Osnovy teorii Galua.

~' 6. I-(eadpupye~lble Aynoqiiu1~

191,

)KeT 6bITb 6JnaroHpH51THbIM. Hccneayem, rHpH KaIKHX 3HaleH1HqX Pk OHOaJeiRCTBHTen~bHo 6AaronpHHTHO. PacCMOTpHIM caieaytyonie BO3MO)KHb1e cJnyqaw
I) X coae~)pw1T HeqeTHbflg MHO.N14HTeJIb t: X t. Xi. Toura 3Ha4eH14ie
MO)KHO npeTLCTaBHTb Talc:
2__ 7- k 2 k- 2 (p'- 1 2(p' - 1) (pX, (t-)+ pXl (t -2) +... + p~l
X4H 1 x(t - 1) + ) (- 2)+. + x1 + 1 COCTOHT 143 HeqeTHoro '4HcJla
He~qeTHbIX cJiaraemb1x, a HOTOMY OHO HeqeTHoe. BmeCTe C Tem npi'i p > 3 oiO,
B c14Jy pm >,(l + 2)f >, 1 + 2m He meHbune, tlem
1 + 271(t- 1) + 1+2 X(t-2) +... + 1+ 2X, ~1
=t+ 2X t (t -1) >- it-t+2~>2X-k
22
a n1OTOMY OHO He mo)KeT HaueJ1o coKpaTJHTbCq C qHCJIHTeJnem. TaKHm o6pa3oM
STOT CgiyqaR He6JaaronpHH~TeH.
I)XHe cQLLep)1HT HelieTHbIX MvHo)HTeJneti, T. e. paBHO CTene~HH JABohKl1:
X = 2'. Torjaa 3Ha14eHF1 (2 MO)KHO HnpJLCTaBHTb Talc
2k- k ______ __27,k-k_
3JlameHaTeuIb COCTOHT 143 9L + 1 MHo}KcHTeJIeh, lcOTOpb1e moIryT couepwam
H HetieTHbie MHO)ICHTeJIH. Ecatt XOTA 6bi jiba 143 3THIX MHO2KHTeJIefl He coaep)1{aT He'leTHbIX MHO)KHTeJneft, TO mbi 6yaem HlmeTb:
p2 + 1-2', p2T 1 +1 29
HpHq1em HH>KHHfI 3HK mO}KeT HmeTb meCTO TOlblco npiif 0. Torita p20 + 1
JJaen11TCH Ha p21 ~- 1. Ho tmniio p219 - 1 - (p21)2P -- T -1 Towe aenli1TC51 iia
p~l ~ 1, a nIOTOMY Hi Hx pa3HOCTb ( p2P + 1) (p2~ 1) 2 eJtIH1TC51 Ha
p21~l. 39ro B03MOIKHO TOnfbKcO B cniy ae p21~1~=2, OTKyJLa p2 - 2:l.
Bepsi BepXHHfl 3HaK, HlOJIyq4M p = 1; 6ep~l HHKHHtI 3Hal, 6yiieM 4meTb.
= 3, T 0. Hlpli p = 3 KawaLbIfl MHO2KITeJIb Bbipa)leHH4SI 32' -- 1=
flpeatrocJne~aHero, aeniHTCH TO1IHO Ha nepByio CTeneHb 2. J'etdCTBHTe~lbHO,
32P + 1 ~- 92P~ + 1 1 + I E2 (mod 4).
HO10TOMy Bce BbIpa)leHHe aLeJtIITC5J TOT4HO Ha 2,+2 '-IaCTHoe (32' -1):2*
AKOJI)KHO., KaKc HeqeTlibffl MHOWfHTeIlb 3Hm~ael Bb1paxceHHH p, 6wlTb JAeJIIIfeJ1em 4LHCJ11HTeJ1 2X-k, OTKyJaa
(32'.- 1):- 2u~+2 < 2X-k;
HO TaK KaK 2 X- k~<X= 2', TO
32' <- 220(+2A
OTKaa 2' < 2a+2, T. e. a. < 3. Ho flti4 a.=2 HmeeT meCTO
32' > 22a ~2,