Osnovy teorii Galua.

~4. rpynnia raAya 8

181

OT Xi H Xr.. LtCe1CTBHTenJbHo, niepeBoxwI B COOTHOIIeHHH (4.9') XI. B Xi
B X. MN~~~~~~~~~~~~~~~~~4X
f(3) = f (Xxi, Xj) (3 Xi) (03 x,) F (Xx8, X,)
3Aecb, jie~aq, a 10TOMY H nipaBasi, tjaCm HmeeT KOPH51MH Xi, x2,.. x2,,n
IIpHqem xi H x. - 1CopHH MHO)KIlTeJIetl - Xi H -X1 n~paBOR iaCTH1.
6. O~~~oa~~atA~~I H~~~C3 f~JX1, 2.. Xki ' ~ HeflpHBOaJT.MblI~i B fIOJI E7,
K(Xl, X2,..-, Xk - ) JelJIHTeJb IIOJIHHOma f( k), HmeIOILm KOPHem Xk= 14
nU1)oILojmaqllpaccy~ixtae~ne ARw 3Ha'IeHHh h 4, 51... ~lmb y6eamuMCfl B i~onie
f(OHILOB, 'ITO A,1 Ball qaCTL) mio6oro COOTHoiIueHlA~ (IVX1, X21, Xv)=0 mo)KeT
65bITb npeaC~aauie~a i cJneapllylnef c3opme:
(4. 10) j + f2('E, 2) 2("l P, J +. +
r~ae E2,...,) (in 1, 2,..., in) - 1OJilHBO~fbI C palLHOHaJlbHbJMH K0o34 -4WflLiieHTaMH. TaHmm o6paaM KawK2asi JaaH1-asi HOJCTaHOBKa memKJLy KOPH-IfMH
XI) X2)..., X. coxpal5IfeT -B c~nie BceBO3MO}K~bie COOTHOweJHH11 mew{JL
xj, x2,...5, X., TOr.LL H TOJlbKO Toraa, ecir oHa coxpaH11eT B CHJIe COOTh
(4.11) 11~~f(Xl) = 0, f2(x1, X2) = 0,
f3(x1, X2, x.3) 0.., fn(x1v, x2,..,Xl, 0.
JReBbie 'IaCTH COOT[iomneHflit (4.11) HOCAT Ha3BaBHH-0 OCH 0B HhI X M 0JayTi, e 9 ypaBl-eHHll (4.1 1). OHH1 xapaKrepH~yior rpynnry Iranya B TOM cmbIcJe,
qTO KawKJIal nIoJCTaHOBRa npmuaaJ.7emKT K rpyrine ['d.iya ToiUa H TOJILKO
Trouta, ecIIH OHa coxpaHll-T B cHJIC BCe COOTHoiuIeHH~1(.1)
1{pome Toro, npolioumal npeuiwaryiume paccywKJeBHHI, MbI y6extHmcs, 4To
B rpylnae raiiya 6yaceT cyuie~CTBOBaTb nOJICTaHOBKa, ne~peBoAiuiasai X, B mooflg
KO 11H x eAr Ol~l 1~) IeflCTBHTeJqbHO, n1OXCTaBJISIFI B f2(-11 ~2)
XtMbI HOJ;IyqHm nOJHHOM, HmoMIflHfl KOPHllMH KaK~e-TO H3 B3ejl14HH
X1, X2,, iX2,, OTJ1H 'biBe oT* x,. Bb6Hpasi HI0OH3BOJnbHO mI6pIo 143 liiHX, HaI1PHMep X 2, H1 HO0aCTaBl1Rff B TpeTHfl AIOaJl~b ~j = Xal %2= Xa',,MbIHOJIY4IHM
UOA'HHOM; HmemuOifH1 tEopHSmm tKaRe-TO 143 BeJ1HqH1Hi X1,x,. OTnlH4 -Hlbie OT X 2 H x.2 [ipOoaoaaall aaJIorIHLIblfl BbI6op, mhZ npitaem K< nIOJCTaHOBKe Xi BV X e Hapywmaul~efl COOTHOUweBHH (4.1 1) H nOTOMY
npiumanewatue K< rpyrrne Fa~nya ypaBHeH14ll (4.1). Floaoito~ne >Ke 3aK.TiroqeH11e mhZ mo)em cateJlaTb, ecdiH OCTaBHM I1eCKOJ~bKO flepBbIX KopHeft, HaflpH-~
mep x1, X2,.-,Xk-l, HeH3meHHb]MH H Ha'I~em ala.jlorlquoe paccy~NgCeie
c k-ro moJayJag.
7. OCHOB~nbie CBOiACTia rpynnub ravaiya
T eo pe m a 5 1. JJ15a Toro, qTO6hI rpynria ypaBBeHH514 (4.1) 6wina
TpaH3HTI4BHa, Heo6xoJLHmO H aTOCTaiOlIHO, tITObOW ypaBneH~e 6wino
H~enpHBOoLIHMO.
t3'le~OTapmB