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鐃緒申鐃緒申鐃�II 鐃緒申鐃緒申 No.2

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{環の標数、9去法、「11去法」}

${\mbox{${\Bbb Z}$ }}/9{\mbox{${\Bbb Z}$ }}$ 鐃叔の計誌申鐃熟￥申鐃緒申鐃祝�(鐃緒申鐃暑い鐃熟￥申鐃淑ワ申肇鐃緒申蠅�)鐃緒申名鐃緒申鐃塾わ申鐃銃わ申鐃緒申 鐃緒申里隆鐃獣縁申鐃緒申鐃緒申鐃緒申鐃粛随申鐃緒申襦�

鐃緒申鐃緒申 2.1   $3457$ 鐃緒申 $9$ 鐃叔鰹申辰鐃緒申鐃緒申泙鐃緒申鐃緒申茵�

(鐃緒申鐃緒申)
\begin{align*}&[3457]_9 \\ =&[3\times 10^3+4\times 10^2+5\times 10+7]_9\\ =&[3... ...^2+[5]_9\times [10]_9+[7]_9\\ =&[3]_9+[4]_9+[5]_9+[7]_9\\ =&[1]_9 \end{align*}

鐃縦まわ申 $[10]_9=[1]_9$ 鐃叔わ申鐃暑こ鐃夙わ申鐃緒申鐃緒申離櫂鐃緒申鐃夙なわけ鐃緒申鐃緒申 同鐃粛にわ申鐃銃￥申10鐃緒申法鐃叔書いわ申鐃緒申鐃緒申 $11$ 鐃叔鰹申辰鐃緒申鐃緒申泙鐃緒申 鐃緒申単鐃祝居申鐃暑こ鐃夙わ申鐃叔わ申鐃暑。( $[10]_{11}=[-1]_{11}$)鐃緒申鐃緒申鐃緒申)

鐃緒申 $R$ 鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃緒申単鐃縮醐申鐃緒申 $1_R$, 鐃夙書き￥申 $1_R+1_R$ 鐃緒申 $2_R$鐃夙わ申鐃緒申鐃緒申 $3_R$, $4_R$ 鐃緒申鐃緒申同鐃粛￥申(鐃緒申鐃緒申 $?_R$ 鐃緒申 $R$ 鐃熟￥申鐃出逸申鐃熟削申廚函鐃緒申舒鐃緒申鐃熟削申鐃� 鐃緒申鐃緒申未鐃緒申鐃塾わ申 $\text{石井}_{\text{浩}}$ 鐃夙わ申 $\text{石井}_{\text{琢}}$鐃夙わ申鐃緒申鐃塾わ申同鐃粛で￥申鐃緒申鐃緒申覆鐃緒申討鐃淑�鐃緒申鐃緒申箸鐃緒申砲肋鐃塾�鐃緒申鐃暑。)

鐃緒申鐃� 2.1 (鐃縦わ申標鐃緒申)   鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃緒申鐃緒申鐃獣て￥申

\begin{displaymath}n_R=0 \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦とわ申鐃緒申鐃緒申鐃塾よう鐃緒申 $n$ 鐃塾なわ申鐃緒申鐃緒申鐃叔最常申鐃塾わ申里鐃� $R$ 鐃緒申標鐃緒申鐃夙呼ぶ￥申 鐃緒申鐃塾よう鐃緒申 $n$ 鐃緒申鐃淑わ申鐃夙わ申鐃祝は￥申$R$ 鐃緒申標鐃緒申鐃緒申 $0$ 鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃暑。 $R$ 鐃緒申標鐃緒申鐃塾わ申鐃夙を、以駕申鐃叔わ申 $\operatorname{char}(R)$ 鐃夙書く￥申

