next up previous
Next: About this document ...

    

鐃緒申鐃緒申鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.4

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{体の拡大次数}

鐃緒申鐃� 4.1   鐃緒申 $K$ 鐃塾鰹申鐃緒申鐃緒申 $L$ 鐃緒申鐃出わ申鐃銃￥申$L$ 鐃緒申 $K$-鐃駿ワ申鐃夙ワ申鐃緒申屬箸鐃緒申討鐃� 鐃緒申鐃緒申 $\dim_K L$ 鐃塾わ申鐃夙わ申 $L$ 鐃緒申 $K$ 鐃緒申鐃� 鐃緒申鐃順次鐃緒申鐃夙呼び￥申$[L:K]$ 鐃叔緒申表鐃緒申鐃緒申 $[L:K]<\infty$ 鐃塾とわ申鐃緒申$L$ 鐃緒申 $K$ 鐃緒申 有鐃渋種申鐃緒申鐃緒申鐃緒申鐃夙呼ばわ申襦�

鐃緒申  
1.
$[\mbox{${\Bbb Q}$ }(\sqrt{2}):\mbox{${\Bbb Q}$ }]=2$.
2.
$[\mbox{${\Bbb Q}$ }(X):\mbox{${\Bbb Q}$ }(X^2)]=2$.
3.
$[{\Bbb C}(X):{\Bbb C}(X^3)]=3$.
4.
$[{\Bbb C}:\mbox{${\Bbb R}$ }]=2$.
5.
$[\mbox{${\Bbb R}$ }:\mbox{${\Bbb Q}$ }]=\infty$.

鐃緒申鐃緒申 4.1  
鐃緒申 $K$ 鐃緒申有鐃渋種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鐃� $L$ 鐃緒申任鐃春の醐申 $a$ 鐃緒申 $K$ 鐃緒申鐃緒申鐃重�鐃叔わ申鐃暑。鐃緒申鐃淑わ申鐃緒申鐃� $K$ 鐃緒申里鐃緒申鐃�1鐃術随申多鐃準式 $f(\neq 0)$ 鐃緒申鐃緒申鐃獣て￥申$f(a)=0$ 鐃緒申澆鐃緒申鐃緒申鐃� 鐃緒申鐃緒申法鐃緒申鐃緒申里茲�鐃緒申 $f$ 鐃夙わ申鐃銃削申鐃純次鐃緒申鐃塾わ申鐃� $f_0$ 鐃緒申箸鐃夙￥申$f_0$ 鐃熟器申鐃緒申如鐃�

\begin{displaymath}\{h\in K[X]; h(a)=0\} \end{displaymath}

鐃緒申 $f_0 K[X]$ 鐃夙逸申鐃竣わ申鐃暑。

鐃緒申鐃� 4.2   鐃緒申鐃� $f_0$ 鐃塾わ申鐃夙わ申 $a$ 鐃緒申 $K$ 鐃緒申虜脳鐃渋随申犲逸申噺討屐鐃�

鐃緒申鐃塾誌申鐃渋はわ申鐃塾講居申鐃緒申鐃緒申的鐃塾わ申鐃緒申砲鐃宿�鐃旬わ申鐃塾種申鐃夙わ申鐃緒申鐃緒け鐃叔はなわ申鐃緒申鐃緒申 鐃獣わ申鐃緒申鐃巡す鐃緒申鐃淑わ申里濃箸鐃緒申鐃緒申箸砲鐃緒申襦�

鐃緒申鐃緒申 4.1 (``鐃緒申鐃緒申悗隆鐃緒申鐃緒申鐃緒申鐃�'')   複鐃叔随申鐃緒申 ${\Bbb C}$ 鐃緒申鐃緒申鐃重�鐃緒申鐃塾でわ申鐃暑。鐃緒申鐃淑わ申鐃緒申鐃�${\Bbb C}$ 鐃緒申鐃叔わ申佞裡鐃緒申竸鐃渋随申犲� (鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)鐃緒申必鐃緒申 ${\Bbb C}$ 鐃塾なわ申鐃祝削申鐃緒申鐃緒申帖鐃�

鐃緒申鐃緒申鐃緒申複鐃叔駕申鐃熟学わ申鐃塾ってわ申鐃緒申覆鐃出削申鐃緒申鐃粛の醐申鐃緒申鐃緒申箸鐃緒申里鐃緒申鐃獣とわ申鐃緒申鐃緒申任鐃緒申鐃緒申鐃緒申鐃�

鐃緒申 4.1   複鐃叔随申鐃塾常申鐃�1鐃術随申 $d$-鐃緒申鐃緒申鐃緒申鐃緒申

\begin{displaymath}f(X)=a_d X^d +a_{d-1}X^{d-1}+\dots +a_1 X+a_0 \end{displaymath}

鐃緒申,必鐃緒申 $d$ 鐃縦の削申鐃緒申鐃緒申帖鐃緒申鐃緒申覆鐃緒申鐃緒申

\begin{displaymath}f(X)=a_d(X-\alpha_1)(X-\alpha_2)\dots(X-\alpha_d) \end{displaymath}

鐃夙逸申鐃緒申分鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 4.2   鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申 $V\in {\Bbb A}^n({\Bbb C})$ 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。

鐃緒申鐃緒申 4.1   $L={\Bbb C}(X,Y)\subset K={\Bbb C}(X+Y,XY)$ 鐃夙わ申鐃緒申鐃夙わ申鐃緒申 $L$ 鐃緒申 $K$ 鐃緒申粒鐃緒申膽￥申鐃� $d$ 鐃緒申鐃夙め、鐃緒申鐃緒申鐃�

\begin{displaymath}{\Bbb C}[A,B]\ni f \mapsto f(X+Y,XY) \in {\Bbb C}[X,Y] \end{displaymath}

鐃淑わ申鐃銃縁申鐃緒申鐃緒申鐃� $\phi$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申多鐃準式鐃緒申鐃緒申 $\Phi={}^a \phi:{\Bbb A}^2({\Bbb C})\to {\Bbb A}^2({\Bbb C})$ 鐃緒申佑鐃緒申董鐃� ${\Bbb A}^2({\Bbb C})$ 鐃緒申鐃緒申鐃重�鐃緒申鐃緒申分鐃緒申鐃緒申 $F$ 鐃叔￥申 $\Phi$ 鐃緒申 $U={\Bbb A}^2\setminus F$ 鐃舜わ申鐃緒申鐃緒申 $\Phi\vert _U$ 鐃緒申 $d:1$ 鐃夙なわ申茲�鐃緒申 鐃緒申里鐃緒申銚鐃緒申弔鐃緒申覆鐃緒申鐃緒申鐃�



2001-10-31