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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{ヒルベルトの基定理}

鐃緒申鐃緒申 7.1   鐃塾件申鐃緒申鐃緒申多鐃準式鐃緒申 $k[X_1,\dots,X_n]$ 鐃緒申任鐃春のワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申有鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。

鐃旬講居申鐃叔わ申多鐃準式鐃緒申鐃緒申導鐃緒申鐃緒申鐃緒申椶鐃緒申鐃緒申鐃緒申鐃緒申鬚△鐃緒申鐃緒申茲�鐃緒申 鐃緒申鐃緒申鐃緒申鐃所が鐃緒申鐃竣でわ申鐃暑。

鐃緒申鐃緒申鐃緒申鐃塾緒申鐃緒申 $\Bbb N$ 鐃緒申 $0$ 鐃緒申鐃緒申鐃緒申鐃塾�鐃緒申鐃緒申鐃緒申鐃淑わ申立鐃曙が鐃緒申鐃暑が鐃緒申 鐃緒申鐃塾講居申鐃叔は逸申咾鐃緒申鐃� $\Bbb N=\{0,1,2,\dots\}$ 鐃緒申 $0$ 鐃緒申鐃緒申鐃緒申鐃塾�鐃緒申鐃夙る。

鐃緒申鐃緒申 7.1  
1.
鐃塾件申鐃緒申鐃緒申多鐃準式鐃緒申 $k[X_1,\dots,X_n]$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申鐃出わ申鐃銃￥申

\begin{displaymath}L=\operatorname{Head}(I)=\{\operatorname{Head}(p); p\in I\} \end{displaymath}

鐃緒申佑鐃緒申鐃夙￥申 $L=\operatorname{Head}(I)$ 鐃緒申 $\Bbb N^n$ 鐃緒申鐃緒申分鐃緒申鐃緒申如鐃�

\begin{displaymath}L+\Bbb N^n \subset \Bbb N^n \tag{★} \end{displaymath}

鐃緒申鐃緒申鐃緒申立鐃縦￥申
2.
$\Bbb N^n$ 鐃緒申鐃緒申分鐃緒申鐃緒申 $L$ 鐃叔￥申$($鐃緒申$)$鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申里鐃宿�鐃緒申鐃緒申有鐃緒申鐃緒申鐃緒申鐃緒申 鐃叔わ申鐃暑。鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃粛�鐃渋個の醐申 $l_1,\dots l_N \in L$ 鐃緒申鐃緒申鐃獣て￥申

\begin{displaymath}L= (l_1+\Bbb N^n)\cup (l_2+\Bbb N^n)\cup (l_3+\Bbb N^n)\cup \dots \cup (l_N+\Bbb N^n) \end{displaymath}

鐃緒申鐃緒申立鐃緒申鐃暑。

多鐃準式鐃縦のワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申砲鐃緒申鐃緒申妨鐃緒申鐃渋逸申鐃緒申襪�鐃宿わ申鐃緒申鐃緒申 鐃緒申導鐃緒申鐃淳なわ申鐃緒申鐃所算鐃緒申鐃銃わ申鐃緒申鐃出よい鐃塾で￥申鐃緒申鐃緒申鐃緒申蠅�鐃緒申劵鐃駿ワ申箸隆鐃緒申鐃緒申鐃緒申鐃� 鐃緒申立鐃緒申鐃暑こ鐃夙を示わ申鐃緒申鐃夙わ申鐃叔わ申鐃緒申箸鐃緒申鐃緒申錣縁申任鐃緒申襦�

鐃述ワ申戰鐃夙の器申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申調鐃緒申鐃重�鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。 鐃緒申鐃緒申呂泙鐃緒申諭鐃緒申鐃緒申弔箸鐃緒申鐃緒申鐃緒申佞鐃緒申儖佞鐃緒申討泙箸鐃銃わ申鐃緒申鐃緒申鐃夙わ申任鐃緒申襦�

鐃緒申鐃緒申 7.2   鐃緒申 $R$ 鐃祝つわ申鐃銃種申鐃塾常申鐃緒申同鐃粛でわ申鐃暑。
1.
$R$ 鐃緒申任鐃春のワ申鐃叔ワ申鐃緒申鐃粛�鐃緒申鐃緒申鐃緒申鐃叔わ申鐃暑。
2.
$R$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $I_1 \subset I_2 \subset I_3 \subset \dots$ 鐃緒申必鐃緒申鐃宿わ申鐃緒申鐃叔止まる。鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃� $N$ 鐃緒申鐃緒申鐃獣て￥申 $ I_N=I_{N+1}=I_{N+2}=\dots $鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃� 7.1   鐃緒申両鐃緒申里鐃緒申鐃緒申鐃緒申(鐃緒申鐃緒申鐃緒申鐃獣て￥申両鐃緒申)鐃緒申鐃緒申鐃緒申立鐃縦よう鐃淑環わ申諭鐃緒申鐃緒申弔箸鐃緒申鐃緒申鐃�

鐃述ワ申戰鐃夙の器申鐃緒申鐃緒申鐃熟￥申鐃緒申鐃塾常申鐃渋随申犲逸申弔魯諭鐃緒申鐃緒申弔任鐃緒申鐃竣とわ申鐃獣てわ申鐃緒申錣縁申鐃緒申鐃�

鐃緒申鐃緒申 7.1   $\mbox{${\Bbb R}$ }[X,Y]$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃縮居申存鐃緒申鐃緒申鐃� $I_1 \supset I_2 \supset I_3 \supset \dots$ 鐃緒申鐃緒申鬚△鐃緒申覆鐃緒申鐃緒申鐃�


Yoshifumi Tsuchimoto
2001-05-25