Osnovy teorii Galua.

162 V.Ypa~tuenuR C 3alOQanuuIA epynnamu
OTKYXIa B CHT1Y (1.10) BbITei~aeT:
B 'IaCTHOCTH,
OTKyJLa cJielayeT, qTO ypaB~eHue X M1- 1 0 HenpemeHHO coaepioHT
rnepBOO6pa3Hbie KOPHH4, q. 1 T. AL.,CqAe ItCT1B 14e. rpynna OTHOCHTuJbHO YMHO)KeHHB1 B KOHeIIHOM
nione Bcer~ta LJ.1KniiqecKasi.
13 CMOM wneB~J14 H~I 2 3 PM-2
camom ene BwqH 1, a 2 a I,, rxte OC BeHe~qHHa HpHHaJVIeKwfLuag K HoKm3TeJ1io PM - 1, Bce pa3JII4MHbJ, a nOT0MY 11c4qepb1BaIOT
CO~OOA Bce OT~llyt1Hbie OT HyJAfI 3aemeHTbl l0J114.
9. 143 cyL1ueCTBOBa1Hlif BeJHq4HH, I ipHHaIaneKaiuIx K H0Ka33uTnJ1 PM1
-T eop e Ma 87. 11onTHHOM X P -X HlenpemeHHO oaCJepKHT HenpipBoatimbie no MOJyl~AO p MHO}KHTiJeIH CTeHeHH m.
]IIROK a 3aTenAb C TBo0. Ecirn 6bi nep~oo6pa3HbIfl KOpeHb OC cmene~Hl M - 1 pJoBAeTBOPRA HeInpHBoA11m0my ypaBHeHHIO CTeneHHH it<in, TO
B MY4A Teopembi 82 OH yILOBAiTBOp5l1l 6bl TaKme ypaBHeiimlo x n-_ 1 = 0,
LIT0 B CHAY pt- < p m HeBO3MO}KHO.
10. flPHpime p 1. Bo3bem~ p 2, m =4. Torxta rioJIHom xP' xX1 x
pa3.qaraeTCSI Ha MHO1KHTeCaf ceApA~owiJm Q6pa3OM (CM. rqa~y III, ~ 2.2):
~ X2 + X + 1) (X8 - XI +X5 -X4~ +XI' x +1)
MqTo6b y3HaTb,, Ha oHHOIRombi icKlKX C~ene~eH OH pa~aaraeTcA no MOILyJUo 2, npH4 -meHHM IC Hawe1my caywao 4bopmyjy (1.7):
2= g(1).
OTciojia g (1) =2, g (2) = 1, g (4). HiHeA~itbe HOAHHOMbl TaIKOBbI:
& x, x - 1.
HenpHlBO~auMblM H10 moJLayo 2 HnAIHHOmom 2-fi CTeneHH-MOWUe 6bImb TOAbKO
X2+X + 1.
HaKo~eu ~Aim nojiyqeHH5 BCeX TpeX HeIIPHBOJXHMbIX fIOJIIIHOMOB 4-Ht CTe1JeHH
Haao paI3AO)KHTb Ha1 MHONK HTCAH HOAHHOM X8 -XI + X5 - X4 + X3 - X + 1:
Xl-X7 +X5 —Xi-+X3-X ---x~I(X4+x3+ 1)(X4+X+1) (tnod 2).
)1A14 n1onyqHH5I HeflpHBOJAHMbIX no moJLyJIio 2 HOJIHROMOB 3-i' CTeneHH p~asOwKHM
nIo.TiHHOM X8-X Ha HenlpHBOAiHMbie no moyjiiAo 2 MHONCHTeJIH:
X8 - X X (X-1 (XI,+ x2~1) (x2 + X~1) (mod 2).
06a IIonyqeHHbie nOnHHoma 341 CTeHeHH Henp~HBOJLmbi no mdojXay 2, TaiK tcaK
s3 4~opmyJIbi (1.7) mb! noJlylMHM:

8 = 8.- g(3)~g(1),
2 = g -

OTfKy2aa g (3) =2.