Osnovy teorii Galua.

____ ________ 3. Teopuqi cpaeneuua____203, 0K a 3a T e Ib CTBO0. HlpoHte 'cero aoKa3aTb 3TY Teopemy, onHiipanb Ha
TO, 4TO cyule~CTByeT Bcero cp (n) rnepI3006pa3HbIx fl-blX KOPHeiR H3 eJLHHH1Jbl.
KaKJJabif KopeHb ypaBHeHHff X' - 1 = 0 ffBJ1ieTcff IepBoo6paaHbIM KopHem
C~eneH i4 d, rje d - HeKOTOpbII4 Jae.TTenlb Mjicja fl. C iiPYr09 CTOPOHbI, xRAR
Ka}KJoro =Re~TeJ1q d 4Hc.Ta n CyHeCTB~eT cp (d) nepI3006pa3HWidx opHefl d-okl
CTei~eHH H3 eJLHHHI~bI, 51RIB)14HO1Ux BmeCTe C TeM KOPHRMH ypaIBneL-HI
paB~a o6uieiy 'IHCJIY KopHefi yOflBHeHH5 X"! —1 0 T. e. 'i~cjiy n, q.
H T. A~.
~ 3. TeopHIr c~paBHeHHiH.
1. EcJiHi pa3HOCT, J1BYX qHceJI a H 1) aIeJHTC9 Ha Im TO, ciieaty% rayccy,
rOBOpHlT, '-ITo a ii b cpaBH HM bl no MOAY.THO in, H 3arii~cbIBaloT 3TO TaK:
a zb (mod in).
Tai~oro po~ta 3arHHcH H3a3b1Ba1OTCS1 c p aB H eH H 1MtH. OHHi BO MHOr14x
OTHOtUeHH51X HOXOKH Ha o6bsKHoBet~iibe paBeHCTBa. HX MOWIHo no4JIeHHO
CKJI3JibIBaTb H YMHo)KaTb. B camom aieJne, ecrnH
ax ---b i-t c-7i-d (mod in),
TO 9T 0O3Ha'laeT, WIO a -1) H c - d ejl~~TC51 Ha in, a nlOTOMY H Hx cymma
(a -b) +(c - -d)-=(a +c) -(b +d) JaeJHTC5H Ha im, T. e.
a+c-3ab+d (modmi).
flyCTb a -12b km H, c-dz=lin. Touaa ac=(b+kin)(d+Iin)=
=bd +(bl +dk +kln)in, B cHiny 'iero
ac p-bd (modtn).
HpHieHffff 9TH 1IpaBBH.T ROCTaTOHHoe liHCJo pa3, MO}KHO y6eaHTbCss
a cnpaBeJLJHBOCTH CJ~e~RysoUiefl TeopembI:XIX. EcTHf f(X) - flpOH3B3OJbHblfl n~HOJHOM c 14eJIbJMH KO,9qj4pI4HeH.
TaMH Ha= ab (modin), TO f(a) & —f (b) (m.o d in).
2. PaccmaTpHBa51 cpaBHeHHSib, B KOTOpbIe BXORJ5]T HeH3BeCTHbje BeJIH4HHbT,
mbi mO)em YCTaiiOB14Tb AJt151 HI4X rnpaBHnla, alHaJIOPHbHIe HnpBHJIm Ja51 ypaBHeH 1 P.
XX. EcJIH B, J1HiHeIIHOM cpaBHelHHH
(3.1) ax.=-b (modni),(a, b) =,TO 3TO cpaB~eHue Bceraa HmeeT peiueHH~I.
JioKa3aTe~lbCTBO. Pewaias Heorspe~tejieHoe ypaBHeHHe au + my =
(CM. V) H1 YMHO>Ka51 Ha b, 6yae~m Hi~eTb:
a (bx) + in(bv) = b,
T. e.
a (bx)E-=b (modin).
I'aycc Hpexi~oxuviJ o6OaHa'IaTb penmeHie CpaBHefHH51 (3.1) HPH 11OMOLIUH
CHMBoJ1a + (mod mn).