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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.12

\fbox{分離拡大(先週の残り)}

鐃緒申鐃緒申 12.1   $k$ 鐃緒申 無鐃渋個の醐申鐃緒申泙鐃緒申里任鐃緒申鐃夙わ申鐃暑。 $k$ 鐃緒申鐃緒申鐃重�鐃淑醐申 $\alpha,\beta$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃銃￥申$\alpha$ 鐃緒申 $k$ 鐃緒申 分離的鐃叔わ申鐃獣わ申鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申
1.
$k[\alpha]$ 鐃塾醐申鐃熟わ申鐃駿わ申 $k$ 鐃緒申分離的鐃叔わ申鐃暑。
2.
鐃緒申鐃緒申 $c\in k$ 鐃緒申存鐃淳わ申鐃銃￥申 $k[c\alpha+\beta]=k[\alpha,\beta]$ 鐃緒申鐃淑りた鐃縦￥申
3.
鐃緒申鐃緒申鐃� $\beta$ 鐃緒申 $k$ 鐃緒申分離的鐃叔わ申鐃緒申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申鐃� $c$ 鐃祝わ申鐃緒申鐃緒申鐃緒申 $\gamma=c\alpha+\beta$ 鐃夙わ申鐃緒申鐃夙￥申 $\gamma$ 鐃緒申 $k$ 鐃緒申分離的鐃叔わ申鐃暑。

鐃緒申鐃緒申鐃緒申鐃緒申 $c$ 鐃緒申鐃緒申鐃俊わ申鐃緒申鐃緒申

$\alpha$ 鐃緒申 $k$ 鐃緒申龍鐃緒申鐃緒申 $\alpha=\alpha_1,\alpha_2,\dots,\alpha_l$鐃夙わ申鐃緒申 $\beta$ 鐃緒申 $k$ 鐃緒申龍鐃緒申鐃緒申 $\beta=\beta_1,\beta_2,\dots,\beta_m$鐃夙わ申鐃緒申箸鐃緒申鐃�$c$ 鐃熟￥申

\begin{displaymath}\frac{\beta_j-\beta}{\alpha_i-\alpha} \quad(i=2,3,\dots,l; \quad j=1,2,3,\dots,m) \end{displaymath}

鐃塾どわ申箸鐃循わ申茲�鐃緒申鐃緒申鐃駿ばよい鐃緒申

鐃緒申鐃緒申 12.1   鐃緒申 $k$ 鐃緒申鐃緒申咾鐃緒申粒鐃緒申鐃緒申鐃� $L$ 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

\begin{displaymath}\{x\in L; \text{$x$ は $k$ 上分離的である。}\} \end{displaymath}

鐃緒申 $k$ 鐃緒申泙鐃� $L$ 鐃緒申鐃緒申分鐃塾でわ申鐃暑。

鐃緒申鐃� 12.1   鐃緒申 $k$ 鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申 $L$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $L$ 鐃塾どの醐申 $x$ 鐃緒申 $k$ 鐃緒申分離的鐃叔わ申鐃緒申箸鐃緒申鐃� $L$ 鐃緒申 $k$ 鐃緒申分離鐃緒申鐃緒申鐃塾でわ申鐃緒申箸鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 12.2   鐃緒申 $k$ 鐃夙￥申鐃緒申鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $L=k[\alpha_1,\alpha_2,\dots,\alpha_n]$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�$L$ 鐃緒申 $k$ 鐃緒申分離鐃緒申鐃緒申鐃塾でわ申鐃暑た鐃緒申鐃� 必鐃竣緒申分鐃緒申鐃熟￥申 $\alpha_1,\dots,\alpha_n$ 鐃緒申鐃緒申鐃駿わ申 $k$ 鐃緒申 分離的鐃叔わ申鐃暑こ鐃夙でわ申鐃暑。

鐃緒申鐃緒申 12.3   無鐃渋個の醐申鐃緒申鐃緒申鐃緒申 $k$ 鐃緒申有鐃渋種申分離鐃緒申鐃緒申鐃緒申 $L$ 鐃緒申必鐃緒申単鐃緒申鐃緒申鐃緒申里任鐃緒申襦�

\fbox{ガロア拡大}

鐃緒申鐃緒申鐃緒申鐃緒申如鐃緒申鐃緒申鐃淑�離鐃緒申鐃緒申任鐃緒申鐃緒申里髻▲鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申 12.4   鐃緒申 $k$ 鐃夙￥申鐃緒申鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $L$ 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

\begin{displaymath}\char93 \operatorname{Aut}_k(L)=[L:k] \end{displaymath}

鐃緒申鐃淑りた鐃縦￥申

(鐃緒申鐃緒申)

鐃緒申鐃縮に￥申$L$ 鐃緒申 $k$ 鐃緒申単鐃緒申鐃緒申鐃淑わ申弌鐃�

\begin{displaymath}\char93 \operatorname{Aut}_k(L)\leq [L:k] \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦のでわ申鐃獣わ申(鐃緒申鐃緒申 8.1)鐃緒申 鐃渋は￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $L$ 鐃緒申 $k$ 鐃緒申単鐃緒申鐃緒申鐃叔なわ申鐃夙も、鐃緒申鐃縦でわ申鐃緒申鐃緒申立鐃縦￥申 鐃緒申鐃緒申12.3鐃薯参照のわ申鐃緒申

