Osnovy teorii Galua.

~ 1. Ocuiowbie nouamuq 1

1 7

HHTepeCHO, WIo ecnii~ 3jie~meHTbl KaKofl-HH6y~tb a6CTpaK(THOR1 rpyruIrb noA~tlHH5UoTcsi acCOLLHaTHBHOMY 3aKOHy, TO iMX mo)Ho paccmaTpHBaTb, KaK HeKoTOpbie upeo6pa3OBaHlig. 3iaeCb MbI He 6yatem AIOKa3UBaTb BTOFO npeianoweHHH1 B o6uweM BHJae.
5. B Ka'ieCTBe eJLmtHHibl (KaK nlpaBOf4, Talc H JneBOA1) B rpyrine npeo6pa3OBaHHf! CIIYNKHT Talc Ha3bIBaeMOe T o e C T Be H H O e n pe O6 p aaO0BaH H e, OCUaBJIII)tomee lKa)KJablif aneImeHT NiHo>KeCTBa T Ha meCTe.
Hpeo6pa3O~aHHe, O6paTl-ioe JaaHHOMY, flojly'aeTCH Taxc: ecJIH XaHHoe ripe-.
o6pa3OBatiHe HepeBOJLtHT AM H M', TO ot~paTHblM nipeO6pa3oBaHl~eM Ha3bIBaeTCR npeo6pa3OBaiHne, nepeBoa~iuiee M' B M. Harp., eciiii A (-=XI X2 X
To rnpeo6pa3OBaHI-e, o6paTHoe A, nepeBoaHTu X2 B X1, X B X2, XI BXS.
TaKHm O6pa3oM OHio eCTb X 3X ilIX 3
I X1 X3/ X3X X2!
3ameTliM, 'ITO B 3ari7HcH nloauaTHOBKH MOKHO pacCTa~i3Tb BePXHHte 3Haq4KH
B nto6o~i nOpgIJKe, HO TOJlbKO npH TaKHX nepeCTaHOBKaX HH)KHIP 313'IOl(
Jaojiwe-H cJiieaoBaTb 3a CBOHM BepXHHAI. Hanipi-mep, (3X iX) (ix 3x)
XI 2 x4 X3 x4 x3x XI
6. Teopema 1. r~pynria coJiep>KHT He 6ojiiee QLIHOfl npaBoiA ejiuiH HUbl.
J.K a 3a TeJA b CT BO. HYCTb J H J, 6yAlyT JBe nipaBble eal1HHH1Lbl
rpylnifb, npH'lem nyCTb J 6yiaeT Ta eJaHHHI~a, KOTOpaff Tpe6yeTcH aKcHomoft IV.
I4meem:
YMHOKHM o6e 'iaCTH cripa~a Ha J.4: J1JJ1.41 J.J4==. YMHO,-cIHM o6e
'IaCTH cnpaaa Ha 9J1emeHT X, npaBbliA o6paTHblAIC ic., T. e. TaKcoft, 'ITO
J1X =J: J11J1JX =J1X, T. e.
(1.2).41=1-.
CpaBHHBaHi (1.1) c (1.2), rlonytiaem: J, =-J, oTxyita BbITeKaeT, 'ITO o6e
eJ~kHHHUbl He MOr-yT 6blTb pa3nIH4'Hbl.
7. T e o p e M a 2. Ilpa~laii exJiHH14a oJUioBpemeHHO 5!BJIfleTCff H
JleBofA eaLHHM~tefl
312 OK a 3 a T enJ b C T B 0. JUiOKa)Ke-m, 'ITO AJI% BCHI(oro 9.TemeH-Ta A rpynrbi
HimeeT meCTO JA = A.
BBeTi~em o6O3HaIeHn~e
(1.3) JA=-B.
YMHOIKHM 9TO paBeHCTBO cripalla Ha BJiimeHT X, nlpa~blfl o6paTHbItl IC
A (AX = J): lAX= BX, T. e. J I- BX. TaKHm o6pa3om AX B. YMHOicaq cnpa~a Ha 3J1emeHT Y, npaBblfl c~6paTliblfl KC X (XY -J), IIOny'IHM:
AXY ==BXY, OTlcyJaaA=r-B, T. e.
JA =A.
)LUoKa3aHHoe jtaeT Ham n~paBO Ha3b1BaTb 3JIemeHT I n~poCTo e A H H H1 i efl
8. Te op e Ma 3. Hlparndg o6paTlabIfl gJ1emeHT SIBJsieITC51 B To )Ke
BpeM% H JleBbim o6paTHMlM 3JnemeHTOM, T. e. 113 AX = J cn'exayeT XA = J.
lll2 a 3a T eA b CTB O. BBeJaem o6O3HaqeHHe
(1.4) XA=-B.
2 'Ie6oTapeB.