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鐃緒申鐃緒申鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.2

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{環の準同型と体の準同型}

鐃緒申鐃� 2.1 (鐃縦に醐申鐃緒申鐃春わ申鐃獣わ申鐃銃でわ申鐃緒申鐃緒申)   鐃緒申 $R$ 鐃夙￥申鐃緒申鐃緒申鐃緒申分鐃緒申 $S$, 鐃緒申鐃緒申鐃� $R$ 鐃塾醐申 $x_1,\dots,x_n$ 鐃緒申 与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申$S$ 鐃緒申 $x_1,\dots,x_n$ 鐃夙わ申泙鐃� $R$ 鐃緒申鐃緒申分鐃縦のわ申鐃緒申鐃叔常申鐃塾わ申里鐃� $S[x_1,\dots,x_n]$ (鐃術鰹申未鐃緒申鐃緒申鐃�) 鐃夙書き￥申$S$ 鐃緒申 $x_1,\dots x_n$ 鐃緒申鐃春わ申鐃獣わ申鐃銃でわ申鐃緒申弔噺討屐鐃�

鐃緒申鐃縮に￥申鐃緒申 $K$ 鐃夙￥申鐃緒申鐃緒申鐃緒申分鐃緒申 $k$, 鐃緒申鐃緒申鐃� $K$ 鐃塾醐申 $x_1,\dots,x_n$ 鐃緒申 与鐃緒申鐃緒申鐃銃わ申鐃緒申鐃夙わ申鐃銃も、 $k[x_1,\dots,x_n]$ 鐃緒申鐃塾とは限わ申覆鐃緒申鐃�

鐃緒申鐃� 2.2 (鐃塾に醐申鐃緒申鐃春わ申鐃獣わ申鐃銃でわ申鐃緒申鐃緒申)   鐃緒申 $K$ 鐃夙￥申鐃緒申鐃緒申鐃緒申分鐃緒申 $k$, 鐃緒申鐃緒申鐃� $K$ 鐃塾醐申 $x_1,\dots,x_n$ 鐃緒申 与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申$k$ 鐃緒申 $x_1,\dots,x_n$ 鐃夙わ申泙鐃� $K$ 鐃緒申鐃緒申分鐃塾のわ申鐃緒申鐃叔常申鐃塾わ申里鐃� $k(x_1,\dots,x_n)$ (鐃楯鰹申未鐃緒申鐃緒申鐃�) 鐃夙書き￥申$k$ 鐃緒申 $x_1,\dots,x_n$ 鐃緒申鐃春わ申鐃獣わ申鐃銃でわ申鐃緒申鐃塾と呼ぶ￥申

鐃緒申鐃緒申鐃縦わ申鐃緒申鐃緒申任蓮鐃緒申鐃緒申任鐃渋醐申澆鐃緒申鐃� $x_1,\dots,x_n$ 鐃緒申 鐃春わ申鐃獣わ申鐃暑こ鐃夙わ申佑鐃緒申討鐃緒申襪�鐃緒申 鐃緒申鐃術随申鐃緒申 $X_1,\dots,X_n$ 鐃緒申鐃春わ申鐃獣わ申鐃緒申鐃緒申里發緒申鐃祝準じ鐃緒申鐃緒申鐃緒申鐃緒申鐃暑。鐃緒申鐃淑わ申鐃緒申鐃�

鐃緒申 鐃術鰹申未抜欒鐃縮の使わ申分鐃緒申鐃熟種申鐃塾よう鐃祝覚わ申鐃銃わ申鐃緒申鐃緒申鐃宿わ申鐃緒申

鐃縦わ申鐃暑→鐃術鰹申鐃�

鐃塾わ申鐃暑→鐃楯鰹申鐃�

${\Bbb C}[X]$ 鐃緒申 ${\Bbb C}(X)$ 鐃夙の違い鐃緒申鐃緒か鐃暑こ鐃夙わ申鐃緒申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 2.1   鐃緒申鐃緒申 $R$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃重�鐃淑￥申分鐃緒申鐃竣わ申鐃緒申鐃緒申

\begin{displaymath}S=\{ a/b; a,b \in R; b \neq 0\} \end{displaymath}

鐃緒申鐃緒申鐃塾わ申同鐃粛関件申鐃緒申

\begin{displaymath}a_1/b_1 \sim a_2 /b_2 \ {\Leftrightarrow}\ a_1 b_2 -a_2 b_1=0 \end{displaymath}

