Osnovy teorii Galua.

54

54 'I.~~~~~- Jrpynnbl

rlepe~ymepyem o6a 3HatiKa TaK, 'ITo5b1 I102CTaH0BKa C1 yBeJ1HqHBaJI
Kawxbifl nepBbIft 3Haq0K Ha ealifHHU~t (ecaHi nHp BTOM oT~pacbIBaTb KpaTHOCTH titcna p); TOlqHO TaK we~ C2 AOtJI}Ha yBeatJHt-H4BaTEb 1aKwubifl BTOpOfl
3Hat4OI Ha e2~uHHy. Toraa flOaCTati0BKt C, H C2 moryT 6blm I1peJCTaBjieHb1
TaK:
H- TaKHm o6pa3oM Kawavu f1OJLCTaHoBica rpyimlw4 2 HnpeACTaBHTC.9 Tax:
C= (ac2P, =O, 1, 2,... p- i).
HpPH TaKom O6O3HaLqeHHH nepememiaembix npeXLmeTOB Mbl M0>Kwem aH3a1HTH4ecKH HnpeRCTaBHTb flOACTaH0Bl(H rpynnrii 6 B BueL A I X )
CP (X, Y) H ~(X, Y) fIO.THHOMbl OT 2LByx fleemeHHbJX X, Y. YCJnoBHe Toro,
'4T0 2 eCTb HOpmaJlbHblit JIaeHTe.b rpyrnniA 0, mo>KeT 6bwTE 3antcano TaK:
C1A = AC12E1C2 1,3 C2A = AC,`C2pll,
T. e.
X*X)(X- ~(XI Y) - X ) t
tP (x+ 1, Y) cp (x, y) + cx,) t(x + 1, y p(x, y) + P, (mod p),
CP (Xly + 1)c(X, Y) + a2, (1Xi Y + )~(Xi Y) + P2 (nmod p).
Paccy>IcxaaH KaK B ni. 9, mbi y6eatimco, MTO CPYHKIWH (p (X, Y) H ~i (X, Y)
wiHeltiHbl OTHOCIHTeJlbHO X, Y:
Tp(XI Y) L-XlX+cM2y + all '(XI Y)==PlX + P2Y + P. (mod p)..Jla~nee Mb! JAOqWiHb HiatTH yCJ1OBHq1, npiH KOTOPIX~ ALOrIOJHHiTealbHaR
rpynrna (,/2 paapewuuma. HeTpyAHo y6ea~MCR, '4T0 B KaqeCTme npeA~CTaBHTenefl rpynnbi SI/, (T. e. MHOIKHTene-4 A, conpsH}KeHHHX, CHCTem 2 Aj) M b
molweM BbI6paTb 0JLH0P0JHbie TH HeflHbie UOJACTaHOBKH, T. e.
notCTaHoBKH THua
Y -> ix +13,y
[lOArHafl rpynria TaKHX U0JICTaH0BQK< (T. e. rpyrnna, B MemeMHTax KOTOPOf4
K094)3~4H1lHCH~ al, az2, PI, fP2 nIpHHi4maIT BceB03MOWKHbie 3Ha'ieHfflq, JHiflb 6w
aI27'P 0 (mod. p), qTo Heokxoluio aqqI o6pamHMoQTH HoACTaHoBoK)
HmeeT IIOpSILaOK (p - 1)2p (p + 1) 1i Hepanpewiuma.
16. Bce BQ3MO>KHbie TlH HEb pa3petuHmbix OJLIOPQZU~bIX JIHHefiHbix rpy'iII
OT 2ABYX llepemeHHbIX (T.x e. TaKHit rpynnbl, 4T0 BCRKa% paapeutm~as rpyrina
R1BafleTCR IueJIHTeJnem 0AHOfl H3 rpyfni-TriQnB) 6wiml HallAeHbI K. >KopaTaHOM.
M4X Bcero TpH:
IT iin. Ho~2rpynlna STorO THITa mo)HeT 6MlTb o6pa3oeal-a KOmno3HILHef caTeJLYoULHx HpOH3BOJgLUIHX HOAICTaHOBoK: