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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.3

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{体の単純拡大}

鐃緒申鐃緒申鐃緒申鐃春わ申鐃緒申鐃処う鐃祝￥申鐃緒申弔鐃緒申里隆屬僚鐃銃縁申鐃緒申鐃宿�鐃緒申単鐃粛でわ申鐃暑。 鐃緒申鐃緒申鐃緒申鐃獣て￥申鐃塾の間の緒申同鐃緒申鐃緒申調鐃駿わ申鐃緒申鐃緒申蓮鐃� 鐃塾の間の「含む、鐃殉まわ申鐃塾関件申鐃竣￥申鐃緒申分鐃塾と鰹申鐃緒申鐃塾の関件申) 鐃緒申調鐃駿るこ鐃夙に居申鐃遵さ鐃緒申襦�

鐃緒申鐃緒申 3.1   鐃緒申 $k$ 鐃夙わ申鐃塾鰹申鐃緒申鐃緒申 $K$ 鐃緒申鐃緒申鐃獣て￥申$K=k(a)$ 鐃緒申鐃緒申鐃緒申立鐃獣てわ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃曙か鐃緒申鐃緒申鐃緒申立鐃縦￥申
1.
鐃緒申鐃緒申鐃緒申鐃緒申多鐃準式 $f\in k$ 鐃緒申鐃緒申鐃獣て￥申 $a$ 鐃緒申 $f(a)=0$ 鐃緒申澆鐃緒申鐃緒申鐃� 鐃緒申鐃塾とわ申鐃緒申$K$ 鐃緒申 $k[X]/(f(X))$ 鐃緒申同鐃緒申鐃叔わ申鐃暑。
2.
$k$ 鐃緒申鐃� 1 鐃術随申有鐃緒申鐃舜随申鐃緒申 $k(X)$ 鐃緒申 $K$ 鐃夙わ申 $k$-同鐃緒申 $\phi$ 鐃叔￥申 $\phi(X)=a$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申里鐃渋醐申澆鐃緒申襦�

鐃緒申鐃峻わ申鐃緒申鐃緒申立鐃縦誌申鐃緒申$a$ 鐃緒申 $k$ 鐃緒申鐃緒申鐃重�鐃叔わ申鐃緒申噺鐃緒申鐃緒申鐃緒申鐃峻わ申鐃緒申鐃緒申立鐃縦誌申鐃緒申 $a$ 鐃緒申 $k$ 鐃緒申超鐃緒申的鐃叔わ申鐃緒申噺鐃緒申鐃緒申襦�

鐃緒申 (超鐃循醐申鐃祝わ申鐃獣縁申鐃緒申鐃緒申鐃緒申)  
1.
${\Bbb C}(X) \supset {\Bbb C}$
2.
${\Bbb C}(X,Y)\supset {\Bbb C}(X)$

鐃緒申 (鐃緒申鐃重�鐃淑醐申鐃祝わ申鐃獣縁申鐃緒申鐃緒申鐃緒申)  
1.
${\Bbb C}(X)\supset {\Bbb C}(X^2)$
2.
${\Bbb C}(X)\supset {\Bbb C}(X^3)$
3.
${\Bbb C}(X)\supset {\Bbb C}(X^3+X+1)$
4.
${\Bbb C}(X,Y)\supset {\Bbb C}(X^2,Y)$
5.
${\Bbb C}(X,Y)(={\Bbb C}(XY,X+Y,X))\supset {\Bbb C}(XY,X+Y)$
6.
$\mbox{${\Bbb Q}$ }(\sqrt{2})\supset \mbox{${\Bbb Q}$ }$

鐃淑駕申鐃緒申鐃緒申 $k$ 鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃重�鐃緒申鐃緒申噺鐃緒申鐃緒申箸鐃緒申鐃� $k^n$ 鐃塾わ申鐃夙わ申 ${\Bbb A}^n(k)$ 鐃夙書いわ申 $k$ 鐃緒申鐃� $n$-鐃緒申鐃緒申鐃緒申鐃春ワ申鐃緒申鐃緒申屬鐃� 鐃緒申屬鐃緒申箸砲鐃緒申襦�

鐃緒申鐃� 3.1   鐃緒申鐃縮わ申鐃緒申 $k$ 鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申 $V\in {\Bbb A}^n(k)$ 鐃緒申鐃緒申鐃緒申任鐃緒申鐃夙わ申 (鐃緒申鐃緒申鐃緒申鐃緒申II鐃叔器申鐃祝わ申辰鐃緒申茲�鐃緒申 $A(V)$ 鐃緒申鐃緒申鐃緒申任鐃緒申鐃夙駕申鐃所す鐃緒申里鐃銃縁申鐃緒申任鐃緒申鐃�)鐃緒申 A(V) 鐃塾常申鐃緒申 $Q(A(V))$ 鐃塾わ申鐃夙わ申 $k(V)$ 鐃夙書き￥申$V$ 鐃緒申 ($k$ 鐃緒申鐃�)鐃舜随申鐃塾と呼ぶ￥申

$V$ 鐃塾関随申鐃塾わ申鐃緒申鐃重�鐃淑醐申鐃祝わ申鐃獣縁申鐃緒申鐃熟種申鐃塾よう鐃淑駈申鐃祝なってわ申鐃暑。

鐃緒申鐃緒申 3.2   鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申 $V\in {\Bbb A}^n(k)$ 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� 鐃緒申鐃塾とわ申鐃緒申
1.
$a$ 鐃緒申 $K$ 鐃緒申超鐃緒申的鐃淑わ申弌鐃� $K\to L$ 鐃緒申 $ V\times {\Bbb A}^1\to V$ 鐃淑わ申 多鐃準式鐃緒申鐃緒申(鐃粛縁申) $p$ 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。
2.
$a$ 鐃緒申 $K$ 鐃緒申鐃緒申鐃重�鐃淑わ申弌鐃� $V\times A^1$ 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 $p:V\times {\Bbb A}^1\to V$ 鐃緒申 $W$ 鐃舜わ申鐃緒申鐃緒申 $\Phi=p\vert _W$ 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 3.1   $L={\Bbb C}(T,U,V)$ 鐃緒申鐃緒申分鐃緒申 $K={\Bbb C}(T+U+V,TU+UV+VT,TUV)$ 鐃緒申箸鐃夙￥申$L$ 鐃塾醐申 $T$ 鐃緒申 $K$ 鐃緒申鐃緒申鐃重�鐃叔わ申鐃暑こ鐃夙わ申

\begin{displaymath}f(T)=0 \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申 $K$ 鐃緒申鐃�1鐃術随申多鐃準式 $f\in K[X]$ 鐃緒申鐃緒申鐃重�鐃緒申与鐃緒申鐃暑こ鐃夙で種申鐃緒申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 3.2   鐃緒申鐃� $L$ 鐃緒申 $K$ 鐃緒申単鐃緒申鐃緒申鐃叔わ申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申(鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申襦�)



2001-10-26