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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.10

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{変数の置換について対称な式}鐃緒申鐃緒申鐃� $k$ 鐃緒申無鐃渋個の醐申鐃緒申鐃緒申鐃緒申里箸鐃緒申襦� (鐃緒申鐃緒申砲鐃緒申磴�鐃緒申鐃緒申 $k$ 鐃緒申 $\mbox{${\Bbb Q}$ }$ 鐃緒申鐃緒申分鐃塾に含わ申仆鐃淑�)

鐃緒申鐃緒申 10.1   鐃緒申 $k$ 鐃緒申鐃� $n$-鐃術随申多鐃準式鐃緒申 $k[X_1,\dots,X_n]$ 鐃塾常申鐃緒申($n$-鐃術随申有鐃緒申鐃舜随申鐃緒申) $L=k(X_1,\dots,X_n)$ 鐃祝わ申 $n$-鐃縦の醐申鐃緒申鐃出称件申 $\frak S_n$ 鐃緒申鐃緒申鐃術わ申鐃銃わ申鐃暑。 鐃緒申鐃淑わ申鐃緒申鐃叔わ申佞鐃� $\sigma \in \frak S_n$ 鐃緒申鐃出わ申鐃銃￥申$L$ 鐃塾醐申 $f$ 鐃緒申 $\sigma$ 鐃祝わ申鐃緒申儡鐃� $f^{\sigma}$ 鐃緒申鐃緒申

\begin{displaymath}f^{\sigma}(X_1,\dots,X_n)=f(X_{\sigma{1}},\dots,X_{\sigma{n}}) \end{displaymath}

鐃祝わ申辰鐃緒申鐃緒申鐃緒申鐃緒申董鐃叔わ申佞鐃� $f,g\in L$ 鐃緒申鐃緒申鐃叔わ申佞鐃� $\sigma,\tau \in \frak S_n$ 鐃緒申鐃出わ申鐃銃￥申
1.
$(f+g)^{\sigma}= f^{\sigma} +g^{\sigma}$
2.
$(fg)^{\sigma}= f^{\sigma} g^{\sigma}$
3.
$f^{\sigma\tau}=(f^{\sigma})^{\tau}$
鐃緒申澆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申砲鐃緒申鐃夙わ申鐃緒申鐃潤し鐃緒申鐃緒申鐃緒申鐃竣わ申鐃緒申鐃緒申竸鐃緒申鐃緒申峇鐃緒申鐃峻ってわ申鐃緒申鐃緒申鐃緒申任鐃緒申襦� 鐃緒申鐃夙わ申鐃出￥申 $\sigma=(1\ 2\ 3)\in \frak S_3$ 鐃緒申有鐃緒申鐃緒申 $f\in k(X_1,X_2,X_3)$鐃緒申鐃出わ申鐃銃￥申

\begin{displaymath}f^\sigma(X_1,X_2,X_3)=f(X_2,X_3,X_1) \end{displaymath}

(鐃緒申辰閥鐃緒申鐃重�鐃祝わ申 $(X_1+X_2^3+X_3X_1)^\sigma= (X_2+X_3^3+X_1X_2)^\sigma$) 鐃緒申鐃緒申.

鐃緒申鐃� 10.1   鐃緒申鐃緒申鐃緒申鐃塾居申鐃緒申慮鐃緒申如鐃� $\frak S_n$ 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

\begin{displaymath}L^H= \{ f\in L; f^{\sigma}=f \quad (\forall \sigma \in H) \} \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申襦�

鐃純え鐃緒申 $K=L^{\frak S_n}$ 鐃緒申 $k$ 鐃緒申鐃�$n$-鐃術随申有鐃緒申鐃緒申鐃塾わ申鐃緒申鐃緒申鐃出称種申鐃緒申 鐃緒申鐃塾を集めた鐃緒申里任鐃緒申襦�

