Osnovy teorii Galua.

184

184 ~~~V. Ypae6nenuq C 3a~aHI-tbAiu zpynnamu

YpaBHeHHII~ (5.4), (5.5), (5.7)H (5.9) y1IOBJIeTBOP,910T yCJ1OBH$1M KpHTepH5~1 93~eHiMefka. '4To6bi y6eaiwTbCfl B 3TOM, nno.TiOHm an, 9T14X C.UyqaeB
COOTBeTCTBeHHO p = 2, 3, 5, 7. Boo6nme ec~ni B ypaBHeHHHI (5.1) n npoCToe 4HcJIo, TO ypaameHi~e ya~eBPe yCJ1OBH5M KPHTepH% 9h3eH~liTetlHa
ripH p — n.
YpaBHeHfie (5.6) timeeT npHr p = 2 cneayioum~e HO~pULKOBbie qiC~na:
4p, 2 +a +3p, 2 +P+2p, 3-F-+ y p, 3,
uae a >- 0, f>0, T >- 0 (nona3aTenHI CTeIneHeIf1, B KOTop1JX 2 BXOaitT B rniciia
g1, g2, g3). BbIqepT14B )uiarpammy (cm. ~ 4.4), Mbl HafJtem jgniq p eaLIHHCTBeH3
Hoe 3HaMeH14~p -i OTCioaa fia oCHOBaHlIH Teopemb1 94 cnieuyeT, 'ITO0
ypaBeHelme (5.6) HerlpHBOaMMO.
YpaBHeHHC (5 8) wimeeT npH p = 2 caieayioiaue HOPR1JKOBbI e l Hcara:
6p, I+c + p,1+P+ 4p, 3+ I+ 3p, 3 + +2p, 4 + 1 p, 4,
rie a >0, j > 0, ~>0, >0, s0. EcJIH ripu aiTom P > 0 (T. e. ecCiH g2
2
JLaeIHTCH Ha 2), TO Mbl I1OJIY'IHM JXl51 p 3Ha'IeHHe p =y3 OTCtoata cieJayeT, 'ITO pa3JIo)KHH51 pa3o6bloTCYI Ba aTBa uwi1.Ta 3-ro nopsLLLKa. ECJH we(
0,TO MbI rIonTyqHM xuJIs p JUBa HaxIeHHSig: p== — =~- T. e.
pa3ilo)KeHH45 pa3o6blOTCSI Ha LLHKJnbi HOP5lJKOB 2 Hi 4.
HpiH p = 5 IlOjPl~KOBbie trncja 6y~AYT TaKoBW:
6p, ~+ 5p, I + p 4p, 1 + y+3p, 1 + +2p, I + +p, 1
3JaeCb flpH ah> 0 mbl 6yJaem HmeTb p = ~,a rpip ci 0 -0, P2 = -5- B nepBOM Ciiyqae HeflpHB3OJHMOCTb olieBHJUHa. BO BTOPOM
>me cniyqae o~a cn'ex~yeT 143 TOrO, WIO B CMAY Teopembi 94 ypaBrnerne (5.8),
C OaHOR1 CTOPOiblb, mo)KeT pacHriaaTbCfl TO.UJbKO Ha He~HPHBOJAtMbie MHOM(~eHTJ4
141 H 541 CTerneHeft, a C JApyrOfl CTOPOMhI, B cixjiy pa3JI-O)KeHFI15 I1pH p =2
HO He moxceT coXUCpKaTb JIHHefHbbix MMOKHlTeneA1.
YpaBneHHe (5.iO ) npii p 2 HmeeT caieayoionie HOP5ULKOBbie 'IHCJIa:
8p, 3 + + 7p, 3+ + 6p, 4 +iT+ 5:, 4 +?~4p, 6+ E+ 3p,
6 + + 2p, 7 + + p, 7.
39T0 JaeT Ham p= - 7, OTKyXLa cpa~y caleJyeT, TITO ypaB~e~iie Henpii
3. O6pai'iiMCs K Teopeme II Illypa. EcviH- Mbl YMHO)A(HM rIOJ1HHom E,,(x)
Ba nt!, TO nlOnyq'Hm nIOJHlOM
KOTOpb1tk yJLOBj1eTBOpsieT YCiIOBIfM Teopemb 1I. Kpome Ti-or, 6ep51 B Kaqe~CTBe
moJayJI( pa33JI14Hb1e npOCTbie 'IIIcJa, Be [1pCwbI1IImiOLLH it, mbi mo)*em y6eAUHTbCS, 'ITo rpyrnna ra.Tya (1ypaBHeHiis (5.11) COaep~KIHT nOJICTaHOP3KH
ix.KJIeHHbIX THrIOB, t1PHCYTCTBHe KOTO~b1X B (8) o6Hapy~xlBaeT, 'ITO (WcoJaep