Osnovy teorii Galua.

130

130 ~~~~Ill. Pa3pewumbze ypameuenw

rHepeimem K LLeJbim qHicjam. Hlonarasi s = - 8.4-.- rxae X, t17 V
ixentie twcnaa, rIony'HM TaK Ha3bIBaeMQe ypaBne~iie 3 n.iie pa (L. Euler):
(3.31) X~~~~4 + (2 v)4 = tL2. V2.,LI4JI9 JAO Ka3aTebCTBa ero HeBO3MO)KHOCTII B0C[0Jlb3yemci4 meTOJJOM 3Bfljepa,
HOCHIILHM Hm3BHi4e descen te infinie (IMH~ a JI rOpHTM HQHH)IKeHlf 5).
HpHmeH5151 OflfTb (430pMYJUbI fHdcIaropa, HlOJyqLUIM:
X2 2- ~2, 4v2 = 2, V- -l
143 BTO0p11l 4)OpMYJIbl c~nexiye, 'ITO
a H3 rnepBoI4, 'ITO X 1LeJ1HTCR1 Ha %. Ho~narasg X = X x, limeem:
Bocno~lb3yemcg 0O1~1Tb c~opmyJTamiH Vheparopa:
C2-=f2 +g2, 2o)2 =2fg,?\1=J2-g92.
1H3 BTOpO1R 43OpMYnbll cUeJyeT, 'ITO
f -k2M, g=P 1M.
Horloaram C tm, [nony'1mm:
(3.32) C2=k 4
TaKHM c6pa3om mbi pti~xoJaHM K ypaBHeHliHO Toro mK Biiaa, 'ITO(31)
1-1 peiueiui KOTOP)Or0 HmeIOT meHbJJIfK BeTiHi'qlHy. H~poaeJIbIBafI HiaL (3 32)
Te xe npeo6pa3oBaH115w, Mbl OfIUITb ymeulibiaeM Be1H'IHlibl peineHHfI. H o
B BH4XL Toro, '-4TO 9TOT [npoLLCC He mo)KeT IITTM AO 6ecKoue'lHocT14, Nlbl 3a -
KJluo'Iaem, '-ITO ypaBl-eH~ie (3 31) He peiuaeTCRi B [Le.TbJx paLtHoHaJ~btbix 'HC-.nax, a HOTOMY ypanieie~e (3.30) Hie momKT iime-Tb pa~f1HaHJlbHbIX Kopieft.
TaKHM o6paaoM ypaB~ieH~e (3.29) He peuwaeTCH B palllaJxaax.
12. Korata ypaB~eHme (3.26) HmmeT xpaTHbie KopHM? B WayI Toro, '4TO
u -v2 9T0, 6ya i4meTb meCTO Toraa H TOJ~bKO TorJaa, ecilH ypaBHeHHie
(3.24) timeT HAll- 1KpTHbie KOPHH1, HAH paBHbIe nio Inul'-ll'He Hi npOTHI3OrlOJlOWKIbie no 3HaKy KOPH!-! MeH%51 3HaK flPl HeH3BeCTHOB V B Bbipa}KeHHH
(3,24), mMi flony1tiMH3meHeHHbIM 3HiaK TOJAbKO nepeia lIJeHom Av, a nOTOMY
CJnyqah1 pa3Iib1X nO BeJlH'IHe H nIP0THBOHOAO0KHb1X HO0 3HaKy KOpHeft mo)KeT
HmeTb meCTO TOJlbKO B CnTy'Iae A = 0, T. e. KoPIaa HlCXOJLHoe ypaBHeHHe
(3.19) HmeeT K~paTHble KOPHH1.
Ecatii we yp iBHeHtie (3.24) iimeeT KpaTHbie KOPHH, TO C HHM HmmeT
o6uwe KOPHH JlpOH3BoJ2amI:
6 v5 - 20 ar3 + 30OC2 V~ A = 0.
HOUITaBJiff OTCioJaa sMatieH-~e A B (3.24), nouay'IiM:
-5v6 + 15caV- 15CX2V2+ 50 3 -5(V2-a)3- 0.
OTCio~aa Ui - -2 c=. HOJICTaBjimIY B (3.26), 6yxmem HmeTb:
3125[P4c -0,