Osnovy teorii Galua.

110

110 HI~~~11. Pa3pewu~mve ypaemenun

So + 81 +. + S. = p 9TO H me T M eCTO TOJI bKO B Ciiy ae Sk p, R~Si 0
npli i/k (k =O,0 1, 2... n), H1 B 3TOM cayqae - =PI 1. n0T0MY MU! HoJnyqHM cpaBHeHie
[a0~alx + a x2 +.. +a,.xnIP ~a + a PxP + apX2p+ +
+ apx' (mod p).
HO 113 Tfeopembi (1)epma cneatyeT:
a a 0,ap =a, apzza V..., a na(nmod p),
OT~y~aa ouoHqaTenlbHo:
9'To H eCmb 4opmy-ia LLI"HemaHHa.
2(l. rHpHBeaeM HecK0JIbK0 BHLI2O 3meHeHHoe aoKa3aTeJnbCTBO HenlpBo~u4 -M1OCT1H. Klilypa (J. Schur)..Ljoyyc'rliM, t-irio f (x) eCTIb HenpHBoalHmb~it noJIHHO0M, K0PHeM KOT0poro si~ji~eTC)1 HeKOTOpbIlt neplloo6pa3Hhht in-it iK0peHb
H3 eaHHfluuJ a. Bo3bMeM HpOI43BOJlbHOC 41l4CJ10 /7, MaHimHo-IlpocToe c mf, H1
AoKaKem, WIO SPeCm TaKmKe K0peCib nIoJIHHOMta f (x). Thrnycxag HpOTHBHoe,
MEJ HaAlIem iApyrofl HenpHB0AlHMbli4 HOJ-IHHOM g (X), R0PHeM K0T0poro
5IBfiffeTCH SP. Torata x"~ -I1, aensich Ha o6a B3aimHio - np0CTbie IlOJUIHOMa
f (X) 11 g (x), pa3JaeaH4TCH1 H Ha iix npouaue~iieue, a n1OTOMY )iHCKPH4MHHaHT D noJIHHoma x - I pa33JeJIHTCff Ha pe3yJlbTaHT H10JIHOMOB f (x) H
g (x) (r~aaBa 11, ~ 1. 9, epopmyJna (1. 15)). Ho ec.'w F (x) = x' -1, o
npOH3BOAHaa F' (x) = mxm a IIOTOMY
rn- 2 (mn-l1) 3 m-1M(M
(rn-i1) m (M +1
= m 2 -~mm
C.ixpyroti CTOPOHbI, 1? (1, g) -f (YI) f (Y2). -. f (Yk), r~xe yi, 1ys
* Y 't - I(OPHH HOJ.IHHOma g (x). PacCMOTpHM BenkTH4H~Y f (yl), uie
=l SP, HlO~rbayfiCb cpaBHeH1em (2.8) Hi flp1H~iimaf BO BH11maHi~e, 'ITO
f (e) =0, nonyt4HiM:
f (y]) 0a (no d p).
3'To O3HaqaeT, tITO BeJ1 1'IHHi f (Yi) MO0)MAO Bbipa3HTb, n<K 4)yHK1JH10
OT B C ixenbimHi K034cpHILlHeHTamHi, aenlJL1l11MHCfH Ha p:f (yl) = p. cp (a).
Hoe MOW0HO11 Ha 0CH083H1414 Teopemh1 65 Bbwpa3HTb, KiK CTe11eHb Yi, TaK
1<aK B CHRYI Teopemh1 64 y1 a~ eCTb rnepBoo6pa3Hblfl KO~eHb Mf-fl CTe-!IeHH 113 eJUHHHUbl. Ho9ToMyf (yI) =p. t (yl), rae ~ (y) nbiHJIHOM c ute-.ib!imH paIJJ4OHaJlbHbIMfH iomt~xIHLuteHf1ami. Flony'eHnoe COOTHowe1HHe OCraHeTC11 cnpaBeatBILIbm, ecCA m Mb!e33M yHM Yio.16b11m KOPHeM HefIPHBOJlHmor'0
ypaBeHeuHs g (x) = 0. FIOJCTaBJI115 nonyqIaembue paIBeHCTBa B Bblpa}KeH11e
onR (,g), nlony'uHM:R Ag) _- kh Tp(Y1) TOO~) CP(k.