鐃緒申   ${\mbox{${\Bbb Z}$ }}/n{\mbox{${\Bbb Z}$ }}$ 鐃緒申標鐃緒申鐃緒申 $n$ 鐃叔わ申鐃暑。

鐃緒申如鐃緒申鐃粛常申鐃竣とわ申鐃緒申鐃緒申里鐃緒申任鐃緒申弔鐃緒申任法鐃緒申鐃緒申鐃銃縁申里痢鐃粛常申鐃緒申鐃緒申鐃緒申鐃緒申鐃所算鐃緒申 鐃祝つわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃塾講居申鐃叔わ申鐃獣わ申鐃叔わ申覆鐃緒申造鐃熟種申鐃塾逸申味鐃塾鰹申蟷誌申鬚垢襦�

鐃緒申鐃� 2.2   $m,n\in {\mbox{${\Bbb Z}$ }}$ 鐃夙わ申鐃緒申$n\neq 0$ 鐃叔わ申鐃緒申箸鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�$m$ 鐃緒申 $n$ 鐃叔鰹申辰鐃緒申鐃� $q$ 鐃夙￥申余鐃緒申 $r$ 鐃夙は￥申鐃緒申鐃塾関件申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申唯鐃緒申里鐃塾とわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃暑。

\begin{displaymath}m=nq+r \qquad 0\leq r \leq \vert n\vert-1 \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申任蓮鐃� $m$ 鐃緒申 $n$ 鐃緒申鐃緒申凌鐃緒申任盥緒申鐃淑わ申鐃夙わ申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃春わ申鐃銃わ申鐃緒申鐃緒申 余鐃緒申鐃緒申鐃緒申砲弔鐃緒申童鐃緒申鐃緒申弌鐃緒申鐃緒申里茲�鐃淑醐申鐃緒申鐃緒申鐃緒申鐃緒申鐃叔緒申任鐃緒申襦�

\begin{displaymath}\text{$m$ を $n$ で割った余りが $r$} \quad {\Leftrightarrow}\... ...=[r]_{\vert n\vert} \text{ かつ } 0\leq r \leq \vert n\vert-1) \end{displaymath}

鐃緒申蟷誌申鐃緒申僂鐃緒申銅鐃緒申侶鐃縮わ申鐃緒申鐃緒申任鐃緒申鐃�

鐃緒申鐃緒申 2.1   ${\mbox{${\Bbb Z}$ }}$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃� $0$ 鐃緒申鐃緒申鐃殉わ申鐃緒申 $n{\mbox{${\Bbb Z}$ }}$ ($n$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)鐃塾件申鐃塾わ申里妨造襦�

鐃緒申鐃藷、誌申鐃瞬わ申余鐃獣わ申鐃塾で種申鐃緒申鐃緒申鐃緒申砲弔鐃緒申峠劼戮鐃�

鐃緒申鐃緒申 2.2   鐃緒申鐃緒申鐃緒申鐃緒申 $n$ 鐃祝つわ申鐃銃￥申

鐃処週鐃述べわ申鐃緒申鐃緒申鐃緒申鐃緒申両鐃緒申鐃緒申呂發�鐃緒申鐃緒申鐃緒申鐃縮駕申鐃叔わ申鐃暑。

鐃殉わ申鐃緒申鐃塾醐申鐃春わ申鐃術意わ申鐃銃わ申鐃緒申鐃緒申鐃緒申

鐃緒申鐃� 2.3  
1.
鐃緒申 $R$ 鐃塾醐申 $x$ 鐃緒申鐃緒申鐃緒申鐃述でわ申鐃緒申箸蓮鐃緒申鐃緒申鐃� $R$ 鐃塾醐申 $y\neq 0$ 鐃緒申鐃緒申鐃獣て￥申 $xy=0$ 鐃緒申鐃緒申鐃緒申立鐃縦とわ申鐃祝醐申鐃緒申鐃緒申
2.
鐃緒申 $R$ 鐃緒申 $0$ 鐃淑鰹申鐃緒申鐃緒申鐃緒申鐃述わ申鐃緒申鐃緒申覆鐃緒申箸鐃緒申鐃�$R$ 鐃緒申鐃緒申鐃緒申任鐃緒申鐃夙醐申鐃緒申鐃緒申

