Osnovy teorii Galua.

192

192 ~~~V. Ypa-szenum C 3adannbimu opynnamu

TaK 'ITO Mbl Hmeem a. ~ 1, X~<2. HO TaK KBK k < 2X~, TO k<3. TaKHM
-o6pa3oM BO3MO)KHbI TOnlbKO ~CayiamH p1.- 9 ii p = 27. B cyiiyae p) k = 9
6hlarorlp1llTHoe 3Ha'ieHHe p paBHO -- B ciiyxiae -~e p = 27,P=
III) X 1, k=~ 1. Torma p= MO)KeT 6bITb 6jiaFor~pH51THbIM
3H3aeHHeMI TOJnbKO TOFLa, ecJ1H p HMmeT dppm 2' + 1, T. e. gmiBefITCfI
ray~ccOBbIM I1pOCTbIM 4IHCJIOM.
10. Oripeazern~m CTeHeHb it H~e1pHBOAHMOFO ypaBHeHHI45 (6.21). B BHJ1Y
TOFO, tITO 6JnarofpHINTHbie uH4KJIbi timeOT HOP51J1OLC p - I npii k 1, 8 npjH
k 2 iH 8 npH k =3(Mb! HpHHHmaem BO BH~M aHlHe, 'ITo nIH k 1 ~Ik =3 'nc~no _ o6JlaCTH paIIHOHaJnbHOCTHIK (jfi) Hm~eeT nopRui1(OBoe 4HcJIo - a npu k = 2 - 'wcjia o6.nacTH pau1LHOHbJIOCTH He MOrYT
4MmeTb lltpo6Hbix FO510UKOBbIX tuiceni), Mb! BHIIHM, 'ITO Ut RLOJI)(HO, IIeJHTCfI Ha
p1' -1 UpH k z1 Hi rpH k =2 14 va 8 npii k =3. C iipyrofl CTOPOHbI,
6narornpn51Tbime LHKn1bi COOTBeTCTBY1OT 3HB'fi~HLO a 1 Hqa'I~JbHOFO qIneHa
k
pa3J1o)KeHHH KOpHefl, a TaK- KaK 3T0 3Ha1TeHi4e HimeeT KpaTHOCTb p H3
KOTOpb1X OA!HO 3Ha'IeHl~e COOTBeTCTBy eT TplHBHliJMHOMY KOPHIO X =1 TO
B C~nyqae k=- 1 H k = 2 HmeeT m eCTO
(6.22) a -! — 1.c
B cniyae wie k- 3, T. e. Pk 27, cyILueCTByeT eaLiiHCTBeHHb~iR 6.naronpH45ITHbIfl LLHKJ! rop~uua 8, a!1OTOMY
(6.23) ui = 8.
11. )IIOnyCTHM, 1ITO mf coae~p)HT HeCIKOJbKO pa3J!WH-mb!x HpoCTb]X MHO~4(Te1eft: mn = p Pi'.. TorLa B Cl-lny COOTHo[Htuefim (6.22) Mb! 6yaem H4meTb:
OTKyaLq pkp - q.,'TO HeBiO3Mo0vHO. To'IHO Tal( we ecjrn p'A=- 27,
TO let ' 9, T. e. p,:: pl. TaKHM Ot~pa3OM MIA BHJLHM, tITO ni J0iTDKilHo 6blmE
CTeneHb1O flpOCTOrO 'IHcjia.
12. PacCMOTpHM pa3JIo)eHW1R KOf)Hef HO CTertieHMm q, ume q -Hl1OCTOPI
MHOKH~I~ tHCJ~1Ii.Mi BW~eII4 'TO B 3TOM C.jiy'ae, Kpome -- 2 k,
MHWKH~enb q~cjia n. MbI B~aejjm, p 2(q, -1)
k
moryT 6blTb 6JxaroHpH5iTHbimH enme 3Ha'IeHHH p =+m~ KOTOpb!M COOTBeTCTB~eT XaBa LumKJa nOpffIKa Pa36epemi morynLI'e BCTpeT14TbC5H 3.aeCb
I. mn -it He eCTb CTeiHeHb JiBOflKH. Toriaa eah1HCTBeHHblm 6JIaronpHR!T~b!M
3a~H~'Hmm p 6yaeT CJ1YWHTb p = 2)- k H Mb! 6yiaem HMCTb: u=q1' 1,
OTKyaia p = q, WIO HeBo3MoWvio. To~fHO TaKweK B Cxryqae p1 = 27 mb! rHoJy'IBM q = 3, 'ITO ofli!Tb HeBO3MO)KHO.