Osnovy teorii Galua.

78

78 Ii~~~~1. rpynna r'a.ya

meTpHLqeCKof rpyrnnbi 441t CTenenHH, mbJ y6eaLHMC$I, 'ITO 113 HH1X He HapyinaioT
COOTHowIeHHAi (4.4) ciieaiytoiwe noRLCTaHoBKH:
1, (X1, x2, x4, X3), (X1, XA) (X2, X-3), (XI, X3, X4, X2).
3mH FJOJCTaHOBKHI o6pa3yloT 1LHKJIH4ecKyI rpynrny.
2. HQICTaBiM BOIIPOC B o6wmeU MaIe. HyCTb mewKJy KOPH1IMH ypamBHMHA1
(4.1) HimelOT meCTO C00THowHIHI4
(4.5) 'Pi(XV, x2,..., xj )=O (i=~ 1, 2,..)
Boo6w~e r~OBOPI1 B 6eCKoHe'IHoM qImcne, HI I1CTb COOTHoweIMII~ MH (4.5) HicuiepbIalbl~OTC5I Bce BO3MOIKHbie COOTHOLUeHH5I MeWKJL 9THMH KOPHRIMH. OTHOCI-.
TeJnbHO xa)KJLOt 11OJCTaHOBKH cHmmeTpI44eCK~fl rpynnbi mewKxy~ x1 x2,,x4
Bceruaa MO)KHO CKa33Tb (uio i~patineft mepe TeopeTHiqeCKH), HiapywaeT All4 oiia
XOTb 01H0 113 STIHX COOTHOIIeHlifl IIJW coxpaHlieT Bce HX B cHJIe (T. e. ne-,peBOAtHT Kawajoe COOTHomIeHlHe (4.5) B Kamoe-HH6y~lb 113 3THX >ice COOTHOT e op e m 50. COBOlyrnHocTb Bcex no0,aCTaHoBoK, He Hapywalonuix
BceX BO3MO)KHbIX COOTHOiueHlifl me)Kai ICOPHRIMI anr-e6pawiecKOro ypaBMimi~, ot~payeT upynriry.,doKa3aTeJflbCTBO. JIlyCTb S ii T 6yxAyT rHOACTal1BoBK, tie Hapywlato11Ie HI! OLHOrO 113 COOTHOIC11.tii (4.5). Ecaii mbi nH~ImeHHm nOJLCTaHOBKY S
K KaKoMy-HHi6y~lb 13 COOTHomefflift (4.5), HanHHmep Kc pj ('xl, x2,..., xJ) = 0,
TO ATOJDIKHbi nolO;y'-IHTb itpyroe COOTHouieHlHe, Jaei4CTBHTeJlbHO mmieioumee
meCTO H [1OTOMY BCTpeqIaioLueecq cpejaH C0oHl-IIweHfI4 (4.5), _cica.)em
Tp1 (Xl, x2,.I.., X?) =: 0. Ecrna Kc 3OMY COOTHo'ieHHimo ~pIImeHHTb noJic-raHOBKy T, TO COOTHoweliHe Hepeftlie OfIRTb B OJIMO 113 COOTHouweHitt (4.5).,
c tmKeM B CPIk (XII x2,, x)=0. Ho nojeOa~~bo nptHmetHiHH K COOTHoiieHH10i (D (XI, X2,. Xj) = 0 CHaqIaaa nOJLCTaHOBKH S, a 3aaem T, paBHO-.
CIHJIbHO Hni4meflHHI{) cpa3y nOJICTaHOBKH S.- T. TaKMm o6pa3oM BbIXOLLHT, 'ITO
pH~1menefHli I COOTHO[eHIH451M (4.5) HOJaCTaiHOBKH S.- T HepeBO)U1T HX B CytieCTBY1OLLLIe COOTHowIMeHH, TaK{ 'ITO HPOh13BaeL!HH S. T TOwKe BXOJaNT B HaI1I'
COBOKyrIHOCTb.
He TPYXIHO TaI(me Hrioama, 'ITO BmeCTe C S BXOJ1HT B COBOK~nHOCTb H
o6paTiiasi HlOACTaHOB3Ka S-1. BCHaii nOJCTaHoBica HimeT KOHe'IHbltI nIopo1JOc,
T1. e. cyateCTeyeT TaKoe m, tITO S' 1. OTCioaa S- = S" -. Ho nouo)RHTeJlbHafl CTeHeHb npHaaJJJew*au~ef K COBOKynHOCTHi n1OJCTaHOBKH TO>Ke
B CHhIY XOKa3aHHoro nHPHHaanewMKT K COBOKynHIOCTH, '1. H T. 11.
rpynna HOItCTaHoBOK, He HapytwaioiiiiX BceX BO3MO)RHbIX COOTHow~eHifl'
Mewty KOBHRMH ypaBHeHHH f(X)= 0, HOCHT Ha3BaHHe r p y 1 Hu riFa ni y a
11A14 nIPOCTo r' p y H H hi 9T01`O ypaBHeHlHff.
3. Ociioimuime MOalYJIH. Tenepb pa36epeM BOHPOC 0 c~aKT11'eCKOM HumO)-.leHHH r'pyHnhi ranya ypaBB~eBH11f (4.1). 143 Bcex COOTHO.Ineli1ik (4.5) paccMOTPH4M Cnaqajia COOTHOuiHeH$I, B KOTOpb1e BXOLHT TOJIbKO OILH H 3 KOpI-efl
Hawiero ypaBHeHH5l, HaHip1mep Xi. HlyCTb (I) (x,) 0 6yJaeT OJLHO 113 TaKIIX
COOTHOUweHHn4 14 f1(x) = 0 6yxaeT HenpHiBOXLHmoe ypawi3eHHe, KOTOPOMY YalO..
BJ1eTBOppeT xl. Torjaa tF(x) =.~- 0 H1 fL(x) = 0 HmeIOT o6wu-e KOpHH4, a nOTOMY
Ha OCHOBaHHH TeopembI 44 L1(x) JAOJDHO JaeJ1HTbC39 Ha f1(x). TaHmm o6pa3oM
(D (x) =:f0 HmeeT meCTO AJI2 icex KOpHeft HeHpHBOtiHmoro MHO>K1TeJ1I f1(x)