����ر齬 I ���� No.8

�����No.8�ˤϡ��ִġפȸ�����ñ�̸�����IJĴ��ĤΤ��Ȥ�ؤ����Ȥˤ��ޤ����ޤ����ֽ�Ʊ���פ�ñ�̸����ݤĤ�Τ�����ͤ��뤳�Ȥˤ��ޤ���

���� 8.1 (��1)   ���γơ��δĤ�Ʊ�����Ʊ���������Ѥ��ƾ������ʤ�����

1. $\mbox{${\mathbb{Q}}$}$$[X]/(X-6) \cong$   $\mbox{${\mathbb{Q}}$}$ .
2. ${\mbox{${\mathbb{Z}}$}}[X]/(X+7) \cong {\mbox{${\mathbb{Z}}$}}$ .
3. ${\mathbb{C}}[X]/(X-\pi) \cong {\mathbb{C}}$ (������ $\pi$ �ϱ߼�Ψ��)
4. ${\mathbb{F}}_{11}[X]/(X-4)\cong {\mathbb{F}}_{11}$ .

���� 8.2   ���̤ˡ��� $K$ ��ΰ켡�� $p(X)=a X +b$ ($a,b \in K$ , $a\neq 0$ ) ���Ф��ơ� $K[X]/(p(X)) \cong K$ �Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.3   ����ǡ�$K$ ���ΤǤ���Ȥ����������� $K$ �����̤βĴ��ĤǤ���� ���ꤷ�����ˤϡ� $K[X]/(p(X))$ �� $K$ ��Ʊ���Ȥϸ¤�ʤ����Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.4   $\mbox{${\mathbb{Q}}$}$$[\sqrt{6}]\cong$   $\mbox{${\mathbb{Q}}$}$$[X]/(X^2-6)$$\mbox{${\mathbb{Q}}$}$$[X]$ �ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.5   ͭ������ $\mbox{${\mathbb{Q}}$}$ ����ʬ���� ${\mbox{${\mathbb{Z}}$}}[1/5]=\{m/5^n; m\in {\mbox{${\mathbb{Z}}$}}; n=0,1,2,\dots,)$ �� $\mbox{${\mathbb{Q}}$}$ ����ʬ�Ĥˤʤ뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.6  

$$(5X-1)$$$$\mbox{${\mathbb{Q}}$}$$$$[X]\cap {\mbox{${\mathbb{Z}}$}}[X]=(5X-1){\mbox{${\mathbb{Z}}$}}[X]$$

�򼨤��ʤ�����

���� 8.7  

$${\mbox{${\mathbb{Z}}$}}[1/5]\cong {\mbox{${\mathbb{Z}}$}}[X]/(5X-1){\mbox{${\mathbb{Z}}$}}[X]$$

�򼨤��ʤ�����

���� 8.8   �� $R$ �Υ��ǥ��� $I$ ���ѿ� $X$ �ˤĤ��ơ�����Ʊ���ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

$$R[X]/IR[X] \cong (R/I)[X]$$

���� 8.9   �� $S$ ����ʬ�� $R$ �� $S$ �Υ��ǥ��� $I$ �ˤĤ��ơ�

$$R+I=\{r+a;r\in R,a \in I\}$$

�ϡ�$S$ ����ʬ�ĤȤʤ뤳�Ȥ򼨤��ʤ�����

���� 8.10   �� $S$ ����ʬ�� $R$ �� $S$ �Υ��ǥ��� $I$ �ˤĤ��ơ�

$$S=R+I, \quad R\cap I=0$$

������Ω�ƤС� $S/I\cong R$ �Ȥʤ뤳�Ȥ򼨤��ʤ�����

���� 8.11   �Ľ�Ʊ�� $f:R\to S$ �ˤĤ��ơ�$J$ �� $S$ �Υ��ǥ���Ǥ���С� $f^{-1}(J)$ �� $R$ �Υ��ǥ���Ȥʤ뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.12 (��1)   �Ľ�Ʊ�� $f:R\to S$ �ˤĤ��ơ�

1. $I$ �� $R$ �Υ��ǥ���ΤȤ���$f(I)$ �� $S$ �Υ��ǥ���Ȥʤ�ޤ�����(�ʤ�ʤ����ȿ���󤲤Ƥ���������)
2. $f$ �����ͤʤ�ɤ��Ǥ�����

���� 8.13   $K$ ���ΤȤ��ޤ������ΤȤ�Ʊ�� $K[X,Y]/XK[X,Y]\cong K[Y]$ �ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.14   �� $S$ �Ȥ�����ʬ�� $R$ �ȤˤĤ��ơ�$P$ �� $S$ ���ǥ��ǥ���ʤ�С�$P\cap R$ �� $R$ ���ǥ��ǥ���Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Ç¥ï¿½ï¿½Ç¥ï¿½ï¿½ï¿½×¤ï¿½Ö¶ï¿½ï¿½ç¥¤ï¿½Ç¥ï¿½ï¿½ï¿½×¤Ë¤ï¿½ï¿½ï¿½ï¿½ï¿½È¤É¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.15   �� $R$ �Υ��ǥ���� $I$ �ˤĤ��ơ� $I\cdot I$ �� $I^2$ ��ά�����ޤ���$J$ �� $R$ �Υ��ǥ���ǡ�$I+J=R$ �Ȥʤ�С�$I^2+J^2=R$ �Ȥʤ뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.16   ���� $R$ �θ� $a,b$ �ˤĤ��ơ����ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

1. $a\vert b$ �Ǥ���(���ʤ�������� $c\in R$ �����äơ� $b=ac$ �Ƚ񤱤�)���Ȥȡ� $aR\subset bR$ �Ȥ�Ʊ�ͤǤ��롣
2. $a$ �� $b$ �Ȥ�Ʊȼ�Ǥ���(���ʤ����$a\vert b$ ���� $b\vert a$ ������Ω��)���Ȥȡ� $aR=bR$ �Ȥ�Ʊ�ͤǤ��롣

�ʲ��Ͻ�����������������Ǥ���

���� 8.17   �������� $a,b$ ��������� $1$ �Ǥ��뤿���ɬ�׽�ʬ���ϡ�

$$a l +b m=1$$

�������� ���� $l,m$ ��¸�ߤ��뤳�ȤǤ��롣����򼨤��ʤ�����

���� 8.18 (��������˸¤äƤ����䤬�ò¤±¤Æ¤ï¿½ï¿½ë¡¢ï¿½ï¿½ï¿½Ê¤ï¿½ï¿½Ë¹ï¿½ï¿½ï¿½é¤º ���η�̤�ȤäƤ褤��(������ˤ���ֵ��Ǥ�뤫�顣)   ) ������Ѥ��ơ� $a,b$ ���ߤ����Ǥʤ�С� $a^2$ �� $b^2$ ��ߤ����ǤǤ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.19   ������Ѥ��ơ� $a,b$ ���ߤ����Ǥʤ�С� $a^2$ �� $b^2$