Osnovy teorii Galua.

194

194 ~~~V. YpameneuR C 3adannblmf U pynnalmu

TaKHM o6pa3oM I1= 2s - 2. B 6Jlaronpm~wrHoM cjiyqae JLOJDIHO H4meTb
mieCTO I= 2t. C itpyrofl CTOPOHbI, Ps+ 1 mo)IKeT TOJ~bKO ToUaa 6um~ IlpOCTbIM '-lHCJOM, ec.TrH S HmeeT 4~OpMY2. FI`3ToMy HmeeT bjeCTO
OTKyJLa
3TO pa13eHCTBO moxKeT i-imeTb meCTO TOJIbKO B ciyqae t - I ==0. OTCioyaa
1, s = 2" 2, p - 2S + 1 = 5. KC a-omy c./iyqaio cnleJayeT ewie npuicoeAIHHHTb cnrytiahf p = 3, flPH KOTOPOM HimeeT Me..CTo 2s - 2 = 0. Bce 0oca~nbHiwe c~nywaia m = p, n 1 He6Jnaronp14IgTHbl.
B c~nyqae m ==9, n =1mbi iimeem:
(6.24) X-1~ = 0,
T. e.
Xs + x7 + XI + x5 + (1~3) x4 + x3 + x2 + X + 1 0.
3T0 BO3BpToe ypaBHeHIoe, TaK Wro er'o miowiio rpeo6pa3oBam- nHp1
UOMOLaLH HOAFCTaHOB1(H X + ~ =Z B cJneXayoIILee
(6.25) Z4+z3-3Z2-2z+(l1~3)=0.
qTOt~bI 9TO ypaBiieHre pemanr~ob 13 KBaaapTHb1X paJLHKanlaX, Heofxoalumo
H JAOCTaTO4HOI 4IT06h ero rpynria fa-aya 6bijia aeurrTentem H3yqeHHOil HaMH
rpynrnu ropuiltia 8, K KOTpoO fl npMHaJaJIeiKHT 4)yHKU1H~1 U =xlx2 + X3X4,
yAoB3.TeTBOaqopuarn cneyioumemy ypa~~etiio (cm. rn~ II, ~ 5.10):
U + 3U2 +(- 6~T12) u-l17+:39=O.
BO3bmeM B 3TO pa~BHeHHH BepXHI~il 3aK. Toraa OHo 6yareT HmeTb
paLLHOHanlbHbrfl KOpeHb u = 4. 3T0 o6CTOflTeJnbCTBO CBsi3aHO C Tem. 'ITO
ypaBi~effie (6.25) pa3,naraeTC]cniCTeXaylofltHm o6pa3oM:
(Z2 -epZ-2) (Z2-_ 32z - 2) = 0,
r~e Ps e 3. TaKtim o6pa3oMI 9TOT Csiyttaft 6Naro1pl151TeHl, T. e. yparn-ie~liie
(6.24) ripH BepXHeM 3Hai~e pewuae~cl B KiiaJpaTlib!X pau4aJwaax. OJ 0,Haw~
jTM lc~yqaio He COOTBeTCTByeT HmmaOfl AIYHO14KHI, TaK KaKBeKOH
Z -X + - 2 cos 8 ypaBHeHH51 (6.25) MHHMbje.
HpIPH HH)KHeM 3aKe c~nyqafl e6,naronpH4,1TeH.
B c~iyqae m iz 27, n = 19 mi 4meeCm:
(X23 -1) (X4- 1) 0O(mod 2).
LlTo6b MM~HT, Ha KaKo1e HenfliBOJiHmbie no0 moJLyJIIO 2 MHO>KH4TeJH pa3-.jaraeTCS] lO.TH4HOM -X 1, Mbl )aOA)KHbl HatITH ilaiimeHb~lllfll HOKa3aTeJb fA
IPH- KOTOPOM X2f - x;tealiTCHI Ha X23 -.,51H 3Toro0 Haa~o oHpeaeJIHTb