Osnovy teorii Galua.

~ 4. Hepwoo5pa3nbie Kopt1u20

207

PewHasi i uenriix qwcniax ypaB~eHue
kut +nv- 1,
__~ — a'" (hu+ntv) — (akn,,) (a')' (m od p).
Ho B cHJ1y (4.3) Hi (4.4) o6a BbitPa>KeHHq B cKo6Kax cpaBHHiMbI c ejAHHHLLefl H MNj [piixoJLHm Kc cpBHeHHIO
a'nLz (mod p)
RPH n1 < n, KoTOpOe nfl~oTHBO,4L4T Haawemy ycAjiofiioO0THOCHTeJlbHo a.
20. HYCTb Tenepb k He B33atiHMo-po~cT Cn, T. e. n1yCTb (k, n) = d > 1.
Tor~aa ak YJOB;1eTBOPHeT CpaBHeHHIO
n
x d _ 1 = (mod p).,LTefICTBptreJlbAo, n1yCTb k -= k1 d. Totaa
(a ) d -a d Z(an)kl- (m od p).
CpaBHeHue x" -1I -O (mod p) He mo)KeT HimTb 6oaTee it Kopnett
(cm. XXIV). Hcna
1,a, a2,.. an'-1
BCe HecpaBHHMbl mewaKyL co6ofl (Iffawe a nplffaxviewaao 6w Kc 6o~nee HM3KOMY
TlOKa3aTeJ io) Hi IOTOMY HcqepniimaiOT co6oft Bce K0PHH cpaBHeHHsi (4.3).
CaetaoBaTellbHo, BC11KO tuicJIo, He cpaBHHmoe CO CTerieHbIO tiHcjia a 110,
MO0RJIIJO p, He m o)1(T 6bl~b KOPHem cpaBHeHHHg (4.3) H In0TOMY He mo)KeT
npliHa3I1Je>KaTb K rio~a~a~eato nt, ti. H T. a.i
TaKUiM o6pa3oM Mbl BJIMM, WIO ecJIH cymeLCTByeT XOTSI 6N OJLHH Bw'ieTI,
flpHHaajLie)KaiHIIt K noixa~anen o nt, TO HX 4HcJIo paBHO p (i). O 603Ha'Ia%
tiepe3 ~ (it) 4Hclio, nH~Haa1jie)K4Hx K loKa3aTeJ1io it Bbl'ieTOB, mNi 6y~em
HMeTb:
(4.5) (it) == p (n) Ha H ~ (it) == 0.
C aLpyroft CTOPOHbl, KawKJUifl H3 p -i KopHefl 1, 2, 3,.., p- cpaBx 1 i 0 (mod p)
flpHnaajie>KHT K KaKOMy-HHt6yJb floKaaTeJHO, H BJ1HoLemycR AJI~eHTeJqeM.
'IHtcJa p - 1. HIO9ToMy, o6o3HatIHB 'lepe3
Bace a1eJTHea'H 'Hcjia p —iIbi 6yem HimeTb
BmeCTe C TerM B XVIII 6bIno )xOKa3aHo, '4TO