Osnovy teorii Galua.

I V. l Rpumuxicetiu meopuu raAya

H5IMH OALHOrO H Tor~o we ieHpHBOAlHMor'o MHO)KHTe.TLH, Ha I(OTOpbIe pa3o6beTC51
ITOJHHOM (x) iiocae flpHcoejHHeHHHR K o6nracTH paiJHOHaJnbhOCTll KOPHR Yj
ypaBHeHHH9 (1.3), HeofxoJaHmo H ZLOCTaTO4HO, 'taobbi nIOJCTaHOBKlH cofnpgfKeHHbWX CHCTeM OliH (831S S O~p nepeBOARTi X1, H<apaBX iX eat
BO)JH1O H TO~w Ke COBOKy nHOCTH aS1~~2 ~O tenle, '-TOb Bo6rcT
pa[LHOHaJlbHOCTH K~y) BejiH'IHHbI H, 4X3. yaL0BnleTB0pflJlH OaHOMY Hi TOmy W~e
HeTIIHBOA~HMOMY ypaBHeHHifo, Heo6xOJLHMO H4 JOCTaTo'IHo, tiTo6bI I1oJaCTaHOBKa
SJ-'S (a TKime AIto6a51 n0.LCTaHOBKa COBOK~I1HOCTHI S, 1W,31S.), neexms
XiB X., JneAwana B rpynne Fanrya nowe npu HOBOA1 o6JiacTAH paiwoHa~biiOCTHI.
HO B CHJIy Teopembi 61 9Ta rpyrnna- ranya eCTb 02. Hlo3TOmy nOj1CTaHOBKH
COBOKyfIHOCTH Si 13I8S AJDOAKHbI.fle)KaTb B (2
OTKyJ~a. cjjel~yeT, '-IO S aJojimaH.iie>aTb B COBOKYHHOCTrH (5lSVM2:
TaKHM o6pa,3oM 1IHCJ1O KOpHefl xj OJUoroF H TOFO wKe HenpnmaiiO~moro
B o6nlacTH K(y1) noJIH~oma, KOPHem KOTOPOFO 2BJI5IeTC5 XP, paBHO 'IHCJ1Y
pa3J1H'IHbIX conIpH}KHIMbIX CHiCTem (i$S1, BX0OJiMIHX B COBOK~HHOCTb (WlSA~2.
B ri. 5 ME! BHaJeJIH, 'IO 9TW qIHcjio pBHO nOP51,LKY rpyrnnbi (~2 ejLieHHOMY
Ha flopmUK nepeceqeisi~ rpri 0. H S7US51Si. O6o3Ha'Ias 3TO iiepece'4eIffle qepe3 St, H BBOaq~ ajn% nHopIjKa rpyunni ~, o~o3atietlime (,Mbf BHJiAHM,
'ITO CTeneHH HeHpHBOaHMbix rloJHHOMOB, Ha KOTOPLb!e pa~jiarae-ica 1(X) B o6 -2 'IHCJIO HX paBuO q-HcJay e COBoKyrIHOCTefl B pa3JO)KHHH (1.9).
Tenepb I1OCMOTP14M, Ha KaKHCle HojiHHOMbI pa3.uiaraeTC51 I1OJ1HOM A~Y).B o6AaCTH K(X). H1O~ib3yHlCb pa3ho~KHHem (1.9), pa3JIO>KiM (i 'aKc:
RIemo noKa3aTb, 'ITo ecJiH SJIemeHT S Co',aep)KHTCH B COBOKYHI11OCTHi (81 Si. - 2
TO gnemeHT S 1 cueLp)KHTCSI B COBOKY1HHOCTH1 -281 0~ 11o6paTHO. Ho0 -3,TOMY COBOKY[1HOCTH B pa3JIO)KeIHHH (1.10) coxtepwa TO -f w iRnc~no BJnemeH'TOB) 'ITO H COOTBeTCTBeliHbie coBOKRI1FOCTH B pa3JIO-KeHlHI (1.9). HO Mbl BHJa-eJlH, 'TO 101HFIMOM g(y) pa~iiaraeTCH B o6JnacT pa~ji4HOaJlbHOCTH h'(x1) Ha
HeflpHBOJLHMbIC MHO}KHTeJIH, CTerieHH KOTOPbIX paBHbl 4HcuIqam coInps1ieHHb1X
~~HCT~ 2 j BXOASfIIL4X B KI)K4YIO 113 COBOKyflHOCTef 6 ' (FS~~10 LoToMy
'IHCJIO!JOJ1yqaeMbIX T1IH 9TOM Henp~HBOJLHMbix rIOIIHHOMOb 6ymteT O1LHO II TO,)e, H flPH 9TOM CTene~HH cooTBeTCTBeHflbIX UOIHHOM0I3 6yalyT 0TH0CHTbCR1,apyr K APYPyj, KK OBK rpn 32 HI (8. HO TaK KaK CTenenHl m, nii
HOJIHHOMOB f(X) H g~y) paBHbl HHJaeKcam (W: W 1) H4 (W: 2 TO Mb!l 3aKJIIqam 'ITO n1OP5AKH HenpHB6,aHmbIX MHOWKHTeJneft OTHOCSITCH iipyr K AZpyry,
icaK m n. TaKmm o6pa3oM Teopema 73 CHOBa JROKa3aHa.
7. H p HmNe p. PaCCMOTPHM ypai~eHflR
(1.11) ~~~~~~~x'- 2=O,
(1.12) x9~3x8~+ 6+ X"+ 1 =O.