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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.9

\fbox{正標数の体}

鐃緒申鐃緒申 9.1   鐃緒申 $R$ 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� 鐃縦緒申同鐃緒申

\begin{displaymath}\phi: {\mbox{${\Bbb Z}$}}\to R \end{displaymath}

鐃緒申鐃緒申佞鐃渋醐申澆鐃緒申襦�$R$ 鐃緒申鐃緒申鐃緒申(鐃夙わ申鐃祝￥申鐃緒申)鐃淑わ申弌鐃�$\phi$ 鐃塾鰹申 $\operatorname{Ker}(\phi)$ 鐃緒申 ${\mbox{${\Bbb Z}$ }}$ 鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申任鐃緒申襦ｏ申鐃緒申覆鐃緒申鐃緒申鐃緒申鐃塾わ申鐃緒申鐃曙か鐃緒申鐃緒申鐃緒申鐃暑。 鐃緒申鐃緒申鐃緒申弔蓮鐃緒申泙箸鐃銃種申鐃塾よう鐃祝書くわ申鐃夙わ申鐃緒申鐃暑。

\begin{displaymath}\operatorname{Ker}(\phi)=n{\mbox{${\Bbb Z}$}}\quad \text{{\rm (}$n$ は素数または0\rm {)}} \end{displaymath}

鐃緒申鐃� 9.1   鐃緒申 $k$ 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緒申鐃緒申鐃緒申鐃緒申鐃� $n$ 鐃塾わ申鐃夙わ申 $k$ 鐃緒申標鐃緒申鐃夙わ申鐃緒申鐃緒申 $\operatorname{char}(k)$ 鐃夙わ申鐃緒申鐃緒申

鐃緒申 9.1   $\mbox{${\Bbb Q}$ }$, $\mbox{${\Bbb Q}$ }[\sqrt{2}]$, $\mbox{${\Bbb R}$ }$, ${\Bbb C}$ 鐃淑どわ申標鐃緒申鐃緒申鐃緒申鐃緒申 $0$ 鐃叔わ申鐃暑。 鐃緒申鐃獣ぱわ申法鐃� $\mbox{${\Bbb Q}$ }$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含わ申鐃塾わ申標鐃緒申鐃緒申 $0$ 鐃叔わ申鐃所、鐃春わ申 標鐃緒申 $0$ 鐃緒申鐃塾わ申 $\mbox{${\Bbb Q}$ }$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含む。

標鐃緒申鐃緒申 $0$ 鐃叔なわ申鐃塾わ申鐃緒申標鐃緒申鐃緒申鐃塾とわ申鐃緒申鐃緒申

鐃緒申 9.2   ${\Bbb F}_p={\mbox{${\Bbb Z}$ }}/p{\mbox{${\Bbb Z}$ }}$ 鐃緒申標鐃緒申鐃緒申 $p$ 鐃叔わ申鐃暑。

鐃緒申鐃緒申 9.2   (鐃塾わ申鐃緒申鐃緒申法)

1.
$k$ 鐃緒申鐃塾で￥申$k[X]$ 鐃塾器申鐃緒申文鐃� $f$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申鐃夙￥申 $k[X]/(f)$ 鐃緒申鐃塾でわ申鐃暑。
2.
$k$ 鐃緒申鐃塾なわ申弌鐃�$k(X)$ 鐃緒申鐃塾でわ申鐃暑。

鐃緒申鐃緒申 9.3   鐃緒申 $k$ 鐃緒申標鐃緒申鐃緒申 $p$ $ {\Leftrightarrow}$ $k \supset {\Bbb F}_p$.

鐃緒申鐃緒申 9.1   鐃緒申 $k$ 鐃塾醐申鐃塾個随申 $q$ 鐃緒申有鐃渋なわ申弌鐃�$q$ 鐃熟わ申鐃緒申鐃叔随申 $p$ 鐃塾巾常申任鐃緒申襦� 鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $n$ 鐃緒申鐃緒申鐃獣て￥申

\begin{displaymath}q=p^n \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 9.2   標鐃緒申 $p$ 鐃緒申鐃緒申 $k$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃塾わ申鐃夙わ申鐃緒申鐃緒申立鐃縦￥申
1.
$k$ 鐃緒申任鐃春の醐申 $a,b$ 鐃祝つわ申鐃銃￥申

\begin{displaymath}(a+b)^p=a^p+b^p \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申
2.

