Osnovy teorii Galua.

21t

Teopu.q quceA

1fl TeovuR uce

H TOIHO, TaK weI
a22 1 (mod 2'), a23 1 (m od 25),.. a2 -'-~ 1 (mod 2")
HO TaK KaKcp (2) =. 2'-1 (2 1= 2-1, TO OTCIOJaa cJaeayeT, MITO
CpaBnei-ne
(4.12) x2"'1' 1 (mod 2"
11PH V >- 3 He HmeeT nepBoo6pa3Hbix KopHefl.
-HiCJuIO 5 HBh1HeTC5I,gAl aTror cpaBHeHHq ronly riep B oo6 pa 3Hbim
K 0p H eM, T. e. 1HPH3XaJ1neKHT K flOKa33Te1IO 2" -.B camom xiejie,
62 +22 2' 1 + 2). 22 + 1.~2 1) 2 +...~i+ 2x (mod 2~)
HlojioKHm ~ + 3 = v. Tor~aa
52"'0~ - I + 2"' ' (mod 2"),
OTK3ULa cJaeJyeT, IlTo 5 H~lHHaxUjie)KT no moxiyrno 2' K noK33TeJio0 p"
TaKHM o6pa3oM Bce Bbl4eTbi no0 moayjit 2" MO)KHo npeAICTaBliTb TaK:
1, 5, 52,...25
-1, 5, 562,, -5 -2
6. B cnryqae m= 4 ~mi Hmeem cp (4) = 2, 14 e CTb IlepBoo6paaHbIRl
KopeHb cpaBHeHHH
(4.13) X2 -I aziO (mod 4).
7. ECJIH M coJLePWHT XOTfI 6b1 /LBa pa3X.nHtlbix npoCTb1X 'iicna H eCJ1H
02W0 143 HHX (B cnyqae JLByx ml~ceii) eCTb 2, HO coa~epWKHTCH B in no
Kpf~f mepe BO BToOPOR CTerHeHH, TO MOayllJb M He HmeeT rnepBoo6pa3Hblx
K{OPHefl. ULeA1CTBHTeJlbHo, riYCm
m =2` pi `pa 1... Pk k,
rJ~e PiD -P21.. Pk - pa3J114qHbie HeeTHbie IlPOCTbwe 'lHcJa. JJA1~J Kawaoro
qHcjja., B3aHMHO-nIpoCTOrO C Mn, k:i~eloT meCTO cpaBHeHiiq
a (2)1 (mod 2"'), a (11z1 (mod plW'), a. a -~k") (mod p."'k)
flO3TOmy eCaHn M eC~b o6alee HaHlMeHbA~ee KpaTHoe q~ceni p (2"'), cp (1l
T (pT),c (pkwk), TO HMieeT meCTO cpaBHeHHe
aM 1 (mod in).
Ho nPH co6.atroxeHHH HaIIIHx yCaTOffHlt cpejRH 'iHcen cp (2"'), y pl)
p(P2'0),...-I T(rPkok) HaflloyTCfI nO KpatlHefl mepe xi~a 'ieTHbIX [cp(20)=
= 2 1 rpiCH o>~2 ii rf( p" -1(i- npi a BCH KOM wj, a I1OTOMY
H~X ofigee Haime~buiee xpaTifOe Al meCFibu1 npc'Il3Beae~t.iWi? (2"') j' (pi""') y (p2")... T~ (pk~k) = cf (2"' pi'P"ps"... p,,'k) = p (in),