Osnovy teorii Galua.

r J1A BA 11.

rpyHHA rAJIYA.
~ I. Hcxoqwi~e (j)aKTbi.
B 3TOM iiaparpa(~e Mbl HaVioMHH4M HelRoT1Tie ()aKTbI 113 o6umero Kypca
azwre6pbi, Ha KOTOpbae HaM BnlocJIeThBH14 HpI4LaeTCff cc-bInaTbCg.
1. KOPHHi IIOJHH~oma. Pa3AoacenHe I1OMIHHoma Ha JIHHeHiHbie MHOW*HTeJIH. B a.Twe6pe OCHOBHOA CqHlTaeTCII caieuyiowaff
Te op e ma 38. BCalKoe ypaBne~iie
(1.1) ~aox" + aixn-l +.+ a ~x+a,7=O
C BeI11.CTBeH~bIMKHAIJH KOMF1J1KCHbIMH4 KO34xPiu~HeHTamvi iimeeT no
Kpatifteil mepe OJLHH H 0 p e H b, T. e. TaKoe 3HaqIeHiie riepemeHHOl x,,
KOTOpoe, 6yJgyqI HOJLCTaBJ1eHO B ypaBHeHHe (1.1), o6palmaeT er'o jie~yio
tjaC'r B HYJ b.
JiInI JLOKa3aTe11bCTBa 9TOtI Teopembi raycc (C. F. Gauss) BUIepBbe fnpeJLJIO)KHI qe~blpe AOKa3aTeqlbCTBa. Bce 3THI 4OaaeiC~ 0CHOBaHb1 Ha H p H1 HII HH eH eH pe pbl B H 0 C T 14. CTpOro r'OBOpa, 3Ta TeOpema npHaHHa1eJXHT He
anrwe6pe, a atiaJIli3y. CoBpemeHHbie aJwre6paHCTbl, >Ke.1a51 136e)fcaTb BBeaTeHiisl
an14gIXJu5 airwe6pbx OCHOBHbIX HIOH5ITHfl, HJIH BB OALIT cy11LCCTBOBafHHi KOPH5I
B BHae~ aKcHombM (RioKa3bJBaq OTCYTCTBHe B Heft HIpOTHBopeqHfl c JLpyF1MH
OCHOBHbIMi4 aKcI~omaMH4), full pacapiaO ypaBieHfll, KaK c, aBFHeim Ho
C;ICTeme 4)YHU[lHOHaJnbHWX moay.nei4. BHocJneJCTBHH (B ~ 4) Mb! OCTaIHOB1MC5]
Ha n1oc11eoHem noJapo6Hee.
T e 0 p e ml a 3 9 (13e3y). Ecurn ypaB~ernie 1(x) 0 HmeeT Kopuiem
4HCJIO 2, TO HOJWHQOM f(X) JteJ1HTC51 Ha x -m.
J 0 K a 3 a T e.B b C T B o. Pa3,aenHB f(X) Ha x -, mi HoliyqHM 4aCTHoe
Q(X) 14 OCTalTOI I?, ROTOpbltt, 6y11y4q CTeHeHH HH)IC, tiem X -2M, He JtojDKeH
3aaBcemb OT A. 1IOJCTaB1l.92 B TOw)ieLCTBO
f(x)=jx- a) Q(x) +R
3Ha'eH~e x -= 2, Mb! HWIyqHM B ripaBoll 'iaCTH? a B niesofI liaCTH' Hiy~b,
TaK( KaI( 2 0CTb KopeHb ypaBHeHH5! f(x) = 0, a HOTOMY f(2) = 0. TaKHM
o6pa3oM
1(x) = (x -COQX
YpaBneHuie Q(X) 0 TO}Ke R.9WOJH0H B CHAJY Teopembi 38 HimeTb KOpeHb
H flOTOMY B C14ay Teopembi 39 TO}KC LeuHTC51 Ha COOTBeTcTv011414flH JIHieflHbUR