��II �� No.11

�� 11.1

�ϴĤǡ��ǿ�

��äơ� $p_R= \overbrace {1_R+1_R+\dots +1_R} ^{\text{$p$ ��}} =0_R$ �� Ω�äƤ��Ȥ��롣 ��ΤȤ�� $\Phi:R\to R$ �� $\Phi(x)=x^p$ ��С� $\Phi$ ��

��餽�켫�Ȥؤν�Ʊ��(��ʽ�Ʊ��)��Ϳ��롣

�� 11.1 �ǿ�

�Υ٥�

�ˤ��ơ� $% latex2html id marker 1195 $ {\mathbb{F}}_q$$ �γ�� $% latex2html id marker 1197 $ {\mathbb{F}}_{q^r}$$ �μ��Ʊ��

$% latex2html id marker 1199 $\displaystyle \Phi_q: x\mapsto x^q $$

�� $% latex2html id marker 1201 $ {\mathbb{F}}_{q^r}$$ �� $% latex2html id marker 1203 $ {\mathbb{F}}_q$$ ��Υե��٥˥��Ʊ��ȸƤ֡�

�� 11.2

�� $% latex2html id marker 1208 $ {\mathbb{F}}_q$$ ��

��δ��¿�༰��Ȥ��롣��ΤȤ��

�κ��ΰ�Ĥ� $\alpha$ �Ȥ��ȡ�

$% latex2html id marker 1216 $ \alpha, \alpha^q, \alpha^{q^2},\dots,\alpha^{q^{n-1}}$$ �� 줾�� κ��Ǥ��롣
�κ��Ͼ�ǵ󤲤��ΤǿԤ��Ƥ��롣

�� 11.3 $% latex2html id marker 1223 $ {\mathbb{F}}_{q^r}$$ �� $% latex2html id marker 1225 $ {\mathbb{F}}_q$$ ��μ��Ʊ�� ( $% latex2html id marker 1227 $ {\mathbb{F}}_{q^r}$$ �� $% latex2html id marker 1229 $ {\mathbb{F}}_{q^r}$$ �ؤμ��Ʊ�� $% latex2html id marker 1231 $ {\mathbb{F}}_q$$ ��¤��ȹ��ˤʤ��)��Τϡ� $% latex2html id marker 1233 $ \Phi_q$$ �ˤ�ä��̿�

�ν�󷲤Ǥ��롣

�� 11.4

�ѿ��¿�༰ $% latex2html id marker 1240 $ f_1,f_2,\dots,f_m \in {\mathbb{F}}_q[X_1,X_2,\dots,X_n]$$ �� $V_1=V(f_1,f_2,\dots,f_m)$ ��Ф��ơ� �֥ե��٥˥�� $% latex2html id marker 1244 $ \Phi_q: V_1(F_{q^r})\ni x \mapsto \Phi_q(x)\in V_1(F_{q^r})$$ �򼡤Τ褦�ˤ��Ǥ��롣

$% latex2html id marker 1246 $\displaystyle \Phi_q(x_1,x_2,\dots,x_n)=(\Phi_q(x_1),\Phi_q(x_2),\dots,\Phi_q(x_n)) $$

�� $% latex2html id marker 1250 $ \Phi_q$$ ��ư�� ( $% latex2html id marker 1252 $ \Phi_q(x)=x$$ )�Ǥ��뤳�Ȥȡ� $% latex2html id marker 1254 $ x\in V_1(F_q)$$ �Ǥ��뤳�ȤȤ�Ʊ�ͤǤ��롣

1�ѿ��Υ��ؿ��ꤷ�Ƥ�� ޤ��ˤĤ��Ƥδ�ñ��꤫��

�� 11.5 $% latex2html id marker 1257 $ {\mathbb{F}}_q$$ ��1�ѿ�

��¿�༰

��Ф��ơ�

�δ�� $% latex2html id marker 1269 $ X^{q^k}-X \quad (k=\deg(q))$$ �Ȥζ��̰��ҤǤ��롣
��Ǥ��뤿��ɬ�׽�ʬ��ϡ� $k=1,2,\dots,n-1$ �γơ��Ф�� $% latex2html id marker 1277 $ X^{q^k}-X $$ �Ȥ��̰��Ҥ�⤿�ʤ��ȤǤ��롣

�㤨�С� $% latex2html id marker 1279 $ {\mathbb{F}}_q$$ ��4��ޤ��5��1�ѿ�¿�༰

�ϡ� $% latex2html id marker 1283 $ X^{q^2}-X$$ �ȶ��̰��Ҥ� �⤿�ʤ��д��Ǥ��롣

�� 11.6 $% latex2html id marker 1286 $ {\mathbb{F}}_q$$ ��δ��1�ѿ�

��¿�༰

��Ϳ��Ƥ��Ȥ��롣��ΤȤ��

�� $% latex2html id marker 1294 $ {\mathbb{F}}_{q^r}$$ �Τʤ��˺��ĤΤ� �� ܿ��ΤȤ��˸¤롣
�� ܿ��ʤ�С� $% latex2html id marker 1304 $ {\mathbb{F}}_{q^r}$$ �Τʤ�� κ��Ϥ��礦�� Ĥ��롣