鐃緒申鐃緒申 10.1   鐃緒申竜鐃緒申鐃塾醐申鐃叔￥申 $\frak S_n$ 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銃￥申
1.
$\frak S_n$ 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銃￥申 $L^H$ 鐃緒申 $L$ 鐃緒申鐃緒申分鐃塾でわ申鐃暑。
2.
$H_1,H_2$ 鐃緒申鐃夙わ申鐃� $\frak S_n$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $H_1\subset H_2$ 鐃淑わ申鐃� $L^{H_1}\supset L^{H_2}$ 鐃緒申鐃淑りた鐃縦￥申
3.
$[L:L^H]=\vert H\vert$.
4.
鐃渋わ申 $L$ 鐃緒申鐃緒申分鐃塾わ申 $K$ 鐃緒申泙鐃緒申鐃�($L$ 鐃緒申 $K$ 鐃緒申鐃緒申鐃緒申鐃�)鐃緒申 $L^H$ 鐃塾件申鐃塾わ申里妨造襦�

鐃緒申鐃緒申10.1鐃塾常申鐃緒申鐃祝は種申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃出よい鐃緒申

鐃緒申鐃緒申 10.2   $L,K$ 鐃熟常申鐃緒申未鐃夙わ申鐃暑。$H$ 鐃緒申 $\frak S_n$ 鐃緒申鐃緒申分鐃緒申 鐃夙わ申鐃緒申鐃淑駕申鐃叔わ申 $M=L^H$ 鐃夙書くわ申鐃夙にわ申鐃暑。 鐃緒申鐃殉￥申 $\alpha\in L$ 鐃緒申 $L=K(\alpha)$ 鐃淑わ申茲�鐃緒申鐃緒申鐃俊￥申

\begin{displaymath}F(T)=\prod_{\sigma\in H}(T-\alpha^{\sigma}) \end{displaymath}

鐃夙わ申鐃緒申鐃緒申鐃緒申鐃塾とわ申鐃緒申
1.
$F(\alpha)=0$.
2.
$\deg(F)=\vert H\vert$.
3.
$F\in M[T]$. 鐃緒申鐃淑わ申鐃� $F$ 鐃緒申 $T$ 鐃緒申多鐃準式鐃夙わ申鐃銃わ申 鐃緒申鐃緒申鐃熟わ申鐃駿わ申 $M$ 鐃塾醐申鐃叔わ申鐃暑。
4.
$L=M(\alpha)$.
5.
任鐃春わ申 $\sigma \in H$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申 $\alpha^{\sigma}$ 鐃緒申 $M$ 鐃緒申 $\alpha$ 鐃夙駈申鐃緒申任鐃緒申襦� 鐃緒申鐃淑わ申鐃� $\alpha$ 鐃緒申 $M$ 鐃緒申虜脳鐃渋随申犲逸申鐃� $\alpha^{\sigma}$ 鐃緒申 $M$ 鐃緒申虜脳鐃渋随申犲逸申醗鐃緒申廚鐃緒申襦�

鐃緒申鐃緒申鐃緒申鐃塾常申鐃緒申鐃緒申鐃緒申鐃緒申鐃処う鐃緒申 $\alpha$ 鐃緒申存鐃淳は器申鐃祝種申鐃緒申鐃銃わ申鐃暑が鐃緒申 鐃純え鐃出￥申

鐃緒申鐃緒申 10.3   $c_1,\dots,c_n$ 鐃緒申 $k$ 鐃緒申鐃緒申曚覆觚居申覆鐃出￥申 $c_1X_1+c_2X_2+\dots c_nX_n$ 鐃熟常申鐃緒申鐃緒申鐃緒申 $\alpha$ 鐃塾常申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申

鐃緒申鐃緒申 10.1   $H=\{(1), (1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}\in \frak S_4$鐃夙わ申鐃暑。鐃緒申鐃塾とわ申鐃緒申 $4$-鐃術随申有鐃緒申鐃舜随申鐃緒申 $L={\Bbb C}(X_1,X_2,X_3,X_4)$ 鐃緒申 $H$-鐃緒申鐃術醐申鐃緒申鐃塾のなわ申鐃緒申 $L^H$ 鐃塾醐申鐃緒申10鐃縦挙げなわ申鐃緒申鐃緒申 (鐃宿のよう鐃淑醐申鐃緒申箸襪�鐃熟種申由鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任覆鐃緒申鐃塾わ申鐃緒申鐃瞬よう 鐃緒申鐃熟わ申鐃銃わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)



2002-01-07