����� II ���� No.5

����ϡ�¿�༰�Ĥ򥤥ǥ���dz�뤳�Ȥˤ�ä��Τ��뤳�Ȥ��Ǥ��뤳�Ȥ�Ҥ٤��� ����ϡ���¸���ΤȤ��Τ褦���ΤȤ���Ӥ�����ˤ��롣���ˤʤ�Τϼ��������Ǥ��롣

���� 5.1   �� $k$ ����� $K$ �ȡ� $K\setminus k$ �θ� $a$ ��Ϳ�����Ƥ���Ȥ��롣 ���ΤȤ���

1.

$$I=\{f\in k[X]; f(a)=0\}$$

�� $k[X]$ ��ñ�।�ǥ���Ǥ��롣

2.

$I$ �������� $f_0$ �ϴ���Ǥ��롣

3.

ñ�ʹĽ�Ʊ��

$$\phi: k[X]/(f_0) \to K: \overline{f(X)}\mapsto f(a)$$

��¸�ߤ��롣

4.

$\phi$ ������ $k[a]$ �Ǥ��롣

5.

$[k[a]:k]=\deg(f_0)$

��� 5.1   ��������� $f_0$ �Τ��Ȥ�$a$ �� $k$ ��κǾ�¿�༰�Ȥ����� (�ä˻��ꤷ�ʤ���С��Ǿ�¿�༰�Ȥ����Х�˥å��ʤ�Τò¤¨¤ï¿½Ö¤Î¤ï¿½ ���̤Ǥ��롣)

�� 5.1  

$$\mbox{${\Bbb Q}$}[X]/(X^2-2)\cong\mbox{${\Bbb Q}$}[\sqrt{2}]$$

�� 5.2  

$$\mbox{${\Bbb R}$}[X]/(X^2+1)\cong{\Bbb C}$$

�� 5.3  
$$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^4-16X^2+4)&\cong\mbox{${\Bbb Q}$ }[\sqrt{5}+\sqrt{3}]\\ \overline{f(X)}&\mapsto f(\sqrt{5}+\sqrt{3}) \end{align*}$$
Ʊ�ͤˡ�
$$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^4-16X^2+4)&\cong\mbox{${\Bbb Q}$ }[\sqrt{5}-\sqrt{3}]\\ \overline{f(X)}&\mapsto f(\sqrt{5}-\sqrt{3}) \end{align*}$$
�Ǥ⤢�롣����ϡ�
$$\begin{align*}& \mbox{${\Bbb Q}$ }[\sqrt{5}+\sqrt{3}]\cong\mbox{${\Bbb Q}$ }[\sqrt{5}-\sqrt{3}]\\ & f(\sqrt{5}+\sqrt{3})\mapsto f(\sqrt{5}-\sqrt{3}) \end{align*}$$
�Ȥ���Ʊ����¸�ߤ򼨺����Ƥ��롣

�� 5.4  
$$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^2-2)&\cong\mbox{${\Bbb Q}$ }[X]/(X^2-2X-1)\\ \overline{f(X)}&\mapsto \overline{f(X-1)} \end{align*}$$

�� 5.5   $\omega=\frac{-1+\sqrt{-3}}{2}$ �Ȥ����ȡ�

$$\mbox{${\Bbb Q}$}[\sqrt[3]{5}]\cong\mbox{${\Bbb Q}$}[\sqrt[3]{5}\omega]$$

������Ω�ġ�

���� 5.1   $\mbox{${\Bbb Q}$ }[X]/(X^2+X+1)$ �� $\mbox{${\Bbb Q}$ }[X]/(X^2+3)$ �Ȥδ֤�Ʊ���������ĺ��ʤ����� $\sqrt{2}+3\sqrt{5}+4\sqrt{10}$ �� $\mbox{${\Bbb Q}$ }[\sqrt{5}]$ ��κǾ�¿�༰�� ���ʤ�����

���� 5.2   �� $\mbox{${\Bbb Q}$ }[\sqrt[3]{2}+\sqrt{3}]$ �� $\mbox{${\Bbb Q}$ }[X]/(f(X))$ �Ȥ�Ʊ���ˤʤ�褦�� $f\in \mbox{${\Bbb Q}$ }[X]$ ���ᡢ�ºݤˤ���Ʊ���������ʤ�����