Osnovy teorii Galua.

206

206 ~~~~~~~Teopun i uceii

EUHr MOJ1yJb He eCTb npoc~oe 'IHlCJiO, TO Teopema mo)KeT y)Ke He iimeTb
meCTa. BO3bmeM cJae.pyoLLif4 11pOCTOI4 [Ipimep: cpaBHeHI4e
x2-3x -O (mod 9)
HlmeeT 3 KXOPHR 0, 3) 6.
~ 4. Hlepeoo6pa3mlihe KopHFI.
1. PaccmOTpI-M cpaBHefHHi
(4.1) x ff "I) -1 z 0 (mod in).
JAns KawKIoro Ht3 ero C? (in) B3aiiMHO-HIpocTbIX C in KopHefl a (To'm~ee
Bbipa}KagCb, KjiaCCOB KopHeft TaI( K~aK Mbl He 6yJaem CqHTaTb pa3J1HqHbIMH
KopHefl, jieKwaL4X B OJIHOM H TOM weKn CJacce, T. e. cp1BHHmbix mewaJy co6oio
no moJayJUo m) cyLue~CTByeT HaHmeHbflUHfl loK[3IITeJlb fl, JVIH KOTporOO HmeeT
meCTO
(4.2) a -zm0(m od in).
Byhlem B 9TOM cnytiyae rOBOPiiTb, 4To 'iHcao, a ~H liHaan eWHT KC flO
K a 3a T en n0f. TaKHM o6pa3oM floxa3aTe.Tb fl eCTb rHopu~oK aiiemeHTa a
rpyfil~bi B3aHMHO-IHpoCTb[X Cmi KJUacCOB OTHocHTeJlbHOvymHo2KeHHSI. Teopema XXI
YTBepwKJaeT, qTO n gBBJIHTC5B BCera JaewnTenem trnCja cp(in).
Ecki~ rpynira I33aHMHO-npOCTbWX c mo~tyJem K.TaCCOB OTHOCHlTehbHO YMHO)KHcHmi eCTb LLH1JH,4ecKasi rpynna, TO Bceraa Haf1JayTCSI B~emeHTbl, npHHaJAniexcaiuie K no~a~aaerno Tp (in), H o6paTHO. 3TH gJIemeH-,bI HOC5IT Ha3BaHHe
H e pBo o6 pa3 Hb X K0p H e A cpaBHieH14Hf (4.1) (HiH InPOCTO rnepB~o6pa3 -HbWX KOPHen MHcsna in). Hcciiexayem, 4V-I KaKI4X THflOB MOJayJefk in Cy11eCTBYIOT
nepBoo6paaHbue KOPHH.
2. PaCCMOTpHM CHaqa J~iCJyqtial in p, ULe p nipoc~oe II4CJIO. ai0Ka2+xew.,
tITO B 3TOM CJay'ae nepBoo6pa3Hbue KOPHH Bcera~a cyltueCTBYIOT. )JASI~ 9TOrG
HnpeaBpHTeJnbHO aoKa)Kem cne.a1OmyIVO Teopemy:
XXV. EC.TH a HpHHaalJe)KH-T K nIouaweJIio fl, TO K TOMY )Ke no~a~aa
TeJHo 6yaeT nH~HaaneJ1CKaTb 'IHCJIO a, rte k - 'IIcIo, I33aHMHO-I1POCTQe
c n. 3)THMH Tp (n') 'IHcJaMH HC'leyni~BaloTCII Bce HHcjia, np~aewu
K HoKa33aTeJ10 fl..UoKaaaTe a EcCT O. 1 0. )i1onycTHiM, 'TO q14CJ1o a', rane (k, n) = 1
IIpHHat1LJI)KHT He K noK3aTeJIIo nl. TaK KaK OHO TO1Ke yXOBJ1eTI3opflT cpaBHeHH10
(4.3) X' - 1 - (mod p),
TO B0HPOC 3aKA10'IaeTC5B TOJIbKO B TONI, He y;XOBJeTBOp5ieT JnH 0110 TaKwe
cpaBHeHI4t0
XN - 1 =0 (mod p),,
rae n1 < n. aLon~yCTHB, 'ITO 9TO, HMdeT meCTO, Mb!!1011yqae3m

(4.4)

(4.4) ~~~~akni -1 -=O (mod p).