�����III ���� No.3

�����Υơ���

$\alpha=\sqrt{2}$ �ΤȤ��� $\mbox{${\mathbb{Q}}$}$$[\alpha]$ ��� $a+b\alpha$ $(a,b\in$   $\mbox{${\mathbb{Q}}$}$$)$ �Ƚñ¤¯¤ï¿½ï¿½È¤ï¿½ï¿½Ç¤ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Â¡ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Ñ¡ï¿½ï¿½ï¿½ï¿½Î¸ï¿½ï¿½ï¿½ï¿½ï¿½ $\alpha^2-2=0$ �Ȥ����ط�����������ñ¤­²ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½È¤ï¿½ï¿½Ç¤ï¿½ï¿½ï¿½Î¤Ç¤ï¿½ï¿½Ã¤ï¿½ï¿½ï¿½ ��äȰ��̤ˡ�����������������������Τˡ��դ��ä���פ��Ȥ��Ǥ��롣 (���� 3.1,���� 3.1)����ˤϡ�¿�༰�Ĥò¤¦¤Þ¤ï¿½ï¿½Ñ¤ï¿½ï¿½ë¡£

��� 3.1   �� $K$ ��ò·¸¿ï¿½ï¿½Ë»ï¿½ï¿½Ä¤è¤¦ï¿½ï¿½ $X$ ���ѿ��Ȥ�����ѿ�¿�༰�� ���� $K[X]$ �Ƚñ¤­¡ï¿½$K$ ���($X$ ���ѿ��Ȥ���)���ѿ�¿�༰�ĤȸƤ֡�

�Ĥ��Ǥ˴Ĥȥ��ǥ���������פ��Ф��Ƥ��������ܤ�������� I�ιֵ��� �������뤳�ȡ�

��� 3.2  

1. $R$ ��(ñ�̸�����IJĴ�)�ĤǤ���Ȥϡ�$R$ ���¡������Ѥ��������Ƥ��ơ� �ѤϲĴ������� $R$ �Ͼ�ˡ�˴ؤ���ñ�̸� $1$ ����ĤȤ��ˤ�����
2. $R$ ����ʬ���� $I$ �� $R$ �Υ��ǥ���Ǥ���Ȥϡ�$I$ �� ��ˡ����ˡ�ˤĤ����Ĥ��Ƥ��ơ��ʤ����ļ��ξ�郎����Ω�ĤȤ��ˤ�����
   $$r \in R, x\in I \implies r x \in I$$

3. �� $R$ �Υ��ǥ��� $I$ ��Ϳ����줿�Ȥ���$R/I$ �ʤ뤢���餷���Ĥ� �������뤳�Ȥ��Ǥ��롣$R/I$ �θ� $\overline{a}$ �� $a \in R$ �θ��Υ��饹�Ǥ��ꡢ ���饹ʬ���ϡ�
   $$\overline{a}=\overline{b} \quad {\Leftrightarrow}\quad a-b \in I$$
   �Ƿ�ޤ롣

���� 3.1   �� $K$ ���¿�༰ $f(X)\in K[X]$ ���Ф��ơ� $I=f(X)K[X]$ �� $K[X]$ �Υ��ǥ���Ǥ��롣

$$R=K[X]/f(X) K[X]$$

�ϴĤǤ��äơ�$X$ �� $K[X]$ �ˤ������;��(���饹)�� $\alpha$ �Ƚ񤯤ȡ� $\alpha$ �� $f(\alpha)=0$ ����­���롣 ����ˡ� $f$ �� $K$ ����� (����ʾ� $K$ ���¿�༰���Ѥ�ʬ��Ǥ��ʤ�)�ʤ�С����δ� $R$ �ϼ¤��ΤǤ��롣

��� 3.3   �� $L$ �Ȥ�����ʬ�� $K$ ��Ϳ�����Ƥ���Ȥ��롣 $a \in L$ ���Ф��ơ�$a$ ����­���� $K$ ��������������ʤ��

$$f(X) \in K[X]\setminus\{0\}$$$$\text { ¤«¤Ä } f(a)=0$$

��ߤ������ �Τ����������Ǿ��Τ�Τ�$a$ �� $K$ ��κǾ�¿�༰�ȸƤ֡�

�ʲ��Ǥϡ��Ȥ����Ǥ�ʤ��¤�Ǿ�¿�༰�ȸ����Х�˥å��Τ�Τ�ؤ����Ȥˤ��롣

�Ĥ��������ϡ����������β���դ��ä��뤳�ȡפ� ����Ū�ˤϤɤΤ褦�ʤ��Ȥ��̣���뤫���������롣

���� 3.1   �� $L$ �Ȥ�����ʬ�� $K$ ��Ϳ�����Ƥ���Ȥ��롣 $a \in L$ �� $K$ ��κǾ�¿�༰�� $f$ �Ȥ����ȡ�

1. $f$ ��ɬ������Ǥ��롣
2. $g\in K[X]$ �� $g(a)=0$ ����­����Ȥ���ȡ�$g$ �� $f$ �dz���ڤ�롣
3. $K[X]/f(X)K[X]$ �� $K(a)$ ��Ʊ���Ǥ��롣

���� 3.1   $X$ ��4���� $f\in$   $\mbox{${\mathbb{Q}}$}$$[X]$ �ǡ� $a=\sqrt{5}+\sqrt{7}$ �ˤ������� $f(a)=0$ �� �������褦�ʤ�Τò¸«¤Ä¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 3.2   ����� $f$ �� $\mbox{${\mathbb{Q}}$}$ �����Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½ (���������ò¤¢¤ï¤»ï¿½ï¿½È¡ï¿½$f$ �� $a$ �κǾ�¿�༰�Ǥ��뤳�Ȥ� �狼�롣�����������ʳ��ǤϤ��ʤ��ñ¤·¤ï¿½ï¿½ï¿½ï¿½ï¿½Ç¤ï¿½ï¿½ë¡£ ���ιֵ��Τ�äȤ��Ȥ��ʳ��ޤǿʤ�С�������������˰פ����ò¤±¤ï¿½è¤¦ï¿½Ë¤Ê¤ë¡£)