Osnovy teorii Galua.

~ 5. Amomopopw5mbi nopMaAbflow floAR

He co6totaanocib. TorJaa BeJ1HtlHH aL = (p (to) 5IBJIS1eTCHl BeJiH4HHIOFl HojISI, H pHHa~ane~aujefl K< rpyrine ~
7. T e o p e M a 57. Hlpiffaxulewauma~ K noatrpynne ~) rpyrnni Fanya
HopMaJnbHOFO noaI% K BeJ1HqHHa aL eCTb K~opel-i HenHI4B0,RHMOrO pa1LHOHanbHoro nioJIHHoma CTerieHH k=(3
140K aa3aTCJqb C T B 0. FLyCTb (0 r=" + ~S2 +....+ ~)8k* Bce BeJIH'4iHHL
ca, ',~..., I Mk pa3J1I44Hb1 mewKJy co6oft.,LIeACTBHTeJ~bHo, ecun1 6bi H1meno
meCTO HaUpHmep CL a L TO, riHmeHM51 K 3TOMY paI3eHCTBY 110JJCTaHOBK~
2 mbi 6b1 noJ1yt1HJii aL CL,F OTKyJaa, cornJaCHOo 0peJaeJneHHi{) BenH414Hbl cL, -cjiejaoBajio 6bi, 4T0 nOLCTaHoBK~a SS2 -I BXOEWT B k), T. e. S,3 Bxo-.awia 6N B S '2 ITO HeBO3MON)KHO.
HOJICTaHOBma S nepeBOH0JIT CL B MC-l' TOrJa 14 TOJnbKO Torxaa ecjiJ4 OHaI coJWp)K[4TCH B coripB)KeHHOl CHCreme ~)S4. HO TaK KaK, YMHowaR~ Bce corip~l)KeHHbie CIHCTembi crpaea Ha 8, Mbl TO.TfbKO n1omefHHim fIX HO0HJJaOK (CM. Teoipemy 22), TO [IPOH3BOJ4CTBO JIio6ofl nOJLCTaHOBKHi S rpynnrii (5$ HaJ COHOKynHOCTWO BeJ1H4HH CL CL'... UL3I TO.TbKO meH5eT fIX RIOpHOK. HO0BTOmy noJ'HHOM
He meiisiTmff OT nIOJCTaHOBOK r'pynilbl (5, OTKyaa cJneJyeT, 'ITo ero K034x14~HUlHeHTbl paIKHOHaJlbHbl.
Jloua}em, 'ITO nIOJHHOM (jRt ernpHBOJLHM. R~tOYCTHiB, '-TO HMeeT meCTO
1(t) (4 p (t). 4' (t), ae (p (t) iH 4' (- IOJH4HOMbl C pau*iOHaJlbHbIMH KO0(1)4)H —
aLLeHTamti, H noiiaraf, 'ITO aL eCTb KopeHb (p (t) H C(3-KOpeHb 4' (t), mbi y6eAumc'I,
'ITO R1O21CTaHOBKa S nepe~exaer HOJIHHOM Cf (t) B nojiHiHOM (p(t, m~eoiuiiii
KOpeHb CLS H HIOTOMY OTJ1HqHb1l1 OT (p (t). fHoa6Hpa% paAOabo 3HaxIeHI~e
t o TaK, 'ITO6bi (p (to) -/ (ps (t) Mbi n7oJIyqHm palLHOHaJlbHoe 'IHCJO (p (to),,
H3me~ff0lLowec'I ip HPH lOI3BOhiCTBe HO0,RCTaHOBKH S, 'ITO HeBO3MO)HO.
8. Te op e Ma 58 (JlarpaHca). EcUH CL flHHJa~)ljeKHT K no1LFpyHne
rpynTli rib ajiya (5$ HopmanbHoro noniij K, a He H3mefHBiTC51 OT
HOJLCTaHOBOK rpyfinbi $) (T. e. niffH~laxieKHT HJIH K ~ HJILI K 6oriee.
06-beMniou~efl rpynne), TO p3 paLjHOHai~bHO Bblpa>KaeTCSI qepe3 xt.,LI~o~a~aen b CTHO. fHyCTb += + )S2+.., S. Paccywaaa
noxto6HG TOMY, "i~K Mbi paccywKJLnJ1 npHi J1oKa3aTenTbCTB~e Teopembi 57, Mbm
y6exu-imcs, WIo nonI4HOM (I)(t) = (t - CL) (t1-C5). (t - as") HmeeT paLLHoHaJnbHb1e KogO~4H1LeHTLA H iie HIMeeT KpaTHL1X KOpHefl. BbupaweHHe
(5.2) 4't=() ~ + + S
11peJLCTaBJ15eT co6ot nolJiHHOM, KOTOpEfl B CHJIY TeX )Ke nplHiqHH HmeeT pa[U4OHaJIbHbie K~o3434IHiLuieHTbl (cm. -~ 1. 5). Holuctm'aBflq5 B 4[opmyny (5.2>
t =Ca H paccym~aff KaK B ~ 1.-5, Mbi 6yJaeM HImeTb:
OTKyJLa
(5-3) ~