Osnovy teorii Galua.

~2. rpynna raflya cpaenienuci 1

163

11. fp H Me p 2. HIyCm p =3, m=-2. 113 4iopmyJHw (1.7) MUl Hmeem~:
9 2 - g(2) +g(l),
CTKCJ~a g (1) =3, g (2) =3.
Paaaojicim noOAHloM XPm - X = X9 - X Ha Hen1pHBOAHMbie MHO)HHTeAHI:.X9 -x= X (X2 -1) (X2 +1) (.x'+ 1).
B STOM P23.110)1eHHiH nillOJIHM X2 + 1 JaoqyKeH 6bITb HenpHBOIHM no MOayjHio 3;
JIOJIHHOM )Ke x4, I 1 Jon0KeH pa3jiaraTbCf9II11 MOAYJIio 3 Ha MHO)KHTeJ]H 24A CTeneHmf.
l1TO6bI HafiTH ST0 pa3.uo)KeHie, pa3~Aemm X4 + 1 Ha InOJIHHOM x2 + ax + b, uje
K094)(HLIHeHTb1 a, b Mbl OCTaBHm noKa lieOnpeAejleHHbIMH. flpiipaBHHBaem Kos4(PIHuHeHWbM noayqaemoro B OCTaT~e iojimHHoma Hyalio:
a(2b -a2) = 0, b2 — a2t' + 1 =;=-0 (mod 3).
lp~paBHHBalI B nepBOM CpaBHeHHH Hyalio nepBbli4 MHo)KIITeJqb: a ~~ 0 (mod 3)
mbi npmHXOA1HM K HeBO3M0)KHOMY cpaBneHnio b2 +=1 -0 (mod 3). flphipaBiHtBan Ke
HYAH0 BTOpOiR MHOKhITe.4b, nlonyia eM:
b - a2, a" +a2 + 1 =0 (mod 3),
0TKYA~~~~ a E~1, b -1 (mod 3),
TaK WIO noHOJHom X9 - x pa~aiaraeTcal Ha HenHP1BOaHM14e no MOA~aIO 3 MHO)KHTeJ1H
TaK:
X, x-x x (X2 - 1)(X2 + 1)(X2 + X- 1)(X2 - X- -1) (mod 3).
HeTpy~aHo IOHqTb, 'ITO Bcqxifi~ n1oJiHHOM, HeInpHBOAUIMNI 110 MOA~J1O p, Tem
6oaqee He1pHBoJAHm B O6bJKHOBeHHOM CMb]CJ1 STOFO cjioBaI (B nioae pLtHOHaJ~b~b1X
iinceji).
Y n p a ic H e H it e 37. )IoKa3aTb, WIO cyme~CTByeT Bcero 10 cneayboul-UX HelipliBOAHMmbx no mo)1Ji1o 5 IIOAHHO1MOB 2-f CTeneHHi:
x+ ~2, X2 -+2x -- 2 x2 ~2x -1, x9~ x +1, X2~+ X 2.
Y n p a w H e H H e 38. J4oKa3aTb: ]ItAq Tror, tITo~b cpaB~eHne n-oii CTen1eHH
f (x) -=0 (mod p) HmeJIo n paU~onaalb~tX KopHeH', Heo6XoAHiMo, 'ITo6bi imeJJo
M~T p + q sq ~ 1 (mod M) a Sm cymma nw-x ceee oef3To
cpaBHeHHII, nonyiyaemasi 113 H3BeCTHb1X peKy~ppHTHb1X 4)opmyal HbIOTona. 3T0 N~e
YCAIOBHe AOCTaTOqHO B TOM cjiy'Iae,- ecaui cpaBnetIle HC HmeeT KpaTHb[X KOpHeft,
T. e. eCJiH ero AHCKptHmihaHHT He Ae-JiTC51 ia p. Eoaee Toro, B E)TOM cqy'iae aocTaTO'IHO y6eAHmbCg B cupaBeJ~nI1BOCTH 4)OpMYqhI 11H q 2,2..,n -~1(C. 0. UlaTyIIOBCKHII)
Y H p a W H e H H e 39. JJoiauaTb, WIO KOHe'IHNi noJi, iimeIOaHe OAHHaKOBbfJi
nlopHIJOK pm, Ii 3 o MO 4i ( H bi, T. e. me)Kity Hix 8JemeRTMH MO)KHO YCTaH'OBHT6
B3aHMHO-OAH03Ha'H~oe COOTBeTCTBHe,- He Hapyilaiollteec)I ripH cjio)KeHHH H YMHo2KeHHH
BJaeMeHTOB.
Y K a 3 a H H e. PaccMoTpeTb Hen~piBOIHhibie no MOAY10JI p MHO)I{HTeaH flO)HHOkIa XP-X.
~ 2. rpyllna raniya cpaBHCeJHI no 11POCTh1M MOaJIYISM.
1. HeUfpHBOJWMble cpaBHeHHSI. M~aCTo rolBOp5T, 'ITO BeqiH'HMbI KOHe'IHoro MARi$ 3U[BJ1MTBP1IOT TeM HJIH Hjibim c 3 a B H e H H H1 M. YnoTpe6.asi% qTy
TepMHHonTornio, Heo6XoJLimo YCJIOBHTbCII, 'ITO mbW 6yImem a33yMeTb nOAi
cp1aBHeHHeM, B K0T0310e BX0JA5IT l13i3pl1H0HaJIbHbie BeIH'qiHHbi. Mwi 6y~em
roBOpliTb, 'ITO BeJIH'IliHa aZ cpaBHHma C BeJINH'HH0 P f3 11 MOAJy.IIO P. ecABH,
6yAy'IqH Bbipa)Kenl tiepe3 nepB006pa3HbIi 3h1emeHT KOHe'IHoro IIoJsi, OHH
pa~BHbI.lpy APIP yry. ToLIHo TaK )Ke nOA KOPHRMH c31aBHeHHfl
(2.1) f (x) - O (mod p)
I I