鐃緒申鐃緒申 12.1   $\alpha=\sqrt{5+\sqrt{2}}$ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申鐃緒申
1.
$f(\alpha)=0$ 鐃夙なわ申鐃祝ッワ申鐃緒申4鐃緒申鐃緒申 $f\in \mbox{${\Bbb Q}$ }[X]$ 鐃緒申鐃緒申与鐃緒申鐃淑わ申鐃緒申鐃緒申
2.
鐃緒申鐃渋随申犲� $f$ 鐃塾削申鐃緒申鐃緒申鐃銃居申鐃淑わ申鐃緒申鐃緒申
3.
$\alpha$ 鐃緒申 $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ 鐃塾醐申鐃叔はなわ申鐃緒申鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申 (鐃述ワ申鐃�:鐃盾し鐃緒申 $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ 鐃塾醐申 $\beta$鐃叔￥申 $\beta^2=5+\sqrt{2}$ 鐃夙なわ申鐃塾わ申 鐃緒申鐃獣わ申鐃夙わ申鐃緒申函鐃� $\sqrt{23}\in \mbox{${\Bbb Q}$ }[\sqrt{2}]$.)
4.
$\mbox{${\Bbb Q}$ }[\alpha]$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含わ申 $\mbox{${\Bbb Q}$ }$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申蓮鐃� 鐃緒申鐃淑らず $M=\mbox{${\Bbb Q}$ }[\sqrt{2},\sqrt{23}]$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含むこ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申
5.
$\alpha$ 鐃緒申 $M$ 鐃祝わ申属鐃緒申鐃淑わ申鐃緒申鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申 (鐃述ワ申鐃�: $M$ 鐃塾醐申鐃熟わ申鐃淑らず $x+y \sqrt{23}$ ( $x,y\in \mbox{${\Bbb Q}$ }[\sqrt{2}]$) 鐃夙書くわ申鐃夙わ申鐃叔わ申鐃暑。鐃緒申鐃緒申鐃緒申鐃処し鐃緒申 $5+\sqrt{2}$ 鐃祝なるこ鐃夙わ申鐃緒申鐃暑か鐃宿わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申茵�)
6.
$\mbox{${\Bbb Q}$ }[\alpha]$ 鐃緒申泙鐃� $\mbox{${\Bbb Q}$ }$ 鐃塾最常申鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申呂覆砲鐃緒申鐃�

鐃緒申鐃緒申 12.2   ${\Bbb F}_p$ 鐃夙￥申鐃緒申鐃緒申 $n$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃� $L$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申
1.
$\operatorname{Aut}_{{\Bbb F}_p}(L)$ 鐃塾位随申鐃緒申鐃緒申茵�
2.
鐃春ワ申鐃駿ニワ申鐃緒申鐃緒申鐃緒申 $F$ 鐃塾位随申鐃緒申鐃緒申茵�
3.
$\operatorname{Aut}_{{\Bbb F}_p}(L)$ 鐃塾件申鐃緒申造鐃緒申鐃緒申蠅誌申茵�

鐃緒申鐃緒申 12.3   鐃緒申 $k$ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $L$ 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.

\begin{displaymath}M=\{x\in L; \exists l\in {\mbox{${\Bbb Z}$}}s.t. x^{p^l}\in k\} \end{displaymath}

鐃緒申 $k$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含わ申泙鐃� $L$ 鐃緒申鐃緒申分鐃塾でわ申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申
2.
任鐃春わ申 $f \in \operatorname{Aut}_k(L) $ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申$f\vert M=id$ 鐃叔わ申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申
3.
$L$ 鐃緒申 $M$ 鐃緒申鐃淑�離鐃緒申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�
4.
$L$ 鐃緒申 $M$ 鐃緒申単鐃緒申鐃緒申鐃叔わ申鐃暑こ鐃緒申(No. 11 鐃緒申鐃緒申鐃緒申11.3鐃薯参照のわ申鐃緒申)鐃緒申鐃緒申鐃術わ申鐃銃￥申 鐃緒申鐃緒申鐃緒申 $\char93 \operatorname{Aut}_k(L)\leq [M:k] \leq [L:k]$ 鐃緒申鐃淑りた鐃縦わ申鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申 (鐃述ワ申鐃�:No.8鐃盾参鐃夙のわ申鐃緒申)

鐃緒申鐃緒申 12.4   $k$ 鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申 $L$ 鐃緒申鐃緒申

\begin{displaymath}\char93 \operatorname{Aut}_k(L)=[L:k] \end{displaymath}

鐃緒申澆鐃緒申鐃緒申覆鐃出￥申$L$ 鐃緒申 $k$ 鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�

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Yoshifumi Tsuchimoto
2000-07-03