Osnovy teorii Galua.

~1. JRoueqwnge no~vI

159,

IO K a 3 aT e nC T BO. i6. H3 Teopembi 82 cnieaye M n'-a'. O. Ec~im
mi = q Itn, TO apm MP n -f = ( apn) Pn (q-1) = ~p n (q-) - ~pn) pit(q-2)MP n q-2) H T. A. a La
20. HYCTb M He aelIJ1TCH Ha fl, H HyCTb m = nq + r, 0 < r < n. TorXaa
pm __ (,pfq)pr p r
r
EcJiH 6b1 HMeAO meCTO a za TO, 1(aK HeTpyXLHO y6eica14b$, xawasxia
/ n-1
BeiI-qHHa BHJ~a ~3 ao + a~cm + +. an 1 a Tox1(e YXOBJIeTBOp5UIa 6ui
ypaBHeHHIO0 Xpr- X = 0. j11eRCTBHTenlbHO, B03BeJLem BTO paBeHCTBO B -0
CTetiiHb H o6paTHM BfflHmaHe Ha TO, qTo noniyqaembie nIpH 3TOm nojiHiHOMHBJ~~~b~~bkol p! k- TOJIbKO ToP.ua He XLeji~lTCS Ha p
ecalH O1LHO 113 k, paiinO p; HO Torua Bce OCTaJ~bHbIe k, paBHbl HyJ11O.
TaKHM o6pa3oM
P~a.' ~' + a *. + a~ n-la~ (Cm. rma~y III, ~2.)
3Xaecb K paLjHOHaJnbHbIM KO3d4)i4iiLeHeITam aD, a,..., a n-l MO)Ktio fpHMeHHTb O6bIKHOBeHHY1O Teopemy 4'epma, B May1 nero
=ao + ajcLP +. + a nl
BO3BOXLH 3TO paBeHCTBO OHfTh B P-IO CTeIneHb H NOBTOP~51H 9TH B03BeJ[eHHR r pa3, mbl 6yJaem nojiyqaib:
-p a_ + ajczP2 +..+ an-1 a
-a_ + c?+..+a1 (-1) p3
Ppr a_+acPr + +av(n-1) pr
1`OXLCTa~illf m noca'eiinee paBeHCTBO BmeCTO aL PrJ1nmeHT aL folyJ3"HM:,_a0 + alot i.. + a7 1c0n = l
TaKMM O6pa3OM KOPHRMH HOJIHo-1ma xY - x,O- CTeneHH 6yXayT C~nynn)KHTb pa~3J1HL1HbIX Benj1HHH a0 + arA +. + a~~ 4T0 HeBO3MO)KHO B
CH1,a Pr < pn Teopema alOKa3aHa.
8. LI0Ka3aHubwe Teopembi y6e)itiaaioT HaC B cflpaBeARJHBOCTH
T e o p e M bi 85. HJOJHHOM X' - x paaiiaraeTcsi 1O MOAY~J110 P
Ha HeUnpHBoixtimbie HIOJIHHOMbl, CTefieFHK KOTO~bIX paB~bEl HJ1H Mn, HMN
JLeaHTensifM mi.