Osnovy teorii Galua.

___ ___ ~ ~ 2. 3adaqa, o6pamnaR tpq a~~~~ 139
llpucoex4~Hiii K o6aaCTH p3LLIoHaJbHOCTH4 BeuleCTBeHHblH' K0~eHb 1/2 paBHfH4RR
(.1,mbi cmo>Kem pa13,tIONKITb ero Ha At~a MHOIKHTeJI'i
x9 + 3x6 ~ X3 + 1-(X3-1/ x+ 1) (X8+1/2 - +2x+1fx +1/x +1)
Moiuio y6ealTwbCfl, 'tTo o6a MHr~ioeimer HellptlBoJU~MbI B none K (12). TaKHM
o6pa3oM, ypaB~eH~e (1.12) pa3aaaraeTCq Ha He~lPHBO)jLHMbie mHo)KllTeJH CTefleHeHi 3 Hi 6.
C Jtpyrfll CTOPOHbl, HOJII4HOM (1.11) pa3aiaraeTCsi B o6iiaCTHpm o3ia0HJMH0CTH I((y),
3
JReftCTBH~nebHo, BeJIHqHHa 1f2_y IBXOJAHT B noJIe K(y). C jxpyroil CTOPOHbl
Y
rnoay~qeHHbie [IOJHHOMbI HenpHiBoHmmbI. M4X CTereHH 2 H 4 11Qponopuh4OHaJbHbl IHoJlyxIeHHbim npe>Kae CTeneH5Im 3, 6, Hi iix o 6atee OTHOmeICHHe paBHO OTHOLLieHio CTeneHeft ypaBHeHfflf (1.11) H (1.12).
~ 2. 3ax~aqa, O6paTHasi.3airqe '-IHpHray3eHa.
1. [yCTb JLaHb RLBa adlre6paH'IecKix nonsi. BO~imraeTr BOnpOC: KiK y3HaTb
CTeneib HIX pOALCTBa? PeLHHm 6o~nee 4aCTHbItt BOnpOC: Korjta JLBa anre6pai-i
q4eCKHx floJgg TO)K1~ecrBeHHFbl? 3roMy BOHIPOCY mo)KeT 6EJTb nlpH1aHa c~neJayotua~f KOHKpeTHas1 c~opma:,LaaHN ABa ypaBHeHHfI:
(2.1)fx)= +an- a a 0
(2.2) f(x) ==x +n "-I ~a 1~a=O
Tpe6yeTCfI y3RaTb, Bbipa}KaeTCH Ai pa ~I4HOI1aJbHO KopeHb oJUioro ypaBHeHHR qepe3 KopeHb JapyrorO, H ecaiR Bbipa)KaeTCR, TO HaflTH 3TO Bbipaweffae.
By.Ttem npeanotrTOll'm ypBHeHH R (2. 1) 14 (2.2) HernpHBOAi14MbAMH. Toruia
eCJIH cyLiLeCTByeT paLMiHOFanbHOe BbTipaKeHHe Ka~coro.Hfi6yJab 113 KOpHeft X.
ypaBHeHHfI (2.2) 'lepe3 KaKOFl-HO4yaTb KopeHb X, ypaBHeHI45 (2.1):
XI % ' + O'lXl + %'2X12 +.. + C_ 1XI
TO, nOJLCTBJAfI B BITO Bblpa}KeHI~e BMeCTO Xi1 KaH(rbifl 143 OCTaJlbHbIX KOpHeR1
Xi, X2. x,, ypa~reHH5i (2. 1), M b! HOIIyqHM BCe KOpiH XI) AT2,..,X"
ypaBHeHHfl (2.2):
n -
(2.3) AT, = % + alX,, + %-X2Q +..+ a_ 1X2ni
An%'i oneeeH ni Ko3cj4IHLLieHTOB 170 ~, a,,..)t-1 M bl TaKHIM o~pa30M no.Tly'aem CHCTemy fl JHHetRHbX ypaBHeKHlit C onee~am paDmium