Osnovy teorii Galua.

_______ ~ ~ ~ S 6. Keae~pupye-mbie AYI-1OqIU __195
-HaHmeHbLUH41 HoKa33TeJlb f TaKoro poiaa, 'To6bI HmeJIo meCTo 2f~nE 1 (mfod 23).
Hafla~eM BbI4eTh CTenenefl qHcJIa 2 no mQoaJHO 23:
2, 4, 8, 16,_9, 18, 13, 3, 6, 12., 1.
HwonyqaeTCHl f =1 1. TaKHMrv o6pa3oMf LLHKJnbI pa3JlOmKeHllfl ypaBHeH141 (6.5)
no CTefleH.M qHcjia 2 HmeIOT B 9TOM Cuiyqae HOP51AJKH 1 1, 1 1, 1, 1, 1, 1, 1.
M143 1mmeioHuHxc 3JLecb 6Jqaronpii~TflbIX LLHKJIOB HeJnb35 COCTaBHTb ypaBHeHHe
841t CTeIneHH.
B C~nyqae m = 27, fl = 1 1 KOPHHI ypaBHeHHHl (6.5) pacna~iaaiTCn no MO-.iRyJIuO 3 Ha LimKJbi nlOPHJKOB 9, 9, 8, 1, a no moayat1o 11I-Ha LLHKJIbi noPRAIKOB 8, 10, 8, 1. B.TiaronpH51THb1fl CfIyqail MOP 6Mi HmeTb meCTO TOJnbKO
TorRa, ecim 6bM ypaBF~eHiie (6.2 1) HmeJ1o CTeHeHb ut = 8. Tor~aa ero KOPHH
no moJLyino 11 6yxiyT COOTBeTCTBOBaTb OJLHOMY 143 kllHKJOB nopuix~a 8; W111
3TorO 1LHKJna WHl Bce 3Ha4eHH51 p nlOJlo}KI4Te.TlbHbl, HJIH4 Bce OTpHl~aTeJlbHbi.
Ho ypaBnieiwe (6.5) B03IBpaTHO, H ecalm Hanp~mep pa3Jo)KeHH51 Kopuell ero
MHO)K14TeJiH f (x) no C~enei-iqM 1 1 HMeIOT fl0J10?K~lTeJibHoe 3H-aqeHile p, To no-.THHOM XSJfy) Hme%.TUI5 pa33Jo)KeHH1 CBOHX KOpHefl no CTenHSHHM 11OTpHltaTeJbHbie 3HateHH5I p, 6yaeT' B3aHMHo-rnpoCT C f (X), OTmyaa 6yJ~eT ciiejioBaTb, 'ITO ypBe~i (6.5) HimeeT BAM MHO)KHTej1iq ceneffli 8 C K09j)4)HU[HeHTamH 113 nojii K (j~.OJHaKo pn3JoxceHHe KopneA1 nO CTeneHRnM
3noi~a3bIBael, 'ITO 9TO HeBo3MO)I{Ho. TaKHm o6pa3oM cjnyqatt m ==27, ni = 1 1
TO.)Ke He6Jaaro[IpHTeH.
B) B 9TOM cnvy'ae paccMOTPHIM ciieapyowiie nOJLCJ~y'aH:
1) m=p, n -q. Tom ~p-~22 + 1, q ==22 +1, a nTOTOmy 22'=
2P+2 OTKyJaa cJneJyeT: s ==2p, 22 = 2.22P., 2 1 + 20, f 0,
~=1, p 5, q =3. 3TOT cjnyqatt 6biji H3BeCTeH RJnay3eHy.
2) tn=9, n q, m n-n 2'. Toraa q =5, T. e. m =9, n =5. Ho
moJ1aJIIo 2 Mbi noJnytHM
(x7-1) (X2 -1)I'm- (mod 2).
'4To6bI HaR~TH HeHpHBOJLHMbIe no moJLxy1o 2 MHOMKHT6JH flOJHHoma X7-1
Hafl~eM BblqeTbI cUenei:ef4 2 no moJ1ynJio 7:2, 4, 1. HO3TOMy KOPHI4 ypaBHeHHAg (6.5) pacnaiaaioTCH no moJ1Jio 2 Ha LLHKJII flOP5IJKOB 3, '3 1, 1, 1.
9To flpOTHBope'IHT TOMY,, WIO u = 9 - 1 = 8.
3)n p, n = 9. 2 2 +1I=23 +1I+2', s=-3, cx 2, p= 17. OTCIoAa
m=17, n=-9.
Hlo moJ4ynio 2 mbi noJny'iHm:
(x3 - 1) (X4 -1) - 0 (mod 2).
BbxiqeTbl CTenenefl 2 no mo11yJIo 13: 2, 4, 8, 3, 6, 12, 11, 9, 5, 10,0
7, 1. KOpHH ypaBHeHHS1 (6.5) pacnaaaaioTCH 11o mOJy2nio 2 Ha UJ4KJbi nopgaxK(OB 12, 1, 1, 1, 1, 1. 3)TO[Ip9TI4BO~peqT TOMY, 'ITO u =17 - 1 16.
4) m ==273, n -q. Torjaa 27 =2 + 2:`1+ 1, 28 +22P =26. Oicio~a
H4111 20 == 1, 1411 S= 1. B o6oHx cjiyIaflx 9TO HeBo3MO)H{H, TaIK 1(K pa3 -IIOCTb 27 - 1 - 2 =24 He pa~ CTen~eHi4 aIBOR1KH.
1 3*