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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.7

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{体の二次拡大}

鐃緒申鐃藷次のわ申鐃夙を示わ申鐃緒申鐃夙わ申弔鐃緒申討鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 7.1   鐃緒申 $K$ 鐃緒申琉鐃緒申竸鐃渋随申犲� $f(X)\in K[X]$ 鐃緒申鐃出わ申鐃銃￥申 $K$ 鐃緒申有鐃渋種申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $L$ 鐃緒申存鐃淳わ申鐃銃￥申$L[X]$ 鐃塾醐申鐃夙わ申鐃銃わ申 $f(X)$ 鐃緒申

\begin{displaymath}f(X)=c\prod_{i=1}^d (X-\alpha_i) \quad (\alpha_i \in L) \end{displaymath}

鐃塾よう鐃祝一次鐃緒申鐃緒申鐃術わ申分鐃薯する。

鐃殉わ申鐃緒申鐃緒申檗鐃緒申箸念鐃緒申竸鐃緒申鐃渋随申犲逸申隆鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申蠅�鐃叔わ申鐃塾で種申鐃塾わ申鐃夙わ申鐃緒申足鐃緒申鐃銃わ申鐃緒申鐃緒申

鐃緒申鐃緒申 7.2   鐃緒申 $k$ 鐃緒申亮鐃緒申里茲�鐃淑逸申鐃術随申多鐃準式鐃緒申 $k$ 鐃緒申鐃緒申鐃叔わ申鐃暑。
1.
$k$ 鐃緒申鐃�2鐃緒申鐃緒申鐃緒申3鐃緒申鐃緒申鐃緒申 $k$ 鐃塾なわ申鐃祝削申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃塾￥申

鐃叔わ申鐃緒申鐃緒申砲鐃緒申弔襦ｏ申鐃緒申鐃緒申鐃緒申鐃藷次鰹申鐃緒申鐃緒申融劼鐃処く表鐃緒申鐃銃わ申鐃暑。

鐃緒申鐃緒申 7.3   $R={\Bbb C}[X,A,B]/(X^2+AX+B)$, $S={\Bbb C}[A,B]$ 鐃夙わ申鐃緒申鐃緒申鐃緒申鐃緒申法鐃�$R$ 鐃塾醐申 $Y,D$ 鐃緒申 $Y=X+A/2, D=A^2-4B$ 鐃叔件申鐃暑。 $R_0={\Bbb C}[Y,D] ,S_0={\Bbb C}[D]$ 鐃夙わ申鐃緒申鐃夙わ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.
$L=Q(R)$ 鐃緒申 $K=Q({\Bbb C}[A,B])={\Bbb C}(A,B)$ 鐃緒申鐃藷次鰹申鐃緒申任鐃緒申襦�
2.
$L_0=Q(R_0)$ 鐃緒申 $K_0=Q(S_0)$ 鐃緒申鐃藷次鰹申鐃緒申任鐃緒申襦�
3.
$L,K$ 鐃熟わ申鐃曙ぞ鐃緒申 $L_0,R_0$ 鐃緒申琉鐃緒申竸鐃粛�鐃緒申鐃舜随申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申
4.
鐃緒申鐃緒申 $L\supset K$ 鐃緒申 $L_0\supset K_0$ 鐃緒申鐃術随申鐃緒申鐃緒申鐃春わ申鐃獣わ申鐃暑こ鐃夙わ申 鐃緒申辰鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�
5.
$R\subset S$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃緒申鐃重�鐃緒申鐃緒申隆屬鐃緒申弍鐃緒申鐃� $D=0$ 鐃緒申鐃緒申分鐃緒申鐃緒申鐃緒申鐃� $2:1$鐃叔わ申鐃暑。

鐃緒申鐃� $D$ 鐃熟わ申鐃緒申鐃緒申鐃� $X$ 鐃緒申鐃藷次種申 $X^2+AX+B$ 鐃緒申判鐃縮種申鐃叔わ申鐃暑。 鐃緒申鐃縮に￥申鐃緒申鐃緒申鐃緒申鐃緒申的鐃緒申鐃緒申 $V$ 鐃夙￥申鐃緒申鐃塾常申虜鐃宿醐申鐃� $A(V)$ 鐃塾醐申 $a,b$ 鐃緒申箸辰鐃緒申箸鐃緒申鐃� $K(V)=A(V)$ 鐃緒申 $X^2+aX+b=0$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $X$ 鐃緒申鐃春わ申鐃獣わ申鐃緒申鐃緒申 $L=K(V)[X]/(X^2+aX+b)$ 鐃緒申佑鐃緒申襪鰹申箸鐃緒申任鐃緒申董鐃緒申鐃緒申亮鐃緒申鐃緒申鐃緒申融劼鐃緒申鐃緒申鐃緒申鐃緒申 鐃処う鐃緒申 $d=a^2-4b$ 鐃緒申鐃緒申判鐃叔でわ申鐃暑。鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申六弔襦�

鐃緒申鐃緒申 7.1   鐃藷次醐申平鐃緒申 ${\Bbb A}^2({\Bbb C})$ (鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申標鐃術随申鐃緒申 $Y,Z$ 鐃夙わ申鐃緒申) 鐃緒申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $V=V(Y^2-Z^3)$ 鐃緒申佑鐃緒申襦�
1.
$V$ 鐃塾鰹申鐃緒申鐃薯かわ申鐃淑わ申鐃緒申鐃緒申( $\mbox{${\Bbb R}$ }^2$ 鐃緒申鐃緒申鐃渋わ申鐃銃書いて刻申鐃緒申覆鐃�)
2.
$A=A(V)$ 鐃緒申 $X^2-Z=0$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃術随申 $X$ 鐃緒申鐃春わ申鐃獣わ申鐃緒申鐃緒申 $B=A[X]/(X^2-Z)$ 鐃緒申鐃宿醐申弔忙鐃緒申弔茲�鐃緒申鐃緒申鐃重�鐃緒申鐃緒申離鐃緒申鐃春わ申 3鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃￥申 $V$ 鐃塾ワ申鐃緒申佞箸隆愀鐃緒申鐃縦器申戮覆鐃緒申鐃緒申鐃�
3.
$B$ 鐃緒申鐃緒申鐃緒申任呂覆鐃緒申鐃緒申鐃�($X^2-Z$ 鐃熟器申鐃緒申任呂覆鐃緒申鐃緒申鐃�)鐃薯示わ申鐃淑わ申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申鐃獣て￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申箸鐃緒申討六箸鐃緒申砲鐃緒申鐃緒申鐃�)



2001-11-16