Osnovy teorii Galua.

112

Teopwq queeA

2. OCTaBHM ce6e 3aaiaqeg HA~TH camble o6utie BbipaBweHHHMIAR5 peweHHR~
ypailHeHHnSi (5.1) TaKoro porta, 4qT0, nOXICTaBJqiH BmeCTO napamlerpoB temaie
3Ha4CHHg, MbN HCteprnaeM Bce peweueir~ ypaBHeHHq (5.1) B3 rtenbix 4qHcjiax.
ECJIrn ixebiwe q~cna x, y, Z, yA0BJieTB0p5HalUHe ypaBHeHHIO (5. 1), coitepaaT
06iitHI1 MHO)KHTe.Ib, TO, CO~paumaf Ha Hero Bce TPI4 4Hcaa, mbi onflTb H0J1It1HM
iteinbe 'q~cna, ytomme~opfloaLHe ypaBHeHH1O1 (5.1). IlQBToMy MU! momKe
cpa~y npeJ~noJaoWHTb, 'W O o6tJ4Ht MHO)KHTemb COKpauJleH, TaK 'ITO XI )', Z
He HmeIOT o6ugtix MHO}KHTeJnel (X, y, Z)!- 1.
L(OKacem., WIO OAtHO H3 4HceJI x, y XLOJDKHo 6bl~m 4eTHbim. B camom
JueJae, HeqeTnoe 'IHcJo MO)KHO Bceraa HnpXtCTaBHTb B B14im 4k 4- 1, a tIOTOMY
ero KBaIIpaT
(4k ~ 1)2 - 16k2+~ 8k + 1 — 1 (mod 8).
ECJIH 6bi X H y 6bu114 He'IeTHbIMH uHCJamff, TO X2 + y2- AOJDHO 6b1 6bJTb
cpaBHHMO C 2 rno mo~ynhIo 8, T. e. JAeJ1HTbC51 TO4HO Ha nepByio CTeIneHb. 2,
4qT0 HeBo3Mfo)KHO, TaK KaK X2 + y9. -= Z2.
flyCmb x 6yiteT 'IeTHbIM 'imcJom. Torma y j Z xAOMIaHbi 6blTb He'IeTHbWMH
tIHc~amH. flepeflHuuem ypaB~eHnie (5.1) TMr
(5.3) X2 = (z + y) (z -y).
'-IHCna Z + Y H z -y (o6a tieTHb1e 'HcJaa) moryT co~tep}KaTb. B K~a4eCTBe
o6luero aeatJITeJI% TOJ~bKo 2, TaK t-a~K B np~OTHBHOM caiygae qHc~aa
x= V(z~y) (z -y), y=-2 (z )I4(-yz
Toae coarepxcarn 6m o6umt~t MHO)*CHeJb, 'iero Mbl He rnpejtrollaraem. '-IHCJa
+Y H B3aHMHo-IlpOCTbl 14 Aaf01T B npOH3Be~teH14H TO'qHIMIl KBaJapaT,
~~1F ~ 2
a nlOTOMY KaKabtb~l ripocTOf MHOB(HTehb Kawaioro H3 9TH4X 'IIceJ a[oJ1)]KeH
z +y z-y
BXOAiHTb B MeTHOtR CTenleHi. 3T 0O3Ha'-aeT, 'iO -- - H HBJI]IOTC51
2 2
TO4qHbIMH KBaJ~paTamHi:
2 ' 2
OTKYJ~a
y11- 2 - V, Z11 2 + V2.
no~itCT3amaIII B (5.3), 6y~tem HMCTb:
x 2uv.
WuTa, HCKOMbIO naaw~~icm Bb1pa)KeHfI5 TaKoBbl:
(5.4) x =2uv, y =U2 - v2~ - U2 +v2.
ECAH we mbi OT6poCHM TpefoBanti, 'ITO6bl HmeJIo meCTO (X, Y, Z) =1
TO napamteTpH'leCKfe BbIlpa(eHH51 IUIH petueHHH ypaBHeHHq (5.1) nlpeJCTaBISTCH
TaxK:

(5.5)

(5.5) ~x =2uvw, y =(U2- V2)W, Z- (U2 +V2) W.