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Âå¿ô³Ø IB No.8(»î¸³)

(º£Æü¤Ï½àƱ·¿ÄêÍý¤Î¼þÊդˤǤƤ¯¤ëÏÃÂê¤Î³Îǧ¤Ç¤¢¤ë¡£ ²¼¤Î²òÅú¤Ç½àƱ·¿ÄêÍý¼«¿È¤òÍѤ¤¤ë¾ìÌ̤Ϥª¤½¤é¤¯¤Ê¤¤¤À¤í¤¦¡¢¤È¤¤¤¦¤«¡¢ ÍѤ¤¤º¤Ë²ò·è¤¹¤ë¤³¤È¡£)

ÌäÂê 8.1   ´Ä½àƱ·¿ $ \varphi:$$ \mbox{${\mathbb{Q}}$}$$ [X] \to {\mathbb{C}}$ ¤¬

$\displaystyle \varphi(X)=10
$

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  1. $ \varphi(X^2)$ ¤òµá¤á¤è¡£(Åú¤Î¤ß¤Ç¤è¤¤¡£)
  2. $ \varphi(8 X^3+2 X + 1)$ ¤òµá¤á¤è¡£(Åú¤Î¤ß¤Ç¤è¤¤¡£)
  3. $ \varphi(p)=356$ ¤È¤Ê¤ë $ p\in$   $ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤ÎÎã¤ò $ 3$ ¤Äµó¤²¤Ê¤µ¤¤¡£ ¤¿¤À¤·¡¢¤½¤Î¤¦¤Á°ì¤Ä¤Ï 3¼¡°Ê¾å¤Î¼°¤Ç¤¢¤ë¤è¤¦¤Ë¤¹¤ë¤³¤È¡£ (Åú¤¨¤È¡¢´Êñ¤Ê³Î¤«¤á»»¤Î¤ß¤Ç¤è¤¤¡£)
  4. $ \operatorname{Ker}(\varphi)\supset (X-10)$$ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤ò¾ÚÌÀ¤·¤Ê¤µ¤¤¡£
  5. $ p(X)\in$$ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤ò $ (X-10)$ ¤Ç³ä¤Ã¤¿¾¦¤ò % latex2html id marker 955
$ q(X)$ , ;¤ê¤ò $ r$ ¤È¤ª¤¯¡£ ($ r$ ¤Ï $ 1$ ¼¡¼°¤Ç¤ï¤Ã¤¿Í¾¤ê¤À¤«¤é¡¢ 0 ¼¡¼°¡¢¤Ä¤Þ¤ê¡¢Äê¿ô( $ \mbox{${\mathbb{Q}}$}$ ¤Î¸µ)) ¤³¤Î¤È¤­ $ p$ ¤ò % latex2html id marker 968
$ q,r$ ¤òÍѤ¤¤Æɽ¤·¤Ê¤µ¤¤¡£(ÀâÌÀÉÔÍס£)
  6. Ǥ°Õ¤Î $ p\in$   $ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤Ë¤¿¤¤¤·¤Æ¡¢ $ \varphi(p)=p(10)$ ¤¬¤Ê¤ê¤¿¤Ä¤³¤È¤ò¼¨¤·¤Ê¤µ¤¤¡£
  7. ¾ê;´Ä

       $\displaystyle \mbox{${\mathbb{Q}}$}$$\displaystyle [X]/(X-10)$$\displaystyle \mbox{${\mathbb{Q}}$}$$\displaystyle [X]
$

    ¤Ç¤Î¿¹à¼° $ p(X)\in$   $ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤Î¥¯¥é¥¹¤ÏÍ­Íý¿ô $ p(10)$ ¤Î¥¯¥é¥¹¤ÈÅù¤·¤¤¤³¤È ¤ò¼¨¤·¤Ê¤µ¤¤¡£



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²òÅú

(1) $ \varphi(X^2)=\varphi(X\cdot X)=\varphi(X)\cdot \varphi(X)=10\cdot 10=100$ .

(2) $ \varphi(8 X^3+2 X+1)= 8 \varphi(X)^3+ 2 \varphi(X) +1=8 \cdot 10^3 +2 \cdot 10+1=8021.$

(3)

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(4) $ (X-10)$$ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤«¤éǤ°Õ¤Î¸µ $ a$ ¤ò¤È¤Ã¤Æ¤¯¤ë¤È¡¢¤½¤ì¤Ï

% latex2html id marker 1025
$\displaystyle a(X)=(X-10) b(X) \qquad b\in$   $\displaystyle \mbox{${\mathbb{Q}}$}$$\displaystyle [X]
$

¤È¤¤¤¦·Á¤ò¤·¤Æ¤¤¤ë¡£ ¤·¤¿¤¬¤Ã¤Æ¡¢

$\displaystyle \varphi(a)=\varphi(X-10) \cdot\varphi(b)=(\varphi(X)-10)\cdot\varphi(b)
=0 \cdot \varphi(b)=0.
$

(5)

% latex2html id marker 1031
$\displaystyle p(X)=(X-10)q(X)+r
$

(6)

% latex2html id marker 1033
$\displaystyle \varphi(p)=\varphi((X-10))\varphi(q)+\varphi(r)
=\varphi(r)
\overset{\text{(¡ú)}}{=}
r.
$

¾Êý¤Ç¡¢

% latex2html id marker 1035
$\displaystyle p(10)=(10-10)q(10)+r=r.
$

¤æ¤¨¤Ë¡¢ $ \varphi(p)=p(10)$ .

¤â¤·¤¯¤Ï¡¢

$\displaystyle p(X)=a_n X^n+a_{n-1}X^{n-1}+a_{n-2}X^{n-2}+\dots+ a_2 X^2+a_1 X+ a_0
$

¤È¤«¤¤¤Æ¡¢

  $\displaystyle \varphi(p)$    
$\displaystyle =$ $\displaystyle \varphi(a_n X^n+a_{n-1}X^{n-1}+a_{n-2}X^{n-2}+\dots+ a_2 X^2+a_1 X+ a_0)$    
$\displaystyle =$ $\displaystyle \varphi(a_n) \varphi(X)^n +\varphi(a_{n-1})\varphi(X)^{n-1} +\varphi(a_{n-2})\varphi(X)^{n-2}+\dots$    
  $\displaystyle + \varphi(a_2) \varphi(X)^2+\varphi(a_1) \varphi(X)+ \varphi(a_0)$    
$\displaystyle \overset{\text{(¡ú)}}{=}$ $\displaystyle a_n\cdot 10^n+a_{n-1}\cdot 10^{n-1}+a_{n-2}\cdot 10^{n-2}+\dots + a_2\cdot 10^2+a_1\cdot 10+ a_0$    
$\displaystyle =$ $\displaystyle p(10)$    

¤È¤·¤Æ¤â¤è¤¤¡£

(7) (6)¤ÎÁ°È¾¤Ë½Ò¤Ù¤¿¤è¤¦¤Ë¡¢

% latex2html id marker 1051
$\displaystyle p(X)-p(10)=(X-10)q(X)
$

(µ­¹æ¤Ï (6) ¤ÈƱ¤¸¤â¤Î¤ò»È¤Ã¤¿¡£) ¤À¤«¤é¡¢ $ p(X)$ ¤È $ p(10)$ ¤Î $ \mbox{${\mathbb{Q}}$}$$ [X]/(X-10)$$ \mbox{${\mathbb{Q}}$}$$ [X]$ ¤Ç¤Î¥¯¥é¥¹¤ÏÅù¤·¤¤¡£
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2008-12-11