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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.13

\fbox{ガロアの基本定理}

鐃淑駕申鐃叔は￥申鐃緒申鐃緒申鐃緒申鐃舜のわ申鐃潤、$k$ 鐃緒申無鐃渋個の醐申鐃緒申鐃緒申鐃塾でわ申鐃緒申箸鐃緒申襦� (鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申侶鐃縮わ申鐃術わ申鐃緒申弌鐃緒申鐃緒申両鐃緒申麓鐃緒申鐃緒申鐃緒申鐃緒申箸鐃緒申任鐃緒申襦�)

鐃緒申鐃� 13.1   鐃緒申 $k$ 鐃夙わ申鐃塾鰹申鐃緒申鐃緒申 $L$ 鐃夙わ申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $\operatorname{Aut}_k(L)$ 鐃緒申鐃緒申分鐃緒申 $H$ 鐃祝わ申鐃緒申鐃緒申鐃緒申

\begin{displaymath}L_H= \{x\in L; \text{$\forall h\in H$ にたいして、 $h(x)=x$ がなりたつ}\} \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦�(鐃緒申鐃緒申鐃緒申里砲覆襪鰹申箸鐃緒申動廚砲錣�鐃暑。) $L_H$ 鐃塾わ申鐃夙わ申 $H$ 鐃祝わ申鐃緒申鐃緒申鐃緒申里噺討屐鐃�

鐃緒申鐃緒申 13.1   鐃緒申 $k$ 鐃夙わ申鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $L$ 鐃夙わ申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $\operatorname{Aut}_k(L)$ 鐃緒申任鐃春わ申鐃緒申分鐃緒申 $H$ 鐃祝わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃塾わ申鐃夙わ申鐃緒申鐃緒申立鐃縦￥申

1.
$L$ 鐃緒申任鐃春の醐申 $x$ 鐃祝わ申鐃緒申鐃緒申鐃緒申

\begin{displaymath}\{h(x); h \in H\} \end{displaymath}

鐃緒申鐃出称種申鐃緒申 $L_H$ 鐃緒申属鐃緒申鐃暑。
2.

\begin{displaymath}[L: L_H]=\char93 H \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 13.1   鐃緒申 $k$ 鐃夙わ申鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $L$ 鐃夙わ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.
$L$ 鐃緒申 $k$ 鐃夙の間わ申任鐃春わ申鐃緒申鐃緒申鐃� $M$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申 $L$ 鐃緒申 $M$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申蝓� $\operatorname{Aut}_M(L)$ 鐃緒申 $G=\operatorname{Aut}_k(L)$ 鐃緒申鐃緒申分鐃緒申鐃叔わ申鐃暑。
2.
$\operatorname{Aut}_k(L)$ 鐃緒申鐃緒申分鐃緒申 $H$ 鐃緒申鐃出わ申鐃銃￥申 $L_H$ 鐃緒申 $k$ 鐃緒申 $L$ 鐃塾間わ申鐃緒申鐃緒申里任鐃緒申襦�
3.
鐃緒申鐃緒申鐃縦わ申鐃出縁申鐃熟互わ申鐃祝居申鐃出縁申鐃緒申与鐃緒申鐃暑。鐃緒申鐃淑わ申鐃緒申鐃� $G$ 鐃緒申鐃緒申分鐃緒申鐃夙￥申$L$ 鐃緒申 $k$ 鐃塾間わ申鐃緒申鐃緒申里箸楼鐃緒申舒鐃緒申鐃出縁申鐃緒申鐃暑。
4.
鐃緒申鐃緒申弍鐃緒申如鐃�$k$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申弍鐃緒申鐃緒申鐃塾わ申 $G$ 鐃緒申鐃緒申鐃緒申鐃緒申分鐃緒申鐃叔わ申鐃暑。

(鐃緒申鐃緒申)

1.
$\mbox{${\Bbb Q}$ }[\sqrt{2}+\sqrt{3}+\sqrt{5}]$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申粒鐃緒申膽￥申鐃緒申鐃緒申鐃処。
2.
$\sqrt{2}+\sqrt{3}+\sqrt{5}$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申虜脳鐃渋随申犲逸申鐃緒申鐃処。
3.
$\mbox{${\Bbb Q}$ }[\sqrt{2}+\sqrt{3}+\sqrt{5}]$ 鐃緒申鐃緒申分鐃塾わ申鐃緒申鐃銃居申鐃処。
4.
$\mbox{${\Bbb Q}$ }[\sqrt{2}+\sqrt{3}+\sqrt{6}]$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申粒鐃緒申膽￥申鐃緒申鐃緒申鐃処。
5.
$\mbox{${\Bbb Q}$ }[\sqrt{2+\sqrt{3}}]$ 鐃緒申佞鐃緒申鐃� $\mbox{${\Bbb Q}$ }$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申里鐃緒申鐃� 鐃叔常申鐃塾わ申鐃� $L_1$ 鐃緒申鐃緒申茵�
6.
$L_1$ 鐃緒申鐃緒申分鐃塾わ申鐃緒申鐃銃居申鐃処。
7.
$\mbox{${\Bbb Q}$ }[\sqrt{2+\sqrt{5}}]$ 鐃緒申佞鐃緒申鐃� $\mbox{${\Bbb Q}$ }$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申里鐃緒申鐃� 鐃叔常申鐃塾わ申鐃� $L_2$ 鐃緒申鐃緒申茵�
8.
$L_2$ 鐃緒申鐃緒申分鐃塾わ申鐃緒申鐃銃居申鐃処。

9.
$\gamma=\sqrt{2}+\sqrt{3}$ 鐃夙わ申鐃暑。 $\sqrt{2}$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申虜脳鐃渋随申犲� $f(X)=X^2-2$鐃緒申 $\sqrt{3}$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申虜脳鐃渋随申犲� $g(X)=X^2-3$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申$f(X)$ 鐃緒申 $g(\gamma-X)$ 鐃夙の削申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申茵� (鐃緒申鐃緒申鐃緒申 $\mbox{${\Bbb Q}$ }(\gamma)$ 鐃緒申鐃緒申鐃緒申多鐃準式鐃夙わ申鐃銃書くわ申鐃緒申) 鐃緒申鐃緒申法鐃�$\sqrt{2}$ 鐃緒申 $\gamma$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申鐃緒申鐃緒申有鐃緒申鐃緒申鐃夙わ申鐃緒申表鐃緒申鐃緒申鐃処。
10.
鐃緒申鐃緒申如鐃� $\sqrt{2},\sqrt{3}$ 鐃塾わ申鐃緒申鐃緒申 $\sqrt[3]{2},\sqrt[3]{3}$ 鐃淑わ申匹鐃緒申鐃緒申鐃緒申鐃緒申襪わ申蓮鐃� $\sqrt{2}+1,\sqrt{3}+5$ 鐃叔はどわ申鐃緒申鐃緒申


Yoshifumi Tsuchimoto
2000-07-06