Osnovy teorii Galua.

32

i. rpynnbi

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3ro paBeHCTBO MOAKHO [LepeInHcaTb TaK:
(SiS71) -1~6(5s 1) =S5
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pa3Jowe~iHmI rpyfirl Ha conrpB)eHiHbIe CHCTeMb-,I.
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nO p %AKa k, KOTOPYFO mbi 6yJaeM Ha3b1H3Tb A 0IOAHOJHHTeJ b Hog rpyri~no A, H KOTopaSI 6yaeT HrpaTb BeCbma Ba}KHYI1 POJUb B q1aJlbHetiUwem. Pa3.Tno.
IKHM ()'- Ha conpW>KeHHbIeC cICTemI rno rioxwpynne Si,
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'HOBOA rpyalfbl, AMJI KOTO~bIX Mb! YCTaHoBHm cnoco6 KOMI1O3L~tlH, BblTeKaioMIAHtI13 ripamiia KOMI1O3H1IHIjR COBOKYIIHOCTeA4. PaccmorpHm npotmuieaieHie
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= j ~:~A AAl = ~)AjAj.
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HH5I (5 no H)1 B CBSI3H c C3THAM Bb1~paKaeTCH TaK: A -A = HA, r~e H3J1emekT rpynnbii kp. Toraa miN, no~ny'ae~i:
SbAl~Aj = S5HA, =- a)j
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CIICTem $)Al EAMJ4qHH'IHM wiiemernom MBJlIeTC5I noaiwpynna ~. O6paTimlfl 3JqeM~eHT TaK)Ke HeTPYJLHO onpealemliTb. TaKcruv o6pa3oM COnlp$IKeHHbie CHCTemw
ipA o6pa3yloT rpyrmiy HopiRA~a k, KOTOPY1O H Ha3bIBaIOT JR 0 II 0 aI H H T e.i bH 0 r py nn OhR H oOo3Ha'IatoT TaK: (W/~).
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3InemewroB, BXOJIIULHX OAJHOBpemeHHO B -'lH B3 S2 q~e~l4AHi o6pa3yeT rpyrrny,
KOTOPYIO H1PHH$ITO Ha3b1BaTb n e pe e LieHiem HJiH 06nUHM Hai6oJnb11 aIJejirmiTenem rpynn - H -jIOII'HO onipeieinqeic~i nepeceqeH~e
6ofibuero qIHcJ1 [pylin.
HeTpyA0anoxaOK3mT cnieiiyiowie TeopembI
T e o p e M a 20. Ecm.IHb H$\ ) - HOpmallbHbie JaeJHTeJIH r'pyUInbi,
TO H Hx nepeceqeHme 9t eCTb Hopmanlb~bIR ej1~iHTeJIb rpynnbi V0.
LI0 K a 3a reAJ b C T B 0. JIOCTaTO'IHO JloKa3aTb, 'ITO flpOii3BOJIbHbIfl 9JnemefT K rpyauRitb, 6yatyqrI rpeoopa3oBaH ffpfl rIOMOUUI nIp0N3BOJ~bHOIO 9Jle