鐃緒申鐃緒申 2.3  
1.
鐃緒申鐃塾個随申鐃緒申有鐃渋の環わ申鐃緒申鐃緒申鐃緒申鐃述わ申必鐃緒申鐃縦逆でわ申鐃暑。
2.
鐃緒申鐃塾個随申鐃緒申有鐃渋の器申 $R$ 鐃緒申鐃緒申鐃緒申覆鐃出￥申$R$ 鐃緒申鐃塾でわ申鐃暑。

鐃緒申檗鐃緒申鐃緒申鐃緒申蠅�複鐃緒申鐃緒申鐃緒申鐃緒申砲蓮鐃緒申鐃縦わ申鐃緒申鐃緒申撚鬚�わ申鐃夙￥申 鐃楯醐申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃処い鐃緒申鐃緒申鐃塾常申鐃祝わ申評鐃緒申鐃熟よい鐃緒申鐃塾わ申里鐃粛随申鐃緒申襦� (鐃渋にはなわ申覆鐃緒申鐃�)

鐃緒申鐃緒申 2.1   10鐃緒申法鐃叔書いわ申鐃緒申鐃緒申 $n$ (鐃緒申: $n=1234567890$) 鐃緒申13 鐃叔鰹申辰鐃粛常申鐃緒申鐃緒申鐃塾は￥申 9鐃緒申11鐃塾とわ申鐃循ど器申単鐃叔はなわ申鐃緒申鐃緒申鐃縦わ申鐃塾よう鐃祝でわ申鐃暑。
1.
$n$ 鐃薯三件申鐃緒申剖鐃緒申擇襦�( $n=\vert 1\vert 234\vert 567\vert 890\vert$ )
2.
鐃緒申鐃准っわ申鐃緒申里鬚修譴常申鐃緒申鐃緒申鐃縦わ申鐃銃加わ申鐃暑。

\begin{displaymath}890-567+234-1 =556 \end{displaymath}

3.
鐃獣わ申鐃緒申鐃緒申鐃緒申鐃緒申 $13$鐃叔鰹申辰鐃粛常申鐃緒申鐃緒申襦� ($556$ 鐃緒申 $13$ 鐃叔鰹申辰鐃粛常申鐃緒申 $10$)
鐃緒申鐃塾最醐申鐃緒申鐃緒申鐃緒申鐃� $n$ 鐃緒申 $13$ 鐃叔鰹申辰鐃緒申鐃緒申鐃緒申任鐃緒申襦ｏ申鐃緒申鐃熟なわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃処。

鐃緒申鐃緒申 2.2   鐃緒申鐃銃縁申佑聞融鐃緒申砲鐃所、 10鐃緒申法鐃叔書いわ申鐃緒申鐃緒申鐃緒申 $17$鐃叔鰹申辰鐃粛常申鐃緒申岼鐃緒申鐃塾件申鐃緒申鐃祝駈申鐃准って￥申 鐃緒申鐃緒申鐃緒申法鐃熟なわ申鐃緒申鐃緒申

鐃緒申鐃緒申 2.3   鐃緒申鐃縮に￥申鐃叔随申 $p$ 鐃緒申鐃出わ申鐃銃￥申 10鐃緒申法鐃叔書いわ申鐃緒申鐃緒申鐃緒申 $p$ 鐃叔鰹申辰鐃粛常申鐃緒申岼鐃緒申鐃塾件申鐃緒申鐃祝駈申鐃准って￥申 鐃緒申鐃緒申鐃緒申法鐃熟わ申鐃縦でわ申存鐃淳わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� (但鐃緒申鐃緒申鐃緒申鐃緒申鐃� $p= 2$ 鐃緒申 $p=5$ 鐃塾常申鐃緒申鐃純外鐃夙わ申鐃暑。)



2002-04-18