Osnovy teorii Galua.

158

158 ~~~V. Ypa,6nen;.q c 3adaflflbmu opynnamu

flOJIHHoma hi (x), aIeSIHTCSI Ha p, TO H KOS44~(HILHeHTbl 'iaCTHoro g (x)
= (R,, - p (x) h (x)): f (x) XLoJDHbi JaeJlffbC51 Ha p, tITo npOTHBope'IHT Hawiemy
npe~noj1oA)KeHHIO OTHOCHTeJabHO K~o9544rHuHeHT0B rIOJHHOMOB g (X), h (X).
TaKHM o6pa3oM I? He aeJ1HTCH Ha p, H cpaBIHeHHe
IR ~==1 (modj)
HmeeT iiejioe pa1LHOHaJlbHoe peuieHne - a. YMHowaH (1.5) Ha a H nOAiCTaIBJIHW Tyata x = OLa, nOJIy4IHM:
OTKYAa cJimeXyT, 'ITo a h (a)- -i- Cb JeHT, O6paTHblfl K?(p(C) OTHOCHTeJIbHO YMHo)xeHHSI.
Tenepb, n~HmeHq Si K HaflJaeHHOMY HOJI[O Teopemy 80, 6yaeM HmeTb:
OL - CZ=O0.
Hp H m e ' a H He. H3 aoaaelC~ 3TOfl Teopembf Tam~ cJneJyeT, 'ITO
cambill O6aLHf IPHeHm JLiH nOCTpoefHHHi KOHe'IHbIX noniell TaKOB. Bepem HenPHBOAHIMtJ 1 HO MOAiYJHO p HOJIHHOM f (X) Ht paccmaTPHBaem COBOI~ynHOCTb,
BCeBO3MO}KHb1X HOJIHHOMOB OT X, CtliHTas x KOPHem ypaBHeHH51I 1(X) = 0.
npH STOM MO)KHO BO BC'IKHX pa~eHCT~ax OTKHJab1BaTb KpaTHOCTH qimc~a p.
TaKoro pouia nonrrn 6biAH BBeJaeHbl B pacO~~ ['ajiya H nlOJIyqHJIH
uHa3saHHe io.Tiefl MHHMOCTefl fanya.
MOWHO He BB0AltHTh Kop~ett ypaBHeHHH f(x) =O, a paccmaTPHBaTb X,
Kaic nepemeHHYIO BeJH'qiHHy; Ho npmH 9TOM JIBa nOTIHFoma (p (X) H 4' (X)
JAOJI}KHbl CtIHTaTbCg paBHbIMH Toriaa H TOJlbKO TO1rLa, eCJIH' HX pa3HOCTb
(p (x) - ' (x) JienJHTC no MOIYAHO p Ha f (x); JapyriHmHi c.ioBamH, mo)KeT
6bITb npenca~ane~a H 4)opme p. g (x) + f(x). h (x). B 3TOM cnryq-ae
rOBOPHT TaKKe, 'ITO UIOJHHOMbI p(x() H ~~(X) c p BHHMbl no JLBOIIHOMY MOJayJHO p. f(X):
B CB5I3H C 9THM KOHeIHbjJe noJIH Ha3b1Ba1OT TaKNKe nogH 0AMHc p a B HeIIHft nO aBOflHIM MOJ~A5IM.
6. 143 Teopembi 82 HeopaTeH BbITeKaeT
T e op eMa 83. BCRKHfl HeriPHBOalHMblrl no0 MOJLIJ p HOJIHiHOM
n-ft CTef1eHH eCTb JIej1HTerlb rIOJIBOma x" - x no MOAYJHIO P.
TaKHM o6pa3oM, %ITo6hI MAtTH Bce HenpHBOJLHmbie no mo2yiiJo p HoAimHOmbi n ift CTerneHII, cjm~IyT pa3JIONKHTJ, nOJIHHOM A~- x Ha HeflpIBOIIHmbie
no MOAJ1IJIO P MH~O1HTe.W. HpH STOM HeKOTOpbie H3 MHO)KHTeJnefl moryT
6ulTb CTerneHH HH}Ke n. CyuieCTBO~aahle )e AMJ1 Ka)KJbix p H fl HenP1HB0 -AH4MbX Ho MOA1-JUO P HOJI4HEOMOB fl-ft CTer~eHH catex~yeT aoKa3aTb oco6o.
7. Teopemia 84. EcJIH qjiemeHT CL KOHeIHor'o MOU yaIOBJIeTBOpfi
psieT HefnpHBOJUHMOMYypa~BHeHHIO CreneHH ni, TO CL CL == 0 HmmeT
meCTO Tor~ia H TOJ~bKO Tor.La, ecaH m AeJIHTC5I Ha ni.