h`pmt /B`B;ûb 8fk
aQB2MiA=
Ñ1 0 0 0 0 1 0 1 0
é
2iB=
Ñ0 0 1 1 0 0 0 1 0
é X
Zm2H 2bi H2 ;`QmT2 2M;2M/`û T`A2iB UQM T`û+Bb2` TQm` [m2HH2 HQB V \
1t2`+B+2 R , :`QmT2 2M;2M/`û
.ûKQMi`2` [m2 H2 +2Mi`2 /m ;`QmT2 bvKûi`B[m2 Sn U+Ƕ2bi ¨ /B`2 HǶ2Mb2K#H2 /2b ûHûK2Mib /2 Sn [mB +QKKmi2Mi p2+ iQmb H2b mi`2b ûHûK2MibV 2bi `û/mBi ¨{Id} TQm`n>3X
1t2`+B+2 k , *2Mi`2 /2Sn
aQBia 2ib /2mt p2+i2m`b /BbiBM+ib /2E 2i /2 KāK2 MQ`K2X
JQMi`2` [mǶBH 2tBbi2 mM2 mMB[m2 `û~2tBQMs/2E i2HH2 [m2s(a) =b2i s(b) =aX JQMi`2` [m2O(E)2bi 2M;2M/`û T` H2b `û~2tBQMbX
PM TQm`` KQMi`2` T` `û+m``2M+2 bm` dim(E) dim(Ker(f) Id)) [m2 f T2mi /Ƕû+`B`2 +QKK2 +QKTQbû2 /2
`û~2tBQMbX
1t2`+B+2 j , :`QmT2 Q`i?Q;QMH 2i bvKûi`B2b
aQBiG mM ;`QmT2X SQm`a/MbG- QM MQi2fa:G!Gi2H [m2fa(x) =a 1xaX JQMi`2` [m2 TQm` iQmia2G-fa 2bi mM miQKQ`T?BbK2 /2 GX
JQMi`2` [m2 :G!Aut(G)i2H [m2 (a) =fa 2bi mM KQ`T?BbK2 /2 ;`QmT2bX .ûi2`KBM2` H2 MQvm /2 X
1t2`+B+2 9 , miQKQ`T?BbK2b BMiû`B2m`b
aQBiG mM ;`QmT2 }MB /2 M2mi`2eX
PM MQi2mH2 TT+K /2b Q`/`2b /2b ûHûK2Mib /2G 2im=p↵11p↵22· · ·p↵dd b /û+QKTQbBiBQM 2M 7+i2m`b T`2KB2`bX JQMi`2` [m2 TQm`i2[1, d]-G/K2i mM ûHûK2Mi /ǶQ`/`2 p↵iiX
JQMi`2` [m2G/K2i mM ûHûK2Mi /ǶQ`/`2 mX aQBiKmM +Q`Tb }MBX
JQMi`2` [m2(K⇤,·)2bi mM ;`QmT2 +v+HB[m2X PM TQm`` +QMbB/û`2` H TQHvMƬK2 Xm 1X
1t2`+B+2 8 , G2 ;`QmT2 /2b BMp2`bB#H2 /ǶmM +Q`Tb }MB 2bi +v+HB[m2
aQBiP =Z[{p
2}X JQMi`2` [m2 H2 bQmb@MM2m /2 R2M;2M/`û T`P 2biA={m+np
2 : (m, n)2Z2}X
*QKK2 TQm` H2b ;`QmT2b- BH bǶ;Bi /m THmb T2iBi bQmb@MM2m /2R+QMi2MMi PX PM MQi2U H2 ;`QmT2 /2b ûHûK2Mib BMp2`bB#H2b /2AX
PM MQi2N(m+np
2) =|m2 2n2|X
SQm`a, bûHûK2Mib /2 A- +H+mH2` N(ab) JQMi`2` [m2z2U bB 2i b2mH2K2Mi bB N(z) = 1X aQBiz2U i2H [m2 z=x+yp
2 p2+(x, y)2N2-x6= 0X JQMi`2` [mǶBH 2tBbi2 mM mMB[m2n2Ni2H [m2 (1 +p
2)n 6z <(1 +p
2)n+1- TmBb [m2z(1 +p
2) n bǶû+`Bi bQmb H 7Q`K2x0+y0p
2 p2+ x0, y0 /Mb NX JQMi`2` [m2U ={±(1 +p
2)n|n2Z}X
1t2`+B+2 e , Z[p 2]
_ûbQm/`2 /MbZ/5ZHǶû[miBQM3x+ 2 = 1
_ûbQm/`2 , ß
5x+ 2y = 1[6]
2x+ 4y = 2[6]
_ûbQm/`2 , ß
5x+ 2y = 3[6]
2x+ 4y = 1[5]
1t2`+B+2 d , *QM;`m2M+2b
R
aQB2Mip2iq/2mt 2MiB2`bX JQMi`2` HǶû[mBpH2M+2 bmBpMi2 ,
H2 ;`QmT2 T`Q/mBiZ/pZ⇥Z/qZ2bi +v+HB[m2 bB 2i b2mH2K2Mi bB p2iq bQMi T`2KB2`b 2Mi`2 2mtX 1t2`+B+2 3 , _û+BT`Q[m2 m i?ûQ`ĕK2 +?BMQBb \
JQMi`2` [m2 bB2↵+ 1 2bi T`2KB2`- HQ`ba 2bi mM2 TmBbbM+2 /22X
SQm` iQmi 2MiB2`n- QM MQi2Fn = 22n+ 1X JQMi`2` [m2 bB m 2in bQMi /BbiBM+ib- HQ`bFm 2iFn bQMi T`2KB2`b 2Mi`2 2mtX
1t2`+B+2 N , LQK#`2b /2 62`Ki
k
