Osnovy teorii Galua.

~ 2. Fpynna Faiya cpaonen-uil 6

167

T e op ema 90. Epynria FanTya cpaBHeHfI4 (2.9) eCTb JaeJHTeJnb
rpynnbi ranTya ypaBHeHHIHB (2.8).
4. H1piM~miasi BO0 BHIiMaHI~e CTY~p rpynairi [an'ya cpaBHeHHHl (2.9)
(Teopema 89), mbi mo)KeM Bbipa3i4Tb Teopemy 90 caeapyoummi o6pa3om:
Teopema 91 (OteJeKIIH~a). Eciii jie~aa%4aCTb ypaBHeHH5I, paccmTHams no KaKOMy-HH6VhIb np~OCTOMY MOJAYJIIO, He i3XOJLBLLemy
B J1HCKPHMHHaHT 3TOFO ypaBHeHHH, paauiaraeTC51 Ha HenpH4BOJUImbieMFIOWHITeJii4 CTeneHe~if nl, n2,..., n,,, To rpyrira ranya 9TOrO ypaBBHeHI
coJIaepKHf-nOJACTaHOBKY, COCTO51IL~jIO 113 [UHKJIOB TOHOR1JKOBfl1, n2 k
5. YpaBuieH]nsi AzejeHHfl ipyra. B Ba~re np~mepa paCCMOTpHM ypaBHeH1Ie aeneHHR Kpyra:.1 ~~~ ~~Xm= 0.
Mu! y)Ke BHaenjH (nMaaa III, ~ 2.9), 'ITO rpynna ranTya 3TOFO ypaBHeHHsi eCTb
a6eneBa rpynna nop51Rn~a p(in). BEMICHHiM, KaKOBbI Aomnicrw 6bl~w HpOCTb1e
moap~ia p., Jat3IOH4IP TO HuIH NHFoe pa3JIo)KeHHi~ HoJIH4HoMa Xm Ha HenpHBOXLHmbie Ho MOLLJY11O p MHOIKHTeuIH. B a6eneBoR rpynnre LLHKjieHHbIII COMTB HUO~CTaHOBKHI BHIOhHe onpeaeigAeTCR ee nOPSJAKOM (TaK KaK B CHiJqy Teopembw 12
ee HIOACTaHOBKH COCT05HT 113 IAHKJIOB paBHbix flOp51JXKOB). HOWoMy Hiawa
3aaiaqa npHBOILHTCH K HaxoH(iKi~HHIO rHOPRiKa nOJACTaHOBKH, o6panyfouwe
rpynny Faarya cpaBHeHHR
(2.12) X.7H 0 (modp).
MUz BHJaeJH, 'ITO B KaqeCTBe TaKOfl nOaCTaHOBKl4 MO)KHO B35ITb HOJACTaHOBKY, rnepemaoim~nyio KawKJbIfl K~opeHb aL CpaBHeHH151 (2.12) B KopeHb a"1.
TaKHM o6pa,3om HOPfIJOK 9TOft -HO,4CTaHOBKlH pa~eH HaI4meHbweImy HOKa3aTe~ino f, ~amoulemy P_-a(o
(2. 13) a1 amod)
BeuwrnqtHa cx yaLoBorieTBop~er ypaBHeHH[O X' - 1 =0; HO3)TOmy ecrn f eCmb
TaKOiI nioKa3aTeJlb, LLJH KOTOPOF HmeeT meCTO mm I (mod in), TO
TaK WIO cpaBHeHHe (3.13) yawBiteTBop-qeTCfI.
C apyroft CTOPOHbi, ecHii CL eCmb KOpeHb ypaBHeHHlil (2.1 1), TO AIpyrHmI4
ero KOPHOMH RBJLIOTC11 BeJHIH4HIbl a0, rae c npo6eraeT Bice 3HaqIeflH4, He npeBbwmatolL~e Mn H B3aHIMHO-npocTbie c in. ECJIH p'r * 1 (mod in), TO p/, 6yalytii
B3aHMH0-HpOCTbIM C in [mbi a1onwIuKb HpeijLHoh1OKHTb (p, M) =1,cpaBHVNIO
C OJLHHM 43 5TI4X 4HceA c. [lyCTb Pt- c (modrn). Tormta
OTKy~a BHJLHO, tITO f-H CTeneCHb HaLueft!1OJCTaHOBKti nepeBAorT Kopet1b CL
B aipyroil KOpenb cpTiBHeHM9 (2.12), T. e. OT.TIHtIHa OT ToKwteCTBeHHORt HOUci'aHOBKH.1HTaK:
T eop e ma 92. EcAHi f eCTb, HaHimeHbwtiHA nOKaaaTeJ~b, )iamou[Wti
TrO nOJIHHOM X,,, pa~auaraeTCi Ho MOJLYJ1IO p Ha HeHpHBOJ1HMbie MHO)KH4
TentK CTeHeHH f.