Osnovy teorii Galua.

~3. Pa3pewuumbe opynnbi u ypaenenwi 2

129

cop'myntbI (3.27) JaaIOT 11J51 KQ0914)HLLHeHT0B ai, [3 ypaBHeHHSI (3.19), pa3 -penmmoro B paas~ajiax, napame~pny~ecmue B )ipawceHiHfq-"J: ]J2 Toro, 4To6bi
ypaB~seHie THHa (3.19) 6buio pa3pe1HHMO B paJRHKxJ1a, He~oJLHomtO H.LjOCTaTO'JHO, qTo6bi ero K034Xq)HI4eHTbI a', [3 M~OMi 6b.ino Ipe,4CTaBHTb
B q~opme (3.27), rue X, H tt pauiHOHaJabHbie BeJiHqHHbl.
10. BO3bMem lpHimep. Ji1aBaBl BeniqH4HHm X, 3HaqeHH5 X =3, ~.=1
3125 *3 3125 -3
24 - (9 -18 +25) 256
H ypaBHeHHle (3.19) [ipiHmeT BHA'.
x5 + 3125.3x + 3125 -3= o.
2 5 256
loac-ramrnl1 x - yi-, 6yaem iimeTb:
(3.28) y5 +15y +12 =O.
3TO ypaBHeHiie, 6y)Iy4H HerHpHBOJ1HMO (KpIMHReH 3ft3eHHJTe1AHa viii
p = 3), B TO }Kce Bpems1 peinaeTC5n B paJtHKahIax.
11. PaccmOTpHM ypaBHeIHHsl TH[in
(3.29) x5 + 5x +S5t ==0,
rmie t -uLenoe paJ4HOHaJ~bHoe qHCJIO, He JteJflIemlC Ha 5, H )aoua)em, '4T0
STH ypaBHeHHl5I He moryT 6MlTb pa3peHiHMbi B pauHmaJax. B camom iteue,
J.OnYCTHM, 'iTO ypa~nefeiie (3.26) npii a = 5, [3=- 5t, T. e. ypaBHeH~e
(3.30) (u- 5)4 (ua2 30u + 625) 59t4u
HmeeT palLHOHau~bHb~t KopeHb U. Honaran U = 53.w H nOJLCTaBJ1H B (3.30),
rlOnyqIHM:
(53.w - 5)4(56.w -6.54.w + 54) 512.t.W
O6o3uaqag 5 t 'iepe.3 s, 6yjtem HmeTb:
25w 6 w + 1 - s w,
OTKyua _ _ _ _ _,M 6+ s4 ~'(S4 + 6)2 ~~ 100
50
HTO6bi STO peuueH~e 6biJIo paIlHOHaJIbHoe, HeofxOJLHMO, 'LTO6bi nOAKOpeHHoe Bbipa}KeHHe 6bino nOJIHbIM KBaJapaTOM:
(S4 + 6)2 - 1 00 = Z2.
BCfIOMHHa~1 H3BeCTHbIe 4~opmyJnbI rH4aropa AJIS peweIHH5I ypaBHeHHII BHAa
X2 +y2= Z2, HMeeM:
CmnaubleaB nep~we xRBe 4,opmyji~bi, nonyuaem:
s4 + 1 6 - ~+ &)2.
'Ie6oTapCX.