鐃緒申鐃緒申鐃�II 鐃緒申鐃緒申 No.12

鐃緒申鐃藷、種申鐃緒申鐃緒申鐃所が鐃縦ってわ申鐃殉ってわ申鐃緒申鐃緒申

鐃緒申鐃緒申 12.1 (鐃緒申鐃緒申11.6鐃緒申同鐃緒申) $% latex2html id marker 942 $ {\mathbb{F}}_q$$ 鐃緒申隆鐃緒申鐃緒申1鐃術随申

鐃緒申多鐃準式

鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�

鐃緒申 $% latex2html id marker 950 $ {\mathbb{F}}_{q^r}$$ 鐃塾なわ申鐃祝削申鐃緒申鐃縦のわ申鐃緒申鐃緒申鐃旬随申鐃塾とわ申鐃祝限る。
鐃緒申鐃緒申鐃旬随申鐃淑わ申弌鐃� $% latex2html id marker 960 $ {\mathbb{F}}_{q^r}$$ 鐃塾なわ申鐃緒申鐃塾削申鐃熟わ申鐃順う鐃緒申鐃縦わ申鐃暑。

命鐃緒申 12.1 (命鐃緒申11.1鐃緒申同鐃緒申) $% latex2html id marker 967 $ {\mathbb{F}}_q$$ 鐃緒申隆鐃緒申鐃緒申1鐃術随申多鐃準式

鐃緒申多鐃準式

鐃緒申鐃出わ申鐃銃￥申

鐃塾刻申同鐃緒申鐃緒申鐃緒申鐃舜随申鐃緒申

$\displaystyle Z(V(f),t)=1/(1-t^n)$

鐃緒申与鐃緒申鐃緒申鐃暑。

鐃緒申鐃縮に￥申鐃緒申鐃緒申鐃緒申鐃舜随申鐃緒申

鐃祝わ申辰討匹里茲�申鐃緒申僂鐃暑か鐃緒申複鐃緒申鐃叔わ申鐃暑。鐃純え鐃緒申

鐃熟鰹申鐃叔随申

鐃緒申鐃出わ申鐃緒申 ${\mathbb{F}}_p$ 鐃緒申鐃渋随申犲逸申噺鐃緒申鐃緒申襪�申鐃� 鐃緒申鐃曙が鐃緒申鐃藷かどわ申鐃緒申鐃緒申

鐃祝わ申辰動曚覆襦� 鐃宿のよう鐃緒申

鐃緒申鐃出わ申鐃銃器申鐃緒申任鐃緒申襪�申鐃夙緒申蠅刻申鐃塾わ申鐃緒申鐃緒申鐃淑のわ申鐃緒申平鐃緒申鐃緒申余鐃緒申鐃緒申箸鐃緒申鐃緒申鐃緒申法則鐃叔わ申鐃暑。鐃緒申鐃祝∥э申両鐃緒申鐃緒申呂鐃緒申鐃緒申鐃緒申鐃緒申里鐃緒申討鐃緒申襪�申鐃緒申鐃緒申旅峙鐃緒申鐃緒申辰留鐃緒申僂箸鐃緒申鐃� 有鐃緒申鐃塾常申離鐃緒申鐃緒申鐃緒申造鐃緒申僂鐃緒申鐃緒申鐃緒申鐃緒申鐃述介する。

鐃緒申鐃� 12.1 鐃緒申鐃叔随申

鐃夙￥申

鐃叔鰹申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃出わ申鐃銃￥申平鐃緒申鐃緒申余鐃緒申鐃緒申(Legendre 鐃緒申鐃緒申)鐃緒申

$\begin{displaymath} {\left(\frac{a}{p}\right)}= \begin{cases} 1 & (X^2-a\text{ .. ...-a\text{ が }{\mathbb{F}}_p\text{ 上既約のとき})\\ \end{cases}\end{displaymath}$

鐃祝わ申鐃緒申鐃緒申鐃緒申鐃暑。

鐃緒申

鐃緒申鐃旬随申鐃塾誌申鐃祝は￥申 ${\left(\frac{a}{p}\right)}=0$ 鐃緒申鐃緒申鐃緒申鐃緒申襦�

鐃緒申鐃緒申 12.2 鐃緒申鐃叔随申

鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃塾種申鐃緒申鐃緒申鐃緒申立鐃縦￥申

$\displaystyle {\left(\frac{a}{p}\right)}= a^{(p-1)/2} \mod p$
$\displaystyle {\left(\frac{ab}{p}\right)}= {\left(\frac{a}{p}\right)} {\left(\frac{b}{p}\right)}$

鐃夙わ申鐃緒申 ${\left(\frac{-1}{p}\right)}=(-1)^{(p-1)/2}$ 鐃祝わ申鐃緒申鐃春わ申鐃銃わ申鐃緒申鐃緒申

鐃緒申鐃� 12.2 $p,\ell$ 鐃緒申鐃緒申曚覆鐃緒申鐃叔随申鐃叔わ申鐃緒申箸鐃緒申鐃� ${\mathbb{F}}_p$ 鐃塾鰹申鐃緒申鐃塾わ申

鐃塾醐申鐃緒申 $\ell$ -鐃処根 $\lambda$ 鐃緒申鐃暑。鐃緒申鐃緒申

鐃緒申鐃出わ申鐃銃￥申有鐃緒申鐃塾のワ申鐃緒申鐃緒申鐃緒申 $\tau_a$ 鐃緒申

$\displaystyle \tau_a=\sum_{t=1}^{\ell-1}{\left(\frac{t}{\ell}\right)}\lambda^{at}$

鐃緒申鐃緒申鐃緒申鐃緒申襦� $\tau_1$ 鐃塾わ申鐃夙わ申単鐃緒申 $\tau$ 鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申 12.3 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申立鐃縦￥申

$\tau_a={\left(\frac{a}{\ell}\right)}\tau$ .
$\sum_{a=0}^{l-1} \tau_a \tau_{-a}=\ell(\ell-1)$ .
$\tau^2=(-1)^{(\ell-1)/2}\ell$ ( $=\ell^*$ 鐃夙緒申).
$\tau^{p-1}=(\ell^*)^{(p-1)/2}$ .
$\tau^p=\tau_{p}$ .

(鐃淑わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃塾よう鐃淑計誌申鐃薯しわ申鐃緒申鐃淑わ申里鐃緒申鐃� 鐃緒申鐃塾逸申弔離劵鐃夙はフ￥申鐃所エ鐃緒申鐃緒申鐃緒申砲鐃緒申襦�)

鐃緒申鐃緒申鐃緒申鐃緒申箸鐃緒申伴鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。但鐃緒申(3)鐃塾常申鐃緒申鐃緒申鐃緒申鐃緒申望鐃緒申襦�

鐃緒申鐃緒申 12.2 (平鐃緒申鐃緒申余鐃緒申鐃緒申鐃祝∥�) 鐃緒申鐃叔随申 $\ell$ 鐃緒申鐃出わ申鐃銃種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申立鐃縦￥申

${\left(\frac{p}{\ell}\right)}={\left(\frac{\ell^*}{p}\right)}$ (但鐃緒申 $\ell^*=(-1)^{(\ell-1)/2}\ell$ )
${\left(\frac{-1}{\ell}\right)}=(-1)^{(\ell-1)/2}$
${\left(\frac{2}{\ell}\right)}=(-1)^{(\ell^2-1)/8}$