Osnovy teorii Galua.

~ 4. rpynna riaflya

HOJIHHOma 1(x). C xApyrofl CTOPOHb[, fl(Xl) = 0 camo nio ce6e qIBJsiei-c5 COOTHowIeHHem, KOTOpoe HapyiuaeTC51 OT Bcex HIOJCTaHOBoK, nepeBORioUMa~ X I
B KopeHib 1(x)' 0, He SIBA1O1Uti4CH Kopreie f,(x) = 0. TaKHM o6pa3oM BCgKasI HOJCTaHOBxa rpynrnbi raiya. mowKT nepe~ottai4Tb K~aabKitbl KCO~eHb B KOPHI4
Tor'o )K Hen1pHBOA1Hmoro ypaBHeHHRl. EcjriH- B 4aCTHOCTH ypaBrneH~e(41
-HeFipIIBOJIMO, TO COOTHouweHHS1, B KOTopbie BXOJUMT O~tmH KopeHb, He HaKnlaJAbIBaIOT HHKaKoro ora~iHf Ha ero rpynny ranya. Ecaii x~e ypaBHetHHe
(4.1) npHiBOJIJMO, TO cpa3y BI'IJHO, EITO ero rpynna raniya HiHTpaH3HTHBHa.
TaKrIm o6pa3oM.neBasi 'IaCT~b BCIKoro COOTHOueHflCH (D (XI) 0, COaep)Kau~ero TOJ~bKO OJLHH Kope~im xI, AiOJnIWIa JaejiHTbCH Ha HeHpHB0QLWMbltI nO0114 —
HOM f1(X), KOTOPOMY yaBe~ose X1. ByJaeM 3ari~cbIBaTb 3T0 TaK:
(D)(x)==0 (mod fj/x)),
rite X Teflepb Haaio C'iIdTaTb nepeMCIIHOI- BeJ1H4qHHOII. [IOJIH4HOM f1(x) 6yiteeM
Ha3bIBaTbHCepBbIM OCHOBHMlM moLayneMi.
4. PaCCMOTpH~M Teriepb COOTHoweCiJII T ri (4.5), B KOTOpble BXOR$IT ItBa.
IKOPH5I., HaIpImmep X1 H X2. Pa-3JowKi~ii JeB103-ia)'-IBCb ypaBHeH~4I (4.1) Ha HeHpHBOJaffible MHO)KHTeCJiI B rionie K(x1) (T. e. B HoJIe paIuIOHaJbHbIX (~)YHKLU1U
OT XI) vi 0603nIaqHM TOT Heri~pHBOJJHMbIfl FIOJIHHO0M, ]KOTOPOIMl YA~iTBPe X2,
'ICO3 h (x1, X). HJycmr (D(x,, x2) 06,tTK C-,~~ COOTHoHIeHHH
mew1y KOPH9MH X1 i x2. PaccmOTpRM HOJIH4HOA 1(DXl, x), KOTOpblfl Mbi 6yaiemi
C4I4TaTb nOJIHHO0MOM OT HepemeHRioft X C KO3a*0uJiweHTamHi H3 rio.,ii K(xl).
Touta B cijay TCeop~m, 4A rinoitivomi (D(x,, x) ThoJIKeH JetCilII~bCR tia _,(jf x):.
(1i(xl, x):=- 0 (mod f2(x1, x)
O6paT~io, ecaiw ~'I"( _X 0 (mod f0 Xx)), TO ),V(Xl, X2) ==0,Ta K
f&x, X2) =0.
flyCTb CTer~ie-lb [IOJUIHoma f1_(x) eCTb Ill H CTerHeHb flO.IHOMa f2(X1, x-)
eCTb n2 B~yJCm CqrJHTaTb, 'ITO B fQ~, x)BJHHH 1BO~T B( OIHM
B ~ 3.8 MbI BHu.e J14, '-ITO aTO B03MO)KHO. CqfmTa5I ~ H ~2 riepemeHHbIMH, pa3 -iie.TIHm ~)~ Ha fn~~ ) o riepemeHHOfl 9 TO MO)KHO BbinOJHHTb, He
BBOARf JlpoftbiX (~YHKLIHA4 OT '~, TaK( KaK Ko34WRLteHTOM n pli ~22 y oal
jioma f?.(', ~2) CJ1M)KT eziciHIua. Torta mb! noJ~yMilm:
rite tIOJIHHOM r(l.1, ~2 mmeT CTeHeHb He BbIHIC11 - 1 OTHOC14TeJIbHO ~2 Ecqii
HO~aCTaBHM cioTUa - XI, TO ypaBBIeHHe r(x1, ~2) = 0 HmeeT KOPHR5MH Bce
iKopHf ypaBHeHH51 fjX1, W ) 0, a rioTomy, 6y~aytiH 6oJnee HI3KO1 CTeneHH,
To>fKtaeCTBeHHO o6paumaeTCH B HyJlb (T. e. o~paawaloTCq B HY.Jlb Bce ero Ko9(p4ni4uj-eviTb fipvi CTeneHOX f2. 1o3ToMy Bce Ko3C(4H~i1HeCHbl HPH CTetieHfIX
noJIHHfoma r(',, ~2), iime5 C HnpHiBOaHlimm noJmi-4ioM.om fj(~,) o6utvie KOPHH4,
JLeji5ITC~I Ha Hero B CtJIy Teopembi 44, 14 TaiKHm O6pa3OM IIOJIBom r(~,, ~2)
mo)ReT 6bITb npeJaCTaBBj1H B 4)opme fi(~,). q1( W, ~) rite qj(~.I, ~2) HeKOTOPMR,
nO.THHOM, TaI( HTO 4POpMYJI13 (4.6) npiio6peTaeT ButA
riteql~, ~) ~'qj~, ~) -HeKOTOpbie 1IlOJiMHOMbl OT "j