0ma f1(x); o6o3Ha4Hm qepe3 x1 KaKoft-HH6y21b
113 KopneR CpaB~eHHSI (p(x) 0 ==(mod p). JIaanee o6o3Ha4Hhi qepe3 (p2(X; XI)
KaKoft-HH6y,4b ttenpHriojxHmbA rno MOaJ1IJO p B HOJ1ie K(x1) AenJHTeJnb nOAIHHoMa fAx; x1), a qepe3 X2-KaKofl.HH6yJ~b 113 tcoptefl ca3BHeHH11 (2(X; XI)~
— 0 (mod p), H1 T. i4. HIPH TaKOM crioco6e napannuehnbHott Hymepa31iHH KOpHetl
(2.8) H1 (2.9) B~C5l(Oe COOTHoUweHHe mex)yK2 1OPHRMH ypaBHeHH5if (2.8) 6y21eT
HmeTb meCTO TaKH{e me)I{Jly KOPHRMH cpaBHeHHF1 (2.9). (3rTO cirle21yeTr HS
Toro, wro., KaK MUi BHJIeRJH B rnatie II, ~ 4, BC51KO aoe Tao OOTHlowef-ite
mo)*(o rlpeAlCTaBHTb, KaK Al41Ieftlty1O KoM6imaLLHIo OT COOTHOwIeHHft
tix Xi 2,. X., x i 0,
H B TO wKe Bpem5I MOALYRH cpaBHeHH5I (2.9) 51JSMIROTC51 AeJ1HTen5MH mo21yneift
(2.10). Ho03ToMy BcasiKS riOjC~aHoaca, tie Hiapywaiouxasi COOTH~weHHft MeWA2Y
KOPH51Mti cpaBeHeHRns (2.9), Hie iiapyuwae Tak}Ke COOTHow~eHHlft mewKJy HopHRMH ypaBHel-HHR (2.8), eCAHr TOJ3bKO. mub 6y~aem co6.ntoiam yKa3aHHwdk napan'nTeJIH3M I1PH HymepallHH KopHeR. OTCio21a MbM 3aKnlioqem: