Osnovy teorii Galua.

~ 6. Hpucoe~uHenuR. Hamy~pGabnb1 uppaquonaabHocmu 9

95

nlOCTPOHM ypaBHeHl~e, KOTOPOMY yAtOBJIeTBJpsieT 4)YHKLLHfl Z1 = XI X2 + XS X4,
npH~aiuiewa~uag K rpynnre (5. 3TO ypaBiieHue TaKOBO (CM. ynaK~~l 29):
(51)z3-a2 Z2 +(ala3- 4a4)z -a4 (a,2-4a2)-a.2=~0.
H4To6bi rpynria ypaBHeHHIN (5.9) 6b.iqa.elieTeJnem rpyrirnbi (5 10) (HItaH CO0
npH)KeHHOfl C Hleft rpyfinbl), Heofxo1Hmo H4 aioCTaTo4Ho, qTo6bI ypaB~eH~e
(5.1 1) ime~ao no Kpa~liHeil m~epe, QA~HH paUtHoHaJlbHblfl KopeHb. B camom jiejie.,
HCEJIIO4HTeJlbHh1]R C.Tlyrafi, KOrJLa ypaBne~i~e (15.1 1) HmeeT KpaTHbie KOPHH,,
BJIeqeT 3a co6ofI paI3eHCTBa THna
XI x2 + 4X3~4 -: X1 x3 + x2 X4,
T. e.
(XI1 - X4) (X2 - X3) = 0,
a I1OTOMY 9TOT C~iyqat4 mo)KeT HaCTydHHTi TOJnbKO TorJaa, ecJiH camo ypaBHeHHie (5.9) u~meeT KpaTHbIe KOPHH.
COBOI~yrHOCTb n7OJLCTaHGBOK, He H3meH5II0H1UHX HH OILHH H3 KOpHefl
ypmeHfHif1 (5.1 1), o6pa3yeT rpynnry
4-ro nOpguuca (Vierergruppe). qro6bI rpynnua ypaBHeHfI4 (5.9) 6wia.eiCiHTeJIem rpyrinbI ~, tieofxoiaHmo H )jOCTaTO4HO, '-To6bI Bice 3 KOPHH ypaBHeHHH
(5.11) 6bumT pauLLoHaIqbHbl.
Y n p a K H e ii H e 31. HaQTVI rpynny BO3B~aTHio[o ypaBHeHHi1 4-o OH CTerneHH
x4 + ax" + bx2 + ax+ +1= 0.
Hlola3aTbh, 'ITO 1vi1i TOFO, liro6bi ero rpynnoii ranya 6wina Vierergruppe (wHJJ
ee IXCAHTeJqb), Heo6XOJUHMO H j~oc raToqHo, 'qTo6bi '4HCAO (b + 2a + 2) (b - 2al + 2)
6buio TOtqllbIM KBaflpdTONI.
YnHp a KH e HiHe 32. Me>[Kny KOPH51MH ypaBHeHHH iueCTof CTeneHH.x6+ a1x5 +a2.x4+ aX3 +a4 2+ a,+ a6= 0
3a2aaHo COOTHOH1e~HHe XI x2 x3 =- X4 r5 - x.- Ha~riir H0OfUI0K rpyrnmi 3T01' ypal3HeHH1R,
COOTHOWCnCHue meway ero K034Xp)iiaeHTamH4 iH HpHBeCTH ero peuie~iie K peweiHHKo
ypaBHeHiHfl Hti3VIHX CTelI1 Hefi.
Y ap a)KHeH H e 33. To }Ke nPH C00THo~uHJCHWX Xl X2 X3 XI - Xs X6.
Y n pa KIt eH He 34. HATH rpynny ypaBHeHIHs9 'x + 199g+ x8~1=.
Y Ka3 a H ite. YpaBneHHe coxpaHHHT cBoio 4Dopmy, eCtiH mbI 6yaem npOH3BOAllfb
B Hem nOACTaHOBKH X f-y- -- H X~zy,
~ 6. HpHCoeAHnetine HOBbIX BeJ1HqHH K o6JtacT paLutOHaJ~bHOCTIH.
HaTypa~nbHbie H I1o6Oqtibie Hppatm~Ha~lbHocTH.
1. O6AaaCm paLAHoHaJlbHoc-TH. B ~ 4.5 Mbi pacaplal IIojiHHOMbf
c p a IH oH an b Hl l M H4 I(O~frHLLHeHramiH H4 oHpeauJfiJii rpynrny 7a~nya, KBKl
COBOK~InHOCTb nOJICTaHOBOK, He HapywiwilHx COOTHOuieHHPI mewaL KOPHRMH4,
Hme[oiIIKx aU H LLHH aJIbHbie Ko09J~4jH[LHeHTbi. Bce BbIBOJ~bI Wmor naparpa4)a mowKHo LelJiKOM iiepeHeCTH iia TOT c~nyliafl, Korua K034)cJ)Ht1HHTM
I7OJIHHOMOB H COOTHOw~eiimti me}!(J KOPHSIMH mbM.0yeiM C~iHTaTb y)Ke He pa1AHOHaJIbHbMMH 4HcjiaMH, a BJneMeHTaMH oco6oro no~na FR, KOTOpoe mbM B 9TOM