Âå¿ô³Ø IB No.8(»î¸³)

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

$$\varphi(X)=10$$

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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{${\mathbb{Q}}$}$$[X]$ ¤ò $(X-10)$ ¤Ç³ä¤Ã¤¿¾¦¤ò $q(X)$ , ;¤ê¤ò $r$ ¤È¤ª¤¯¡£ ($r$ ¤Ï $1$ ¼¡¼°¤Ç¤ï¤Ã¤¿Í¾¤ê¤À¤«¤é¡¢ 0 ¼¡¼°¡¢¤Ä¤Þ¤ê¡¢Äê¿ô( $\mbox{${\mathbb{Q}}$}$ ¤Î¸µ)) ¤³¤Î¤È¤­ $p$ ¤ò $q,r$ ¤òÍѤ¤¤Æɽ¤·¤Ê¤µ¤¤¡£(ÀâÌÀÉÔÍס£)
6. Ǥ°Õ¤Î $p\in$   $\mbox{${\mathbb{Q}}$}$$[X]$ ¤Ë¤¿¤¤¤·¤Æ¡¢ $\varphi(p)=p(10)$ ¤¬¤Ê¤ê¤¿¤Ä¤³¤È¤ò¼¨¤·¤Ê¤µ¤¤¡£
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      $$\mbox{${\mathbb{Q}}$}$$$$[X]/(X-10)$$$$\mbox{${\mathbb{Q}}$}$$$$[X]$$
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  $$X^2, 8X^3+2X+1, X-10, -7X^3+123/57 X^2+3/2 X-1/127$$
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• ´Ä $A$ ¤«¤é´Ä $B$ ¤Ø¤Î¼ÌÁü $\varphi$ ¤Ï¡¢Ï¤ÈÀѤòÊݤÁ¡¢Ã±°Ì¸µ¤ò ñ°Ì¸µ¤Ë¼Ì¤¹¤È¤­´Ä½àƱ·¿¤È¸Æ¤Ð¤ì¤ë¡£

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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)

• $356$ (Äê¿ô¿¹à¼°)
• $3 X^2+5 X+6$
• $\frac{1}{10} (3 X^3+560)$
• $\frac{356}{2008}(2 X^3+8)$

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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

$$a(X)=(X-10) b(X) \qquad b\in$$   $$\mbox{${\mathbb{Q}}$}$$$$[X]$$

¤È¤¤¤¦·Á¤ò¤·¤Æ¤¤¤ë¡£ ¤·¤¿¤¬¤Ã¤Æ¡¢

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

(5)

$$p(X)=(X-10)q(X)+r$$

(6)

$$\varphi(p)=\varphi((X-10))\varphi(q)+\varphi(r) =\varphi(r) \overset{\text{(¡ú)}}{=} r.$$

¾Êý¤Ç¡¢

$$p(10)=(10-10)q(10)+r=r.$$

¤æ¤¨¤Ë¡¢ $\varphi(p)=p(10)$ .

¤â¤·¤¯¤Ï¡¢

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

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$$\varphi(p)$$

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

$$= \varphi(a_n) \varphi(X)^n +\varphi(a_{n-1})\varphi(X)^{n-1} +\varphi(a_{n-2})\varphi(X)^{n-2}+\dots$$

$$+ \varphi(a_2) \varphi(X)^2+\varphi(a_1) \varphi(X)+ \varphi(a_0)$$

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

$$= p(10)$$

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(7) (6)¤ÎÁ°È¾¤Ë½Ò¤Ù¤¿¤è¤¦¤Ë¡¢

$$p(X)-p(10)=(X-10)q(X)$$

(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¤·¤¤¡£