Osnovy teorii Galua.

2nd

Teopun quceA

204 TeovuR ucevl

3. ByateM~ Ha3UBaTb coB~oKyUIIOCTb 4HceJI, cpaBHHmbix n HoaJyAIo, m,,
K jia c c mc Cpa BH eHH1 HA O MOfIYJ1I m. RJUL5 KawaToro moJayJI5 m oqeBHA4HG
CYa~eCTmyeT m pa3AilHblHX KJ<aCCOB. B K~aJLom Kiiacce MOWHO HaRtTH n&,
QALHOMY H TOJlbKO OAHUOMY npealCTaBHTeJio, X, KOT0ObIA 6bin 6w noaquineH
O <X <iM.
TaKHe HnpJtCTaBHTeJIH KslaCCOB flOnJ14IiH Ha3BLIHHe B blqe T 0B HG
Bce Kiiaccmi cpaBHe~ffifl rno mo~ayJio in o6pa3yWoT rpynny m-ro, nopsaLKa
OTHOCHTeJ~bHo CJIO>IMH-5I. fHpu onepauwusix c qTIemeHTamtH kTOl rpynnw MoKHG
HMeTb aeno, H-e c LLeJbAMH KJnaccaMH, a C Bbl14eTamiMK AMir KJaacca. Ho npti
cJIoKeHHH AJBYX BMlqeToB mo)I(eT y)iKe He n1OJIyqHTbC5I BbliqeT, H TOr~a noCJIe
cJIow*eHHLI Haio OT6pOCHTb (Himn pHlaa~b) OT cymmbi I{pTHOCTb MOAYM15 in
TaK, '-ITo6bi oHnITb nwjlyqtlHJCR BbltqeT.
ECAH IBblqeT B3aH4MHO-HPOCT C MOJL3Je~M M, TO H Bce BXOJL5ILLHe B TOT we~
Kicncc MI~cna B3aimHio-rnpoCTbl C M, ii o6paTHO. 1`OBTOMy Mbl mo)Kem rPOBOPHTb
0 KJiaccax, B3aHM H0-rpoCTbIX C moayJIem I. TaKHX KiiaCCOR
o4eBHJLHO cp(i) (T. e. CTOJlbKO0 w*e, CKOJlbKO B3aH4MHO-rlpoCTbIX C M
BwlqeToB). OHHl o~pa3yIOr rpynniy OTHOCHlTeJlbHO YMHo}KeHHSI. aLeflCTBHTeJlbHo,
eArHHCTrneHaoe, B 'em HiyiKOo oco6o y6eauiTbCsI, 9TO cyiueCTBouaniie o6paTHwx
9JnemeHTOB. Ho oHiI nioqyqa'ioTC5 KaK( peawensil cpaBliieHHSI
ax 1 (mod in),
rae a - 3aaaHnbti~ BblqeT. H~pli 9TOM pew~eHH5I COcMBJUIOT eJLHHCTBeHHbIA
KJia~cc TaK KaK 143 ax 1 (mod m) ii ay 1 (mod m) mu lo~nytiaem
a (x -y) -O (mod m), OTKyJaa B CHAY VI HMeem x- y =O (modin).
IlyCTb a HlpO143BoJfbHoe LIUCJO, B3aHMFIO-npOCToe c in. Ero, C~erneH o6pa3Y!0Tr UIHKJIHqecKyIo noiwpynriy (OTHOCHTe~bHO YMHoHweHHII) KjiaccOB. HJOPHIaOK
3TOIR HoJarpynb1 pai~eH HaimeI-bHmemy 1oKa333eJIio f, iawoaemy
(3.2) a' 1- (mod tn).
BCHOMHiHaff Teopemiy JlarpaH>Ka (y Hac - Teopema 7), Mwi y6eawmcg, 4T0
IIOPR1JOK n 3Tofl awHJIJHqeCKOfl rioarpynnu eCmb lejiHTe.Tb nopsu~ia (p (in)
Bcef rpyrinwl B3aHMHO-npOCTbIX C Mn BblqeToB. HlyCTb cp (in) = n. q. BOMBARs
ole tjaCTHi cparnienimi (3.2) B q-io C'reneub, Mbl B cwny XIX 6yaem HmeTb:
Hs TaKIHM o6pJ3OM Mbl flHXHOIWM K cJaeapyoueL Teopeme:
XXI. (TeopeMa 9flaeplI). Ecmis a B3aHMHo-npOCToe c mf LIM10I,
TO H-meeT MeCTO ~ paBHeHHe
Bepff B KaqeCTBe Min HOCTOC qH1C]IO p, TaK t1To 6y.zeT liMeTb MeCTO
p (p)=p - 1, Mb! noJny4im:
XXII. (Teopema 4epma). Ec~ni (a, p) =,TO imeeT meCTO:

(3.4)

p - I -
a = 1 (mod p).