Osnovy teorii Galua.

~1.- 0 OeiruMocmu quceA 9

199!7pum~ep. PeiuHTb ypaP~eHue 34x + 28y = 2. HMmeem:
34 28*1 4-I6,
28c 6 -I+-4,
6= 441+2,
4 = 2.2.
OTcMoaa 2-6-4.1 =6-(28 - 6.4).1 = 6*5-28*1 =(34 -28.1)5 -28.1=
=34-5 - 28-6. TaKHM o6pa3oM 3Ha'IeHH5I
x =5, y =-6',
5IBJISIOTC51 petueff~eM 3aw~HHoro ypaBHeHI~a.
5. EctiH o6nwfl HaH6onbmuiH aTeJHTenb aLBYX 'wceii a 14 b eCTb ea~nimnaa.,
TO a 14 b Ha3bBaB[OTC5H 63aUtMH-(-npoCMblMtt qUcula.4. ULIA5 B3aHMH0-rnp0CTblX
qttceni a. b ypaBueH~e
ax + by = I
KmeeT pemieHmH B LueXbix hLmcnrax.
HmeI0T meCTO caieapyione Teopembi:
VI. EcnKI a - b JLeJIHTC14H WIC, a a B3aHMHo-rnpOCTO C C, To b JaeqfHTC5I
Ha c.
l 0K a3 are JI b C T B 0. H~yCTb X, y 6yprerpeuieiWie ypBe~ ax + cy = 1
YmHoHmm 310 paBeHCTBO Ha b: b ab -x +b -cy. 06a 'tineHa npaBoM qaCTH
JIaen1STCff Ha c, a 11OTOMY H CHJy ii H HXcymma, T. e. b, aeaiHTCHI Ha c, 'i.
H T..
VII. EcJIH a H b B3aHMHO-rlpOCTbl C C, TO H HX npHBae ~- b
B3aHMHO-rIpoCTO C C.
tOK a 3a T e bC T BO. Y"aBHeHHli ax + cy H bU +CV ~r-Aq Hme1OT
peineuwi B iuenbIX q-HcJax. IHepeMHOIKHM 3TH ypaBn-eHHnI:
1=(ax +cy) (bu + cv) =qab -xauJ c, (axv + bya+ cyv).
3TO paBeHCTBo n01a3bfflaeT, qT0 ypaBeH~HH aZbx + Cy ) 1 umeeT peUIeHH9
H iaenjmX qHcJIax, Tr. e. qTo ab H C B3aHMHo-rnpOCTbl, 4i. H- T. it.
VIII. EcJiH a teiiTCRc Ha a~ B3aHMHO-rnpOCTbie wHcnTa b H C, TO
OHO JeiT~JWCR H Ha HX HP014I3BeaeH~e bc..jOKa3aTeJnb CT B. IlyCTb. a = bq, rite q HeKOTopoe iueuioe 4Hicjio.
HO YCJ1OBHiio bq aJIlJITC$1 Ha c, [1pHLem b B3aHMHo-npOCTO C c. Torita
143 reopembi VI cJneayeT, qITo q aeahHTC51 Ha C, T. e. 'iTO a = bq JteJIHTCH
Ha bc C. H4T. it.
6. r3yitem pacaP1B~ Taioe o6uue HaHi6ojibHIe iteJHTeJIH HecKOJHjMIX qi4ce)1. Hycmb a, b, c Hime1OT 06M1UM HaHi6oJnbHHm teniHrenrem d, H HyCTb
(a, b) =- 4,. Torita.
IX. d eCTb TaK>Ke o6UAHH~ HaH6O~bWHR4 itejiHrejb qHceui di H C.
o K a3 a TeJ b C T Bo. YpaBHeHHe ax + by = di~ B cHJIy V iimeeT peweH~e
B iJnbix qHcJI3x. H3 ae1lJ1MOCTH a H b Ha d cneityeT aetJIHMOCM Ha d f4HcJ1
4i = ax + by, OT~yita BHitHO, TITO d eCTb tejiH4Te~ib d1 H c.,Tt~rYCTHM, 'ITO
(d1, c) d2 > d. Torita BblflieT., WIo a, b, c xtejiTCH Ha d., 'ITO HeHO3MO)KHO