Osnovy teorii Galua.

If 1. Ocnodnuie nonH,vw us

19

'l 2 4 5 1 2 3 4 5 6\
YnHp aW HeHHii 3. flpoiBepllib: ( I1 ~ =( 4 3 2),( 3154
~(I 2 3) (4 56). (
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HOBKa eCTb flpOH3BQ1~HHi C0CTaBJ1511O11u4X ee Ui4HRiIB.
Y ip awKie H 14 e 4. ToaO~aTb, tITO Lt14K31bI, COCraBIisuoniye OALHY 1 Ty we noxaCTaHomyK (T. e. apweLHUiCHje o6uuix iutiqp), nepcCTaHoBO'IlbI.
co6oft nlOaCTaHOBKY, y~ejuituaj4Bomyio nw}jLpo widipy Ha eatHHHLy, rnpu ycJIoHHHI,
eCAH KpaTHOCTH 'IHcJ1 m mbi 6yiaem oT6paCbIBaTb: X — > X + 1 (m od in).
KnaipaT 3TOR H7OJLCTaHOB1H eeTb nO)1CTaHoBxa x -> x + 2 (mod in).
He TpyJLHo IOHHTb, 'ITO 9Ta rnocjieausiq I1OJCTaHHBKa COCTOIHT fl14 HCHCTHOM MZ
1(3 OA1HOrO LUMKJI, a npH 'ICTHOM M 14H3 )BYX [LHKJOB. '-ITo6bi COCTaBi4Tb
9TOT ILHRAJ (H1311 UHKnbl), HaAtO B pRuAY 1, 2,-..., in 6pamb Uw14puI qepe3 oiAHy.
TaKHtm o6pa3oM (1 2 34 5)2 -zz (1 3 52 4), (1 2 34 5 6)2 z1 3 5) (2 46).
Ky6 uiunuia (1, 2, m.,i) eCTb B1OjCTaHOBKa x — *> x + 3 (mod th), T. e. B
YAHKJIax 3TOPI nIOJCTaHOBK1 HaJLo 6paTb Utiepbl 143 paa 1, 2, m.,i xiepe3
ABCe 1 T. J1.
KaRie cUevncui m-jineHH-orO uiMKaa jxaAYT T>eJ4CTHeHH~fO noJIcraHOBKY,
T. e.- 6y31yT oc'raBJ114Tb Bce lL4lpbi 1, 2,..., in Ha Me-CTe? TaK KaIK n-ag
GTeneCHb 1114KJIa (1 2 3... m) n1CpCHOJ14T X H x ~ ni, npiW1eM KpaTHOCTH
'IHcaa Mn oT6paCbm~aroTC', TO JAl5 3'fOrO HeoftxoJLmo 14 JUCTaT'O'IHO, HITO6bI ni
AeJHIHIOCb Hia m. TaKUim o6pa301M n1opHuoK m-tmJeHHOrO 1114KJ1, T. e. noKa3IiTeJb Hai~ime~mefl CTeHeHH mrn-'~eHHorO ttiiKza, Jawo~uef TOwKLeCTBeHHYIO
nOJACTaF.oBKY, paBeH Mn.
Y~ a p a )KiiH CHe4 5. HlyCTm 11OJCTHOBica COCT0OHT 148 rni-, in2-,, MA-qnJeHHbIX ILHKJIOB.
XLOKa3aTb,,I1o nopqAoic 5TOii IJOCTaHOBKH CCM Hal4meHbuiee paT~oe tm1ce1 Mn,,
Y n p a)HC HeHCf 6. IlapalaofcC. Bce l'py~lflbl KOMll~yTaTfIBHbl..LI 0 K a 3a Te a bC TB0. flhpeaWapHTelrblIo AOKa}em, 'ITO eci aemeHIMCHb A., A2,
An C"I3a~bi COOTHOw~e'nteM Al A2... A,,=J, TO IIPOH3BeACH14 9TIX anleMeHTO13 B.'11060M HOP5UAKe paBHO TaR>Ke J. 9TO YTBep)KJ1eHi4e cIIpaH~I3CUIHB iipi n 2
(Teopema 3). llpeanO.10OCHM, 'ITO 0110 cHnpaCeaAHBO JRJ15 'l4l ejq, He 60JlbIfl~x n, H O
Ka~iem er0 A~ns n -j- 1.- TaK E~aK Mbl yi~iiJUm HimeKi ~ 2), 'ITO BCIIKaI HOJUCTaHOBKf' MO* eT 6blTI npeJACTaB~esa, KalK npOl3BCJeH~e Tp)a H C H 0 S 1H 11 H-14 T. e. nepe~CTaHoBOK
ABYX Llrn4)p, TO YTrBepA.afl~Hie 6yaieT A01a3aHo, eCC3H MhZ HOKalwem, 'IO 143
c3TeAyeT
BBOJ0AI 0603Ha'IeHHS Al.. Ai- 1 B, AfJ. A~ CAj... An+ = D Mb
NO-AY'IIM COOTHOluiHHe BAjCljD -J 143 MeHbUwero qiiCJza saemeHT0OB, a nOT0MY B
CHAY Haulero npe)U1AnoaoN(H1451 BA, CAiD =J~, 'I. 4 1. XL
Tenepb H1eTpyRH no Oila3ITb KOMMyTaTIHBHOCTIb HaUIC2 rpyTflnl. llyCmbAB C.
Tor~a AB80' J, H B ctuy1 AxoKa3aHHoro (no31arast n 3) nflyJq'Hm BAC'1 J,
*TkcyAa BA =C, T. e. AB = BA.
Mbi Bii~eJ1H OJ~aKO Ha npOCTblx npi~mepax (3), 'ITO MT YTBep)KJcif1e HeBepno.
B~ 1 paccyw(seHHH iJe}KHT OWn6ka?
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