Osnovy teorii Galua.

176

176 ~~~V. Ypaimenun C 3adautibimu epyninamu

8. Hpo6Jaema paunOHanblioro MKHHHmaJbHoro 6a3Hca. 3ai~aqa noCTpoeHHRI ypaBHeHHfl C Hanepea 3aJ~aHHofl rpyrrnoil mo1KeT 6wm~ pewenaav~i
Trex rpynrx ($, KOTOpwje.aonyciiaioT TaI( Ha3b1IaaembIPI p a iL H 0 H a a b H W f
MdH H m a A b ~wbI 6 a 3HC. HOxR STHM fIOHITiiem pa~ymeeTrC5i caeatyioulle.
i-IyCTb HaM 3aaaHo nt + 1paaiLHOUJlbHbIX (~YHKLIHA;
(3.13) a1 Af1(xi, X2,..,XJ), U2 f2(XI) X2,.,xn)l..
OT fl He3aBHCHMblx nepemeHHbIX X., X2,.., x. Me>KAy Be1H4HHamH
li U2, U- +I aJ OJ1)HO h4MeTb meCTO dalre6paHqeCKoe COOTHO1uH~HH
(3.14) 'I' (ul, a2,...,un~ 0
3a naqa 3aK~iflOiaeTCS] B Haxow~aemm n a i3aBHCHMb!X 4!YHI(LHfl OT
'iqepe3 KOTopb1e mo)KHo 6bJu o 6b1 B CB01O o4epe~b paUllOHaJ~bHO Bbipa3wr'b
a1 = I (VI, V2,.,Vn), 112 = T2 (Vl, V2,... 'V),.
3rTy 3ajaaqy MO>KHo 6ojiee 9aeraHTHO Bb1a33i1Tb B TepMI4HaX TeoplHH
lonefl. JiJAf sR Toro paccMOTPHM nione K (x1, X2,...,I x,) paitRoHaibibwX
4~YHK1LH4I 0? f He33BNCHMbIX flepemeHHbX X1,X2,..., X.. flo~ne
K (111 U, 11g *..,P a + ) 51BI151TCsI ero aLeJHffeneM.. '3ajaqa Haxom<JLe11HH CHCTeMbi)
4~YHKLLHtI(3.15) paBHOCHnbHa BOHPOCY, H430MOP43HOAH no~ne K(a1, 112., a-4-1
c nIonem K(x1, X2,..,X~).
3-ra 3axtaqa 6bina nOCTaBJieHa ii pewei~a J110pOTOM (P. Lilroth) Janv c~y'iasi n - 1. KaCTenmHyoBo (G. Castelnuovo) penwim ee uiuig n == 2.,lJsi
n =3 3HpHKBec (F. Enriques) H-awean rlpHimep, Koriia 3a~aqa He H4meeT
9. HlpHmeHHM peweienie 9TOR 3aa~aqH (KoHeqHo, B C~nyqae, Korta OHO:CyaeCTayeT) K Haxo}K)aeHHIO ypaBHeHHlf C 3aJ~aHHOtI rpyniroWt Bo3bMeM
B Ka4eCTBe 4)yHKatLHt (3.13) CHCTemy BJnemeHTapHO-cHimmeTpHqeCKlX 4?YHKRHfl
'OT XI) X2,... of, a Tra~}e 4!YHKLkHIO (P, npHrnaxLiee~anXYIO K 3aJ1aHHOi#
rpynrie (0. Me)Kly HH4MH HmeeT meCTO COOTHOUIeHHe (3.7). H1peamo~wr~,
WIO 9114 4)yHKLLHH Hme10T paLtHOMablbit M4HHM~anbHbfi 6a311C VI, V2, O.., Ivn
Torx~a, HCXOAH 113 Bblpa)KeHHfl (3.11), Mb! cmo)KeM Bb1qlHCAHlTb 3HateHHn
Jv5 VIV2..., v,,n. HIOaCTaB~fi9q HX B CBOIO o4qepeJb B BbIjpaKeHH5(
A,,I A2,...,2 An' Cp, Mbl n!OAl~HM JARR HHX BMatiHisr, Boo6ate He coanaiamoaiHe c nepBOHaqanlbHbIMH, HO cpaBHHMb1e C HH4MH HO MO.yAIMO P. 9TH 3Ha4eHH5! YJOBAieTBOP51IOT COOTHOU1eHHo (3.7), a n1OTOMY 3aaatiay MO)KHO B 9TOM
~Cay~ae C4HTaTb peuieHHOiR.