Osnovy teorii Galua.

50

I. rpynnbt

rHoACraBA'IH5 cio~a k = — 0, 1, 2,..,p-l noiyqtmM:
8. rlp H1m e p. p = 5. 3aHymepyeM nepemeuwtaembieUHm~b1 TaK: -2, - 1
0, 1, 2. Toraa F(x) =-(x +2) (x +1) x(x- 1) (x-2) ~-x5 - 5x3~
+ 4x =-x5 - x(mod5); F'(x) - 5X4 -15X2+ 4z~ 1 (mod 5). HlpeJLCTaBHiM
aHlh1HTH4eCKH TpaHcI1O3HLHIO (-,1)IJIRI Hee mbI Hm~eem
f(X) =(X5 X){x ~ = 4X5 -1 2
+~-4-~}=~(x5-) {X'.~-1 + 2'x3 x) + 2(xc3 + XIV-=X3
(mod 5).
TaKHM o6pa3oM (- 1, + 1) = (x -> -x3). Hloio6HbzM we o6pa3oM Mblu
y6eIaHmc, tITO (- 2, + 2) - (X ---- X3). Hpon3Bexaeffe 9THX TpaHcHo3HLLHiif
eCTb (X- (X3)3 -= (X- -X9). AIen1H X9 Ha X5 -x, nIoji4Hm x. lo-3ToMy
9. MeTaU.HKuIHqecKHe rpylnbl. BepHemcfl orfiTb Kpa3peiuHmofl rpynne (
HPOCTOA CTefleHll P. rpynna SP, o6pa3oBam-Iasl nOJLCTaHOBKOfl C = (X -~ X +1)
jaofiu)*a 6bJTb HopmaJnbHbIM ejiHejr~eiem rpynnbi (3, a HOTOMY BCHicas HOJLCTaHOBKa A (x — f(x)) rpynnui (b JOJDKHa npeo6pa3oBUBaTb C B KaKpIoHH6yJab CTeHeHb C, T. e. AJ1OI)f{HO HmeTb meCTo A' CA = C~, IH~i CA AC~.
Ho C~(x -'>X +ct), OTK~XJa mbi iojly'aem:
(x — >x + 1)(x -— f(x)) = (x — >f(x))(x -— x + a);
HO OTCioJLa cileJayeT:
A~x+ 1)if(x) + a (m o d p).
flpuJaaBaff nepemeHHofl X nocJe ~~J~~ H4HHR 0,1,2..,P- 1,
noJRyqHM:
f(1) f(0A) + 9., f(x);: —f(0) + 2 a.) f(3) -; (0) + 3c,.,f(p- 1)z=f(0) +
TaKHM o6pa3om aJlJH BceX 3Ha'IeHHtt X Mbl Hmeem:
Afx) = otx +1f(0) (mod p),
T. e. noniyqaeTC5u, 'TO Bce n1OJCTaHOBKCH Irpynnlbi ( MO)KHo, HpeJLCTaBHTb, KaK
a HHeftHbie flOJCTaHOBKH Tunra (x ax + b). Iti TOro, 'ITOGIA
alaITHqco npeo6pa3oBaHue (x — > ax + b), riae a ii, b - juejiwie pauiioHaJlbHbie 'ucjia,, AIaBa~o, JaefCTBHTehIbHo HnLLCTaHoBKY, T. e. J-Hi pa3J1H4HbIX
3HaqeHHit X ja~ano, pa3.JHLIue 14 HecpaB3HHmbie 1n0 MOJyl~O P 3HaqeHHR,
Heo6xoJJaHMO H JtOCTaToqHFO, tITo6bi a He AenlM~OCb H-a p. ~JaBan K094)CI)HuHeHTy a 3Ha'IeHHI 1, 2,..., p -1, a KOB4)4PHiiHeHTY b -3HalIeHHH 0, 1,