鐃叔わ申鐃緒申鐃夙￥申鐃緒申鐃緒申鐃緒申 $S/\sim$ 鐃熟わ申鐃淑わ申鐃淳の誌申法

\begin{displaymath}a/b \pm c/d= (ad \pm cb)/(bd), \quad (a/b)\cdot (c/d)=ac/bd \end{displaymath}

鐃祝わ申辰鐃緒申里砲覆襦�

鐃緒申鐃� 2.3   鐃緒申鐃� $S/\sim$ 鐃塾わ申鐃夙わ申 $Q(R)$ 鐃夙書き￥申$R$ 鐃緒申鐃緒申鐃緒申鐃塾と呼ぶ￥申

鐃緒申鐃緒申 2.2   鐃緒申鐃緒申 $R$ 鐃緒申鐃緒申 鐃緒申鐃緒申 $S$ 鐃舜の緒申同鐃緒申 $\phi$ 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�
1.
$\phi$ 鐃緒申単鐃粛なわ申弌鐃� $Q(R)$ 鐃緒申鐃緒申 $Q(S)$ 鐃塾緒申同鐃緒申鐃緒申鐃緒申鐃祝鰹申張鐃緒申鐃緒申襦�
2.
鐃春に￥申 $\phi$ 鐃緒申 $Q(R)$ 鐃緒申鐃緒申 $Q(S)$ 鐃塾緒申同鐃緒申鐃緒申鐃緒申鐃祝鰹申張鐃緒申鐃緒申鐃淑わ申弌鐃� $\phi$ 鐃緒申単鐃粛でわ申鐃暑。

鐃緒申鐃緒申 2.3   鐃緒申鐃緒申 $R$ 鐃緒申鐃緒申 $k$ 鐃緒申弔箸鐃緒申鐃粛�鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃� $x_1,\dots,x_n$ 鐃夙わ申鐃緒申 $R$ 鐃緒申有鐃渋個の醐申鐃緒申鐃緒申鐃獣て￥申 $R=k[x_1,\dots,x_n]$ 鐃緒申鐃緒申鐃緒申立鐃獣てわ申鐃緒申箸鐃緒申襦� 鐃緒申鐃緒申法鐃� $k$ 鐃緒申鐃緒申鐃緒申鐃� $S$ 鐃夙￥申$Q(R)$ 鐃緒申鐃緒申 $Q(S)$ 鐃舜わ申 $k$-鐃緒申同鐃緒申鐃緒申鐃緒申 $\psi$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃緒申 $S$ 鐃塾醐申 $f$ 鐃夙環緒申同鐃緒申鐃緒申鐃緒申

\begin{displaymath}\phi_0: R \to S_f \end{displaymath}

鐃叔わ申鐃獣て￥申$\psi$ 鐃緒申 $\phi_0$ 鐃塾鰹申張鐃祝なってわ申鐃緒申鐃塾わ申存鐃淳わ申鐃暑。

鐃緒申鐃緒申 2.1   ${\Bbb C}(X,Y)$ 鐃緒申鐃緒申 ${\Bbb C}(X,Y)$ 鐃舜の緒申同鐃緒申鐃緒申鐃緒申 $\psi$ 鐃緒申鐃緒申

\begin{displaymath}\psi(p)=p(X,X/Y) \end{displaymath}

鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦�($\psi$ 鐃緒申鐃塾の緒申同鐃緒申鐃叔わ申鐃暑こ鐃夙わ申 鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒け鐃叔わ申覆鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任呂鐃緒申鐃緒申認鐃緒申討鐃緒申鐃緒申鐃緒申箸砲鐃緒申襦�) 鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃緒申2.3鐃祝わ申鐃緒申茲�鐃緒申 $f$ 鐃緒申 $\phi$ 鐃緒申鐃緒申茵�

鐃緒申鐃緒申 2.2   ${\Bbb C}(X,Y)$ 鐃緒申鐃緒申 ${\Bbb C}(X,Y)$ 鐃舜の緒申同鐃緒申鐃緒申鐃緒申 $\psi$ 鐃緒申

\begin{displaymath}\psi(p)=p(X,1/X) \end{displaymath}

鐃緒申与鐃緒申鐃暑こ鐃夙はでわ申鐃淑わ申鐃緒申鐃緒申鐃緒申呂覆鐃緒申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�



2001-10-02