Osnovy teorii Galua.

1. Fpynnbi

YMHO>KHM 3T0 PaBeHCTBO crnpa~a Ha X: XAX = BX, T. e. X BK.
YMHOKHiM cnpa~a Ha 3JiimeHT Y, HpaBbffi o6paTHb~l ic X(X Y I ): X Y = BX Y
T. C. J= B, OTKyJaa B CHh1y (1.4)
XA =,
'ITO H Tpe6OBanlOcb JaoKa3aTb.
9. T e o p e M a 4. He moYKCT cyIIICCTB0BaTb JaBYX pa33JiH'HbIX
ripaibix o6paTlibIX ij~emeHTOB.
oEtoxa 3a T eJb C T BO. LtoIOFycTM, '-ITO HMCCT meCTo AX =A Y =J. ALlOIxa)Kem, ']TO X = Y. YMHOMHKlM paBeHCTBO c an e B a iia X: XAX = XA Y, OTKya B CwnY TeopembN 3: JX=JY. Owcio~a B CHi.'Iy Teopemb1 2: X= Y,
4T0 H Tpe6oBanTOCb LioKa3aT!,.
10. KoiieqHbie rpyalrbI. B Au3JbH~iieIICm mN 6yJaem 3aHHimaTbCfI TOJ1bKO
TaKHmI4 rpyHinami, KOT0pbie coJtej3YKn KOHeCMHoe MiHcJo 3JCmeHTOB. TaKue
rpytrinbi H-OCRT Ha3BafHHC K 0H etl4H h XFp y Hn, a 'IIICjIO coaeCp}KLLumxc5 R
TaK0g r'pynne 3.eICmeHTOB - HO p 51 JL 10 M r p) y H H bi.
Kpome nopsiuca rpynrIH~, mL! 6yaem HiMeTb aeno c nOHIITHicM n o p %A.i K a
a a1 e M e Hi T a r p y H H bi. Ymfiowasi 3JICmeHT A HeorpaH-iieHHoe 'uicqio pa3
caM Ha ce6.q, T. e. B,03Boa%5 A B C T eH ei14i: A, A2, A3,7..., Mbl B ClJ1y
KOHCMIHOCTH~ fpyi7Irlb1 He cMio)Kem nofayqaTb BCe BpCMS1 pa33jIlIHbIX gdiemeHTOB.
TascHm o6pa3oM rnonyiyaembie 9JemeCiITbl ftOJDHbl HOHT0PSiTbCSI, T. e. mN 6y~tem
nony'IaTb paBeHCTBa TIM3 Al"1 + h~ Al, riqe k > 0. YMHow~a5I cnpa~a Ha gnemeHT, o6paTHbltt K Am, HO.TUy4HM: Ah I. HaHmCHblI~ee 143 4HCceii k, )aioiu~x Ah~ =1J, Ha3bLHaeTC5In0 HO51JLKOM 3JICmCeH T a A.
Y np aWH e H He I. ]IloKa3aTf., 'ITO BcC 3Je~meHTbJ J, A, A2, A.. Ahi, uie k?
IIopHJLoK 3aCmeHTa A, pa3JIlH'HbI.
Y ni p a 'K i e ii H e 2. RLoKJ13ab, 'ITO paBeHCTBo A' = A'MO>KCT HmeTb mCCTO B
TOM H TOJ~bKO B TOM cllytlae, ecJiH pa3H0CTb rn-n iLeJIHTCSI Ha H0psiILoK epneUeHITa A.
1 1. flpez~lCTaBJneHHe flO)CTaHOBOK B BHae IWAH~OB. PaCCMoTpHiM KaKytoHH16yiab nIOiCTaHOBKY, Hanp. 1 1 4 8 6 9 2 5) (3iaeCb BmeCTo 6yKB CO
3Ha'IxaMH, HaI1pHmep xl, X., X3,..., Mbl HIPOCTO CTaBHM [lHJpplI, 1, 2, 3,.
B Heft LLI4pa 1 nepexo1JMT B 3, 3-B 4, 4-B 8, 8-B 2, 2-B 1
3Jteci6 Mb! npiHH1H OIIIITb K IICXOJIHOtI lLcppe, 'ITO mL! 6yiCM BL1paWKITb CJ!OBamH UHL 1IK A 3 aMHK H y AC$ 9 Ta Kpyro~a5i 3aMeHa iLH4p Ha3blBaeTC5I
IH KnAoM H o6o3Ha'IaeTC51 TaK: (1 3 4 8 '2). fHpo~on~xaH paccy)KaeCHHC Hfia
uiHc~amH, BC 3aTPQHYTINmH npeC2LUbLajjiHmH LL4KIaMH, MNl Hony4Hmee CflCUKJAb
(5 7 9) Hi (6). TaKHM o6pa3oM mbi 6yiaeM 3aflHCbl1HTb Hawy~ HOLLCTaHOBKY B
IXH K ia X TaK: (1 3 4 8 2) (5 7 9) (6). B 3TOM CJ1iyIaC FOBOPH-T, '-TO fTOiCTaHOBKa COCTOIIT 143 5-'IaeHHoro, 3-'i~eHHo ro H4 1-'IACHHoro wLHKAoB.I-JICHHble UHKJILI, OCTaBAI5ioIJlHe uHJ4Iy Ha MCCTe, o6b1'IHO HC riHHflyTC5I.
HeTpyiLHO TaK}Ke nepeCITH OT LLIIKJIiHHOrl 4j!OpMbf H0iLCTaHOBKH K HepBOHaqanbHO~n. TnyCTb Ham jaa~a HanHtHmep rIOlCTaHOBKa (1 3) (2 4) (5 6 7).
1-43 oHpeJLenIHIH5 iHKia~ ceaCJyCT, 'ITO B Heni 1 nepexoIHT B 3, a 3 B 1 (noceAC~LHSI 1Uli4pa imHKa~ HepeXOalHT B HepBylO), H T. It. HOaTOMy (1 3) (2 4) (5 6 7)
(1 2 3 4 507
~3 4 1 2 0 7 51