Osnovy teorii Galua.

~3. Pa3pewu biu4e -epynnbi u ypaenenui72

121

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KOPHH - ypBHeHH$1 un -a. = 0, ec.ini C'qHTaTb KOPHH I1epBb1X S - 1 ypaBHeHHAf (3.4) npiP4coeJHHe~HHbIMH K o6AaCTHI pauIoHaJ~bHoCTH. rpynna ypaBHe-~
HHI It - a 0 B 3TOf1 o6JIaCTII paHlHOHaJlbHOCTR p'a~peW~ma (cm. ~ 1. 7).
[rpyrina ypaBF~eH1H$I, KOTOPOMY 3'LOBJIeTBOpHieT C, 51BJ~lleTCf ee 1o riOJIHHTeJILHofl rpyinoft 14 IIO9TOMy TO1Ke pa3pewn1ma. Ho, c apyroil CTOPOHbl, 9Ta
rpyfrina B CHJIY Teopemb1 62 H,30M0P(1)Ha c rpyrnnoil ( rS a ue A~ -nepece'ie-~
Hi4e Bcex rpynin, corljp41KeHHb1X C T aKHM o6pa3oM Mbl BH4JLIM, 'ITO
H4meeT OTJ1H4qHblfl OT ce6% HOp.Maj1bHbi4 xaezrnTe~lb T3aKoro poxaa,
WIO J1OHOJH11TeJbHafI rpynria (5i/S9, pa3peUJlHma.
HlpHcoexulHRHM K o6.iaCm paiLHOHaJ~bHOCTHI BeJI441Hy, nHPHaaneI~KaLLyi,
K rpynne q,, a 3aTem CTaHem OI1~ITb 11OCTeHeHHo l1pHCoulfU4H1b K o6JuaCTH4
pa1iJoHaJnbHOCTH KOPIIH ypaB11eHHA~ (3.4). HlpI4 HOBom nIOHii)KeHHH rpynimi7,
KoTopoe HeripemeHHO A0J1)KHO nIP0430RTH, mu, nOBTOp5WB upebiayiyutee paccy>Kaelifle, y6eummcsi, 'ITO rpyrira Al H4meT 0TJ1W1qHbIPI OT ce6g HopmaJlbHblll J1eJ1TeJlb,
N42 TAKOF(0 poJaa, 'It-O LOHIOIIHHTeJnbHaI rpyrina R2 pa3peulHma. fHpoJao.~uiaq
flpoitecc, Mu npHJaem K p~iiiy iioarpynnr
(3.5) W) RD r-1 1
Taicoro, po)~a, 'ITO KawKan5 rpynna Ri 51 BJlf1TC~1 HOpmanb1H1]M aernlenem,1~
npeabi~ynieft rpynnI Ai 14 AOnOnffiffeTj.1Hasi rpynna Ipa3peuilma. BCT9 -BH4M B psuA (3 5) meWKAY Ka>bitmMH JIBymsi rpynflamf S, pgu Taimx
rpyIIH, '-TO6bI J1OfilOJ1HHTeJlbHbie rpyrinbi, COCTa1BAiHHb1e 143 coceJ1HHx rpynn,
6MuiH LLHjcJIjqecit(14m1 rpy[1[IMH rnpOCToI'o n~p51JaKa (3T~o BO3MOKHO B C14J1y
pup3flUHMOCTti rpynn RS~jS~). Toraa p~UL (3.5) HpeBpaTHTC51 K KOMHO3I14HOHHblfl p%) C P~LTOM HHJLeKCOB 143 nlPOCTbIX qiicea'. Cn1e~ioeaTen1bI1o rpynna ($i
pa3peIuli4 ma, 4. 14 T. R.1.
[1 p1i4 m e ' a 1114e. Ha camom a~ejie rpyrina ip BJI14eTCHl h;opmaJlbHb1M aeJIHT~enem rpyninbl 03 H HOTOMY COnriaaiaeT C ~J, HO MbI He nOffb3OBJJIIHCb 9TIIM
4~aKTOM B BHJLY CIO)KHOCTHi er'o YCTaHOBJaeHI45I. BO3HHKaeT BOflpOC, MOXKCT.111 cjnyqHiTbC$1i, KaK MM 9T0 npeanHoJIo}KHJI, 'ITO rpyrrna ypaBHeHnH'I (3.1) Hie
[IOHII>KaeTCH rip" nHHcoeJLHHeHRR K OGJnaCT14 paUKOHaJ~bHOCTH KOpfiefl He~o-~
TOpbMX 143 ypBeH (3.4). Mbi JLonycTiH1Hx 1'XcyHueCTBOBaffHi onflTb-TaKI4,LIunq ynpotiieHHH JROKa3aTe.TlbCTBa; Ha camo OM ke jieue TaKH4MH ypaBHeHHHfM14.
mor~yT 6bITb ypaBHeHHHl, KOTO~bIM yJIOBJieT8t3OjMHOT KOPHH14 3 eJAjHHL~bl, 1(0 -TO~bMe Ham OAtHaKO HeofxOJIHmo npHcoeaHHHTb.
GCnea CTBi4e. YpaBHeHfle, CTene~Hb KOTOPOFO > 5 H rpynna Iranya
KOTOPOI'O eCTh 14.11 c14mmeTpH'iecKasi 1414H 3HaKonepemeHH514 rpynnia, He
mo)KeT pemua~bC5 B paziiwauax (CM. Teopembi 27 14 28).
2. Paapeulmmbie ypaBHeIHfl HpOCTOfl CTeneHH. HIYCTb
(3.6) f x)=- 0
HienpHBOJ1Hmoe ypaBue~ie P-09l C~erneHi, rie pf - npoC~oe 'IHcjio. Mu
TOjlbKO-4T-10 y6ext4JIHmb, 'IO XTtJI pa3pel11HMOCTH ypBe~i (3.6) B paJLHKaqaX neofxoiiu-io Hi AtCTaTO'rHO, t4To6b ero rpynna 03 6bina pa3pei1mma.