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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.5

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{体の拡大次数II}

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 5.1   鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申 $V\in {\Bbb A}^n({\Bbb C})$ 鐃緒申鐃緒申鐃獣て￥申 $V$ 鐃塾関随申鐃緒申 $K={\Bbb C}(V)$ 鐃緒申単鐃緒申鐃緒申鐃� $L=K(a)$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃殉￥申$a$ 鐃緒申 $K$ 鐃緒申鐃緒申鐃重�鐃夙わ申鐃緒申函鐃緒申鐃緒申里箸鐃緒申鐃�
1.
鐃塾の鰹申鐃順次鐃緒申 $[L:K]$ 鐃緒申 $a$ 鐃塾最常申多鐃準式鐃塾種申鐃緒申 $d$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申
2.
$V\times A^1$ 鐃緒申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $W$ 鐃緒申鐃緒申鐃獣て￥申$K\to L$ 鐃緒申 鐃粛縁申 $V\times {\Bbb A}^1\to V$ 鐃緒申 $W$ 鐃舜わ申鐃緒申鐃緒申 $f$ 鐃緒申鐃緒申 鐃緒申泙鐃舜随申鐃塾の種申鐃緒申鐃緒申同鐃緒申襪居申鐃暑。(鐃緒申鐃緒申)
3.
鐃緒申鐃緒申 $f$ 鐃熟「ほとわ申匹両鐃緒申如鐃� $d:1$ 鐃叔わ申鐃暑。 鐃緒申鐃淑わ申鐃緒申鐃�$V$ 鐃緒申鐃緒申鐃重�鐃緒申鐃緒申分鐃緒申鐃緒申 $V_s$ 鐃緒申鐃緒申鐃獣て￥申$V_s$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃淑�鐃叔わ申 $f$ 鐃緒申 $d:1$ 鐃塾種申鐃緒申鐃叔わ申鐃暑。

鐃緒申   $K={\Bbb C}(X^2) \subset {\Bbb C}(X)=L$ 鐃夙わ申鐃緒申鐃夙￥申 $L$ 鐃緒申 $K$ 鐃緒申 $2$ 鐃緒申鐃緒申鐃緒申鐃塾でわ申鐃暑。 鐃緒申鐃塾鰹申鐃緒申鐃緒申弍鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申隆屬亮鐃緒申鐃�(鐃塾逸申鐃�)鐃緒申

\begin{displaymath}{\Bbb A}^1({\Bbb C}) \ni z \mapsto z^2 \in {\Bbb A}^1({\Bbb C}) \end{displaymath}

鐃叔わ申鐃暑。 鐃緒申鐃塾種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申茲�鐃祝￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $2:1$ 鐃塾種申鐃緒申鐃叔わ申鐃暑。

鐃緒申   $K={\Bbb C}(X^3) \subset {\Bbb C}(X)=L$ 鐃夙わ申鐃緒申鐃夙￥申 $L$ 鐃緒申 $K$ 鐃緒申 $3$ 鐃緒申鐃緒申鐃緒申鐃塾でわ申鐃暑。 鐃緒申鐃塾鰹申鐃緒申鐃緒申弍鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申隆屬亮鐃緒申鐃�(鐃塾逸申鐃�)鐃緒申

\begin{displaymath}{\Bbb A}^1({\Bbb C}) \ni z \mapsto z^3 \in {\Bbb A}^1({\Bbb C}) \end{displaymath}

鐃叔わ申鐃暑。 鐃緒申鐃塾種申鐃緒申鐃熟醐申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $3:1$ 鐃塾種申鐃緒申鐃叔わ申鐃暑。

鐃緒申   $K={\Bbb C}(A,X)$ 鐃塾鰹申鐃緒申鐃緒申 $L$ 鐃緒申 $L=K[Y]/(Y^2-X^3-A)$ 鐃緒申鐃緒申鐃緒申鐃緒申襦� $L$ 鐃緒申 $K$ 鐃緒申 $2$ 鐃緒申鐃緒申鐃緒申鐃塾でわ申鐃暑。 鐃緒申鐃塾鰹申鐃緒申鐃緒申弍鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申隆屬亮鐃緒申鐃�(鐃塾逸申鐃�)鐃緒申

\begin{displaymath}({\Bbb A}^3({\Bbb C}) \subset) V(Y^2-X^3-A) \ni(x,y,a) \mapsto (x,a) \in {\Bbb A}^2({\Bbb C}) \end{displaymath}

鐃叔わ申鐃暑。 鐃緒申鐃塾種申鐃緒申鐃熟駈申鐃緒申 $V(X^3-A)$ 鐃緒申鐃緒申鐃緒申鐃� $2:1$ 鐃塾種申鐃緒申鐃叔わ申鐃暑。

鐃緒申鐃緒申 5.1   鐃緒申鐃緒申鐃祝緒申鐃獣て￥申鐃塾わ申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃緒申鐃所、鐃緒申鐃塾鰹申鐃順次鐃緒申鐃緒申 鐃緒申鐃淑わ申鐃緒申鐃緒申



2001-10-31