Osnovy teorii Galua.

~ 3. Ypa~neuiiu C 3adaHnblMu opynna.41a17

1 71

pa3J1araeTCS1 Ha HeflpHBOA~HmbIe no M0JAYJII0 3 UOJIHHOMbI 5-il CTeneHH. B camom
X'0 ~ X9 + x8 + X7 + XeO + Xe -F Xe + XS + X2 -4- X + 1 -EE(X5 + X'-X3 + X2 - 1) (X5 - Xe+ x2 - X-i) (mod 3).
B icaqeCT~e P2 Mb! He MO>K —M B311Tb tlHcAa 2, TaK KaK Jais p = 2 HmeeTr meCTO
g (2)'= 1, B TO BpemSI KaK AnA1 HaC Heo6XoJU4Mo g (2) >- 2. Bo3bmem p2 = 5. Torua
Ham I1pHJUTC5I HaaHTH HenpHBOIJJMbIl 110 MOJIYJHIO 5 MH0)KHTeJlb 2-11 CTe11eHH 110O2IHHOMa X52 x - XX (X24 -1). Bce JIIHHerlHble MHO)KHTehIH RJUO.TKItbJ BXOJAHTb AJUIHTeJ151Ml
B 1OAIHHOM Xe x, a [IOTOMY HIOJIHHOM X= x'4- x2a -1 1 ojUOI)KeH pasnaraT b~fl Ha
He11pHBOAUHMbie 110 MOAY3UI[O 5 110)21HH1Mb! 2-11 CTeneHH. B camom Jteae,
X4 + X2-j- 1 = (X2+ X + 1) (X2- X-4- 1).
TaKHM o6pasom YCJIOBHSI HamiefI 3aULqH 6yhlyT BbflnoJHeHbi, ecmi~ Mb, HaUnpHMep,
nOJIO)KHM:
f (x) x5~+x4 -x3 +x2 mjod 3), f (x) x (x + x2 +1) (mod 5).
f (x) =5 Xe Aix4 + A2X8 + A~x2 -H A,,x + A,5,
InOAqy4[M JRl1 KOS(4pHLuHeHTOB Ai czieayiounse cpaBHeHIIq:
Al 07O A2~ 1, A3 O, A,, -1, A5~sO-=0(mod 5).
Peniag 3T CHCTemy CpaBHeHHHif, floJly'IIIM:
T. e.
f(X)=X5+lox'+11X3+10x2+6X +5(mod 15).
BcnqxttHl HOJHHOm f(x), yJU0!IJeTBOPRIOLLHA 3TOMY cpaBHeHHIO, A~aeT pemueH~e
nocTaBneHHo11 3aJ~aqH.
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cpaBHHTeJlbHO.nerKO UOCTPOHTb ypame~iie 6 e aa 4 4)e K Ta, T. e. c cHimmeTpHqeCKOII rpynirott ranoya. B ca1mom Jteae, cHimmeTpHqeCKa11 rpynna co~uep)KHT 11OJICTa1HOBKH BceBO3MOIKbIX IbX HKJ1eHHbIX THRIOB. C iApyro11 CTopoHbl:
eCJIH rpynna CO11ePM(T n10XCTaHOBKH4 Tpex cJ1iJLy!oLL~xU4X HKJIetiHfX THn1OB
1) LU4K) fl-co nopexLa, 2) tuiKRo (n - 1)-co nop9,Rna 14 3) TpaHCrIO3HtUHIO,
TO 0Ha1 Aojmi~a 6b!Tb cHmmeTpHqeCKofl rpyrinott. B caMOM un~ie, oHa TpaHi3HTHBHa, TaK KaIC couaepWHT LU1K)! l-ro, nOpHJUKa. CoJaepwa TpaHcroMIHIuO,
OHa B CHJIY TeopeMbi 14 JAOJIKHa 6wlTb H4J1H CHMMeTpH~eCKO11, H4J111 HM11PHMHTH4BHO11. JiOfIyCTHIM, 'ITO OHa I4mnpHmHTHBLa. Kpome Toro, oHa coJUep}KHT
ILHKJ1 (n - Il)-ro nopluua, KoTopblft 11PH HaJUJeKaiLaem cnoco6e Hymepa~uii
KOpHefl mowKn 6bITb npucrawi~eH Tam: (1) (23..n). I~YCTb 1, cx2 %M**. M
65yJeT Ta. ee CHCrema lIMHPHMHiTHBH0CTJ4, B KOTOpOil J1eWHT 11114)pa 1. Torjia,
cornraCHO onpe~ejieuuIoI lHlMfIHMHTHBfHOCTH, n0oICTaHOBKa (1) (23...n) xaoRi>a
nepemeulamb Uli4)pbl %,) on..., am, mexmiy co6oft, T. e. coJLep)KTb LU4KJab
nopnaJKoB< <i, B TO BpeM11 KaK B camom jAeje oHa COJUeP)KHT eIIHHCTBeHHblil
LWKJ (n - l)-ro llop9AJKa. 3T0 11POTHBope414e IoKa3b!BaeT, tITo rpynrna.EOqDKHa
6WlTb cHmmeTpHtIeCKoft.
BmeCTO Troo 'iro6ii CTaBHTb, B KaqeCT~e YCJ1OBH51 o6naJlyaHHe UJ4KJOM
n-ro!1OpffIKa, mb! mo)Kem o6ecne'rnTb HenptiBOUHMOCM ypaBiieHH~ii RlpyrHM
cnoco6OM, l-alpH~Mep n1PH InoMOIUB KPHTepHR 3113eHLHTeflHa, qTO npaKT!4qecKH
CUHJ0H o6Jiec'IaeT 3aJaaqy. Kpome Toco, ecum n npoc~oe 'iiiCiO., To rpyntaa