Osnovy teorii Galua.

~S 6. IKwapupyembie AyHio'Ku19

193

II. m - n = 28. Toura KOPH11 ypaBHeHH1~ (6.2 1) COOTBeTCTB~l0T HAH11
A) Ha'-eHmigm k HA11iB) KaK3Ha'IeH11qmp 2X-k1, Ta
mr-n' 2(qx - l)
k
H 3Ha1qeH11fm p=~
rn-n
A) B neIpBOM cny'lae 1H.11 tt rn-n, Mn.11 u= 2 EcrnH p f27,
rn-n
TO 11.711 n1, 11.711 m 1 2 'IO HeBO3MO>KHO B BHAY1 n>1
ECJH Y~e m =27, TO HaHI~ n ~m -ua=19,9 wrHn n=rn- 2u= I11.
HlyCTb nl = 1. PacCMOTpIHM cpaBHeH11e
(x - 1)- 2(x'- 1) -0 (mod 2).
3T0 cpaBHeHHe MO}KHO B CH4.TY ~ 1 (mod 2) npeaJCTaBHTb TaK:
p~+1 p-I
(x 2 1)(. 2 -1) -zz0 (mod 2).
Ecnii k 1, TO U p- 1, 11 B 6naronp11flTHOM ciiyqae Bce i11Knbi
JAOJ1)KHbl 6bITb, 6it~arofpHSITHEJ. PaccMOTp11M MHO1K1eTmA cp (x) x -2 1
OH He HmeeT KpaTHb1X nTo moJay.7i 2 KOpHett, K~aK TaK npHMO13OJHas
to P~1 2 __ 2 (mod 2) B3aIHMHO-IHpocTa c (p(x). H3 Teopembi
2
85 ca7eay~eT, t[TO BCBKHAI HeripHBOa7HMIMl nTo mo.l~yni1o 2 1IOAH1'HOM CTeHCHH f
5IBJsieTCH ejwC71TeneIm 1onlIHHOma x -~ X, HO He mo)KeT 6bITb AeIHe~iu~iem no-.YHHoma T11na x2fl - x n4)11 f, < f. EcaHr Bce HeaH~1BOJL1mbie 110 moaywJo 2
XLeJHTeJ1H nlORfIHOma cp(X) XLOJIKHbI H4meTb CTe1TTH5MH CTe1eHH JLBOIRKH, TO
HaHH113WIHff nO.1IIHOM B14JLa X2'1- X, aenf1lumHics Ha cp(x), aoioNmeH HmeTb
n1oKa3aTeJnb I B eIpome f2t. Ho flO.7IIHOM X21_.X - X(X2 - 1-1) JaeJIHTCSl
Ha cp(x) - x2S9+i 1(ecnHr p:-289+ 1, TO -2s +1 O
R ObOTorJaeCJTIH 2 -1.te.7IHTCHl Ha 2' - + 1. Ho 22 (s - i)~ 1 AeJiHTC9
Ha 2s-1 + 1, a flOTOMY Il~2s - 2. C japyrort CTOPOHbI, I He mo)KeT 6bITb
meHbwe1 2s - 2, TaK KaK B 9TOM cjiyqae MO)KHo nIOJIO}KHTb 1 (s-1 + r,
rjie =0 HAH 1, r < s - 1. Torjta 6wl Hmen7o meCTO:
Ho 2 2- 1, 6yJayqti me~bwe no BeJIH4HHe, q4eM 2,HeM)8Tx
AIITbCHf Ha 2s - + 1.
13 'qe6OTlpeB.