Osnovy teorii Galua.

94

94 II~~~~~1. rpynna raAya

Kea~apaT KOTOporo eCmb J.UCKpHMHHaHT ypaBHeHH14 (5.8). TaKHM o6pa3oM,
4lTo6b1 y3HaTb, He qBJH1eTC$1 AH rpyiina raiiya yfIa1BHfHH51 (5.8) 3HaKoriepeMeHHoft rpyniiot W. HaJao BbI4HCTIHTH ero.LHJCKPHMHHaHT D H~ nOCMOTpeTb,
He gBB.ISeTCfI.IH OH TOLI~bIM KBaJapaTOM. ]Jajiee HaLo InpoBepH4Th, He HMmeT AiH
ypaBi-eHne
2- D ==0
K~paTHb1X KOpHefl.:3TO HmeeT meCTO T0JlbKO B Cniyqae D = 0 HO TOraa~ H
11epBOHaqIaJbHoe ypaB~e~itie (5.8) HimeeT K~paTHble KOpHH, tiero mbl He flpeAnouiaravIH.,UA51 BbI4q4cJeHHfI J11CKPHMHHafITOB HaH6oJnee ripocT caieapyomHig crnoco6 *
LI2ICKPHMHHaHT, KaK H3BeCTHO, pa~eH -f' (XI)f' (x2)f' (x3.), rae f' (x) -
HPOH3BOJLHa5l OT HORiHHoma f(x); apynFHmH CJIOBami4, OH pae CBo6oJUIomy,qyleHy ypaBeliH14, 1HOTOPOMY Y~JtBJIeTBOP5110T f'(xl), f'(x2)i, f1(x3). Haoaxem
3T0 ypaBHeHI~e. nlOJlb3y5ICb meTOaOM ~ 3.9: z =1', (x),
z x -3 q -- 2 px,
zx2- -3qx-2 pX2,
OTKy~aa
p - z 09 + 3,
-3q, -2p-z, 0, =0,
0, 3 q, -2p- z
T. e.
Z3+p2 pZ ( p -72 0.
TaKHM o6pa3oN1= p3 7 2
15. YpaBHeHHe lieTi~epT0i CTeueHH. )I1aHO ypaBe~ie~~
(5.9) x + a, x3+ a2 X2~a~x +a4 =O.
qro6wi y3HaTb, He saBafITCg aiH ero rpynlna ranya JtC IHTeJaem 3aKaOnepemeHHOfl rpyrnnbl, JROCTaTOqHO BbISICHHTEb, tie s1BjifITCSI Jil ero, JUCKpHMI4HaHT
D = (x -X2)2 (XIL - X3)2 (1-x)2 (2-x)2 (2-x)2 (X,
TO4lHblM KBaJ~pa-TOM. JiefACTBlHTeJlbHO, 4~YHFCLHR1
(XI - X2) (X1 - X3) (X1 - X4) (X2 - X3) (X2 - X4) (X3- X4)
iie mC11eTC5I lnPH IIO.CTaHOBKax 3H~npeeHf rpynnbi, a l1PH BCHKHIX
JLpy[Hx nlOJCTaHOBKax meH5IT 3HaK. 3Ta 4~YHKLUFl moweT cJaew~BaTeJ~bHO tie
meHqTb '1HcneHHOR4 BeJiH4MHb flpH HeqeTHbIX HORCTaHOBKaX TOJabKO TorJaa, ecniH
oHa paB~a HyMiO- HO Torga HcxoJ4Hoe ypamieiine (5.9) JtOMLKHO 1MAIM xpaTinmie KOPHI4.
'-ITo6bI MAtTH nPH3HaK Toro, Wro rpynria ranya ypmBHeHH5 (5.9) eCrb
JLeJiHTeJib rpynInbi
+ (13 24) +(1 4 23) +(I 2) +(3 4),
*)Coo6teH a3BTOPY npocp. B. H. Ae'1oHe.