\begin{displaymath}F: k\ni x\mapsto x^p \in k \end{displaymath}

鐃緒申 $k$ 鐃緒申 ${\Bbb F}_p$ 鐃緒申亮鐃緒申塀鐃銃縁申鐃緒申鐃緒申鐃緒申鐃粛随申鐃緒申襦� (鐃緒申鐃塾種申鐃淑緒申同鐃緒申鐃緒申鐃緒申鐃緒申侫鐃緒申戰縫鐃緒申鐃緒申鐃緒申鐃緒申噺討屐鐃�) $k$ 鐃塾位随申鐃緒申有鐃渋なわ申弌鐃緒申侫鐃緒申戰縫鐃緒申鐃緒申鐃緒申鐃緒申鐃� $k$ 鐃塾種申鐃緒申同鐃緒申鐃薯あわ申鐃緒申鐃暑。

3.
$f(X)=X^p-c$ 鐃緒申 $k[X]$ 鐃塾器申鐃藷元でわ申鐃獣わ申鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申 $K=k[X]/(f)$ 鐃緒申 $k$ 鐃塾鰹申鐃緒申鐃塾でわ申鐃所、$X$ 鐃緒申 $K$ 鐃叔のワ申鐃初ス鐃緒申 $a_0$ 鐃夙わ申鐃緒申鐃夙￥申 $K[X]$ 鐃叔は￥申

\begin{displaymath}f(X)=X^p-c=(X-a_0)^p \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申 鐃緒申辰醗鐃緒申未法鐃�$k$ 鐃塾鰹申鐃緒申鐃緒申 $L$ 鐃緒申鐃緒申鐃獣て￥申 $L$ 鐃塾醐申 $a$ 鐃緒申 $a^p=c$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申弌鐃�

\begin{displaymath}f(X)=X^p-c=(X-a)^p \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 9.1   鐃緒申鐃緒申鐃瞬刻申鐃緒申鐃緒申僚鐃緒申鐃緒申砲蓮鐃� $\epsilon=1$ 鐃夙わ申鐃緒申鐃緒申鐃緒申僚鐃緒申鐃緒申砲蓮鐃� $\epsilon=-1$ 鐃夙わ申鐃暑。
1.
$f(X)=X^2+\epsilon X-1 \in {\Bbb F}_3[X]$ 鐃熟器申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�
2.
$K={\Bbb F}_3[X]/(f)$ 鐃祝わ申鐃緒申鐃緒申 $X$ 鐃塾ワ申鐃初ス鐃緒申 $\alpha$ 鐃夙わ申鐃緒申鐃緒申鐃夙にわ申鐃暑。 $K$ 鐃塾醐申 $a+b\alpha$ ( $a,b\in {\Bbb F}_3$)鐃緒申鐃出わ申鐃銃￥申 鐃緒申鐃塾フワ申鐃駿ニワ申鐃緒申鐃緒申鐃緒申鐃祝わ申鐃緒申鐃� $F(a+b\alpha)$ 鐃緒申 ${\Bbb F}_3$ 鐃緒申鐃緒申鐃緒申 $\alpha$ 鐃塾一次鐃緒申鐃叔わ申鐃緒申錣件申覆鐃緒申鐃緒申鐃�
3.
$K$ 鐃塾醐申 $\beta$ 鐃叔￥申 $\beta^2=-1$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申里鐃� 鐃緒申鐃駿てわ申箸瓠� ${\Bbb F}_3$ 鐃緒申鐃緒申鐃緒申 $\alpha$ 鐃塾一次鐃緒申鐃叔わ申鐃緒申錣件申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 9.2   鐃縮随申 $q=p^r$ 鐃緒申鐃緒申 $k$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申 $k$ 鐃塾フワ申鐃駿ニワ申鐃緒申鐃緒申鐃緒申同鐃緒申 $F$ 鐃塾位随申鐃緒申鐃緒申茵ｏ申鐃緒申覆鐃緒申鐃緒申 $F^l=(\text{恒等写像})$ 鐃夙なわ申茲�鐃淑最常申鐃緒申鐃緒申鐃緒申鐃緒申 $l$ 鐃緒申鐃夙わ申茵� (鐃述ワ申函鐃�$k^\times$ 鐃緒申有鐃渋件申鐃淑ので￥申鐃緒申鐃緒申鐃緒申鐃銃の醐申鐃緒申 $x^{q-1}=1$ 鐃緒申澆鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃獣て￥申 $k=\{x\in k; x^q=x\} $ 鐃緒申鐃淑りた鐃縦￥申鐃塾常申鐃� $n$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃縦以駕申鐃塾解しわ申鐃盾た鐃淑わ申鐃緒申鐃夙にわ申鐃緒申鐃春わ申鐃処。)

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Yoshifumi Tsuchimoto
2000-06-21