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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.6

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{標数 $0$\space の体の有限次代数拡大は単純拡大である}

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鐃緒申鐃緒申鐃緒申鐃緒申鐃� $n$ 鐃緒申鐃出わ申鐃緒申 $K$ 鐃塾なわ申鐃緒申 $K$ 鐃緒申単鐃縮醐申 $1_K$ 鐃緒申 $n$ 鐃縦わ申鐃緒申

\begin{displaymath}n_K=(1_K+\dots+1_K) \quad (\text{$n$ 個の和}) \end{displaymath}

鐃緒申佑鐃緒申襦� $n_K$ 鐃熟醐申鐃緒申龍鐃緒申鐃塾なわ申鐃緒申鐃祝わ申単鐃緒申 $n$ 鐃夙書かわ申襦� 鐃緒申鐃春わ申鐃駿わ申鐃淑のは￥申鐃緒申鐃緒申鐃夙わ申鐃緒申 $n\neq 0$ 鐃叔わ申 $n_K\neq 0$ 鐃夙は限わ申覆鐃緒申鐃緒申鐃� 鐃叔わ申鐃暑。

鐃緒申鐃� 6.1   $K$ 鐃緒申鐃出わ申鐃銃￥申 $n_K=0$ 鐃夙なわ申脳鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃渋醐申澆鐃緒申鐃出￥申鐃緒申鐃緒申鐃� $\operatorname{char}K$ 鐃夙書き￥申鐃緒申 $K$ 鐃緒申標鐃緒申鐃夙呼ぶ￥申 鐃緒申鐃塾よう鐃緒申鐃緒申鐃緒申鐃緒申存鐃淳わ申鐃淑わ申鐃緒申鐃祝わ申 $\operatorname{char}K=0$ 鐃緒申鐃緒申鐃緒申鐃緒申襦�

鐃緒申   鐃叔随申 $p$ 鐃緒申鐃出わ申鐃銃￥申 ${\mbox{${\Bbb Z}$ }}/p{\mbox{${\Bbb Z}$ }}$ 鐃緒申鐃塾でわ申鐃所、鐃緒申鐃緒申標鐃緒申鐃緒申 $p$ 鐃叔わ申鐃暑。 $\mbox{${\Bbb Q}$ },\mbox{${\Bbb R}$ },{\Bbb C}$ 鐃淑ど￥申 $\mbox{${\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 $0$ 鐃叔わ申鐃暑。

鐃緒申鐃緒申 6.1   鐃塾わ申標鐃緒申鐃緒申必鐃緒申鐃叔随申鐃緒申鐃殉わ申鐃緒申 $0$ 鐃叔わ申鐃暑。

鐃縦わ申鐃緒申鐃緒申鐃緒申鐃宿醐申鐃� $0$ 鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃緒申鐃出わ申鐃銃わ申必鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申劼戮鐃緒申茲�鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申能鐃叔わ申鐃暑こ鐃夙を示わ申鐃銃わ申鐃暑。

鐃緒申鐃緒申 6.2   鐃緒申 $K$ 鐃緒申標鐃緒申鐃緒申 $0$ 鐃淑わ申弌鐃�$K$ 鐃緒申任鐃春わ申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃� $L$ 鐃緒申 単鐃緒申鐃緒申鐃叔わ申鐃暑。

鐃緒申鐃緒申鐃緒申鐃緒申楼鐃緒申辰討鐃緒申鐃緒申里亡愀鐃緒申覆鐃緒申鐃緒申鐃塾�鐃縦わ申鐃緒申 ${\Bbb C}$ 鐃緒申鐃緒申鐃緒申多鐃緒申鐃塾の関随申鐃塾の常申鐃祝は器申鐃緒申鐃緒申的鐃緒申鐃緒申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 6.3   $K\subset M\subset L$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃縦わ申鐃緒申 $K,M,L$ 鐃緒申 与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申鐃緒申鐃緒申鐃�

\begin{displaymath}[L:K]=[L:M][M:K] \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 6.1   $L={\Bbb C}(X,Y,Z), K={\Bbb C}(X+Y+Z,XY+YZ+ZX,XYZ)$ 鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申
1.
$X$ 鐃緒申 $K$ 鐃緒申虜脳鐃渋随申犲逸申鐃緒申鐃淑わ申鐃緒申鐃緒申
2.
$M=K(X)$ 鐃夙わ申鐃緒申鐃夙わ申鐃緒申 $[L:M], [M:K]$ 鐃薯それぞ鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 6.2   鐃緒申鐃緒申鐃緒申鐃叔￥申$L=K(X+cY)$ 鐃夙なわ申茲�鐃緒申 $c\in {\Bbb C}$ 鐃緒申鐃緒申鐃� (鐃緒申鐃緒申6.2 鐃塾常申鐃緒申鐃薯参考わ申)鐃緒申弔鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�



2001-11-05