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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.9

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{体の二次拡大・三次拡大}

鐃緒申鐃緒申 9.1   $p,q$ 鐃緒申複鐃叔随申鐃叔￥申 $f=X^3+pX+q$ 鐃緒申判鐃縮種申鐃緒申 $0$ 鐃叔なわ申鐃夙わ申鐃暑。 鐃緒申鐃塾とわ申鐃緒申 ${\Bbb C}[X,Y]/(Y^2-X^3-pX-q)$ 鐃塾常申鐃塾わ申 $k$ 鐃夙わ申鐃緒申函鐃� ${\Bbb C}(X),{\Bbb C}(Y)$ 鐃熟とわ申鐃� $K$ 鐃緒申鐃緒申分鐃塾と醐申鐃暑こ鐃夙わ申鐃叔わ申鐃銃￥申 $[K:{\Bbb C}(X)]=2$, $[K:{\Bbb C}(Y)]=3$ 鐃緒申鐃淑りた鐃縦￥申

判鐃縮種申鐃緒申鐃緒申鐃緒申楼焚鐃緒申暴劼戮襦�

鐃緒申鐃� 9.1  

\begin{displaymath}(X-\alpha_1)\dots(X-\alpha_n) \end{displaymath}

鐃緒申 $X$ 鐃祝つわ申鐃緒申展鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申多鐃準式鐃塾件申鐃緒申鐃塾わ申鐃夙わ申 $\alpha_1,\dots,\alpha_n$ 鐃塾器申鐃緒申鐃出称種申鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申 9.2   $\alpha_1,\dots,\alpha_n$ 鐃緒申鐃出称種申鐃叔わ申鐃緒申茲�鐃緒申多鐃準式鐃緒申 必鐃緒申 $\alpha_1,\dots,\alpha_n$ 鐃塾器申鐃緒申鐃出称種申鐃緒申多鐃準式鐃夙わ申鐃緒申表鐃緒申鐃緒申襦�

鐃緒申鐃緒申 9.3   鐃緒申 $K$ 鐃緒申鐃� $n$ 鐃緒申多鐃準式 $f(X)$ 鐃緒申鐃出わ申鐃銃￥申$f$ 鐃塾削申鐃緒申 $\alpha_1,\dots,\alpha_n$鐃夙わ申鐃緒申

\begin{displaymath}\Delta=\prod_{i<j} (\alpha_i -\alpha_j) \end{displaymath}

鐃夙わ申鐃緒申鐃夙￥申 $D=\Delta^2$ 鐃緒申 $k$ 鐃塾醐申鐃叔わ申鐃暑。鐃緒申鐃緒申 $D$ 鐃塾わ申鐃夙わ申 $f$ 鐃緒申 判鐃縮種申鐃夙醐申鐃緒申鐃緒申

鐃緒申鐃初か鐃祝￥申$f$ 鐃緒申判鐃縮種申鐃緒申 $0$ 鐃祝なるこ鐃夙わ申 $f$ 鐃緒申鐃重削申鐃緒申鐃緒申弔鐃緒申箸鐃銃縁申佑任鐃緒申襦�

鐃緒申鐃緒申 9.1   $f=X^3+pX+q$ 鐃緒申判鐃縮種申鐃緒申鐃緒申鐃重�鐃緒申 $p,q$ 鐃緒申多鐃準式鐃緒申表鐃緒申鐃淑わ申鐃緒申鐃緒申



2002-01-06