Osnovy teorii Galua.

168

158 ~~~V. Ypa,6enenw C 3adanntbIu opynnawu

14HcAo f BnonTHe onpuexqe~niec apwmeTH qecKoll nporpecCffet4 1 + m
B KOTOPOR JIeWHT npoc~oe 'qicao p. B camom JaeJie,
(b + mY bt ~V(mod p).
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eCJIH p -= I + M TO Xm paunaraeTCHi no MO]aAilO p Ha AHHeftHbie MHO)KV,Meuf; ecJiff we p g -1- m ~, ivie g OJWH ff3 nepBoo6pa3HbIX KopHeR cpaBHeH111 X' 1' (mod in), To nOJIHHOM Xm OCTaeTCSI H nO MOILYMO p HepHpfBoJaHMb1M. H~pH 9TOM Heo6XoJaHmoOoTmeTHTb,7 qT0 repBOO6pa3Hbie KOPHH CyluwCTBYIOT He AL~n Bcex moatyJeft mn, a TOJIbKO.Ruif: 1) C~erne~ef HieqeTHbjX npoCTbIX qHceii, 2) yaBOeHHbIX OTefleHefi HeqeTHbix npoCTb1X qwicen u 3) Lcuica 4
(cm. ]Zjo6aane~ffle). B CBfI3H C STHM ypaBneH~ie (2.1 1) moNKeT 6MlTb ljlHKJIHq1e_
CKHM TOJ~bKO ARl5 9THX 3HaqeHHfi qmcnla mn.
6. lpeJRCTaBJiSqeT HHTel-eC BOHlPOC, Ha HenIpMBOJW~mbj MHONKHTeJ1H KaKH~x
CTenefefet mo}KeT paa.narambCf JleBafl qaCTb ypaBHeHH51, paccmaTprniaemasi KaK
cpaBHeHHe nO BceBO3MO)KHhJm np~oCTb]M mQayaHqm. 111J5 cniyqasi ypaBHeHffff
jaenieHHRs icpyra OTBeT rionyqaeTCi ff3 cae~aytoui~eft TeopembI UthpHxhe (G. Lejeune-Dirichlet, 1837):
BO BC5KOR1 apff~meTffqectoil nporpecciiH 1 + in ~, rite b H in
B3aHMHOHlPOCTbie, iieJIbIe paijHfOHaJlbHbue qlfcjia, 1ie>KHT 6ecT4icaneHoe
MHo}iKeCTBO rHPOCTb1X qHceJI.,flns ypaBHeHHtk we o6at~eFo Tfffna BOnPOC peineH- (Dpo6eHffyCOM (G. Fro.
benius), aloicaaaBWHfM cnextyioityio Teopemy (1896):
ECIH rpynna ypaBHeHH5I 1(X) =0 coaepw)HT noatCTaHOBKY, COCTOSIIIIYIO ff3 ILHKJIOB HO051JKOB fl1,n2 ** 1nk' TO cyI1e2CTByeT 6ec'lfcJieHHoe MHO)KeCTBO TaKHx npOCTb1X '4ffcei p, 'ITO cpaHHfH e
f(x) 0 (mod p)
pamaraeTcsi Ha HefHpHBOJLHmbie no MORtYuO p MMHO~fTejnff C~eene~e
n,z1 n2, nkc
3Tm TeopembI JLoKa3bIBa1OTcsi nPH lomoLtiH nHeHmOB aHaJ1HTff4qeCKofl TeopK14 qffce.T. 3JaeCb Mbl He HmeeM BO3MO)KHOCTHI 1pHBeCTH ix iLoKa3aTejlbCTBa.
YuHp a x H e H Hi e 40. ]viao ypaB~eHsse Atieneffl~ Kpyrax, X15. Onpeigeaffb,
KaF.HM JnoJCTaHOB~am COOTBeTCTBYIOT nJpOCTble 'qHcjia, jie)Kcaatie B cauptYoULx apHi4PmeTH4qeCKHX rlpor'peccHIx:
~ 3. Hoc'rpoeitie ypaBHeHHii C 3aaaffHiblitf rpyulfwaMH.
1. B CoBpemeHHOfA TeopH4H Fanya o'iepeAHiOR 9BJI51TCH cJ1efLp0LLLaf
npo6n'ema, pewueH~e KoTOpOfl npeaCTcaBAt9T ito CHX nop Henpeoitonrwmbe
TpyJIHOCTH:
3aitaala rpynna 0, npe~iCTaBnlsiemas, K!1a( rpytina IIOJLCTaHOBOK n4k
CTerieHHf. Tpe6ye'rcs Hat4TH ypaBffefffe, rpynna KOTOpor 6una 65i
H3omopfHa c ~3.
3Ta npo6Jlema B HaCTosiuee Bpemsi pewiena 1AAlR HeKOTO~bNX liaCTHb1X
BHJAoB rpynn Q3, Hanpnmep Jal5 CffMmeTpff'iecKffx, 3HaKonepemeHliblx, a6ejie