Osnovy teorii Galua.

~ 1. I-Icxo~nbie _Oa~cmbl 5

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1(x) = g(x).- q1(x) + -r,(x),
(13) ~~~~~g(x) =r1(x). q2(x) + r2(x),
r2(x)== rm1(k~)qj(x) + rm,(x),'
rm 1(x) = ral(x)qm + 1()
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meHUHO JAejiHTeiiem f(X) H4 g(X), qBBJI5eTCf TaKweC LeJH4TeJ1em rm(x). LIjwi 9Toro,
flOJlb3y5ICb noc~JeaJOBaTeJlbHO BceMH4 paBeHCTBaMH (13,Ha'HHa~i C B e p XH e rO, Mb y6extHMC5 B TOM, 'ITO 9TOT nIOJHHOM 5IBJIqeTC5I J=IHTeJnem nOnHHOMOB r1(x), r2(x),..., rmi,_ (x), rm(x).
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Bbipa3H~m rmx) Ha OCHOBaHHIH np~ice~r paBeHCTBa (1.3)
'Iepe3 r ~1(x) H r___X)
rmX) = rm - 2(X) - qm(x-) ' r., - ()
110OXCTaBHM B 3TO paBeHCTBO Bbipa}KeHHBs rm, - 1(x) qepe3 rm - 2(X)H
rm,, _ 3(X) Ha OCHOBaHHH4 npajC~i~tr paBeHCTBa; Torjaa flOJIT'Hm OAIHOPOJ~LHoe J1HHeflHoe BbipaxcH{HH rrnx) tiepe3 rm._2(x) H rm, _(x) C HOJIHHOmaMH OT X B KaqeCT~e K0OBHWluIeHTOB:
rm(x) = 1+ q,, _ (x)q,.(x)] r - 2(x) -qm(x)rm
rHOCTyfIaqI Ta K~ Ce c peaweHCTBYLO[1HM paBeHCTBOM, BMbi3Hm rm(x)
qepe3 rmn 3(X) H rmn - 4(X), H T..iX. H1puonwoaf rpoiugcc, MM B KOHLLe KOHILOB rHOhiy'HM 4~OpMYlY
(1.3') r,.,(x) -u(c).1(x)+ v(x).g(x),
rM u(X) Hi V(X) - OAHHOMb] OT X.
B 'IaCTHOCTH, ecaHr f(x) u4 g(x) BMaHMHQ-rnpOCTMl, TO mbi 6yJ~em HmeTb
(1.3"t)
u (X).f (X) + v(X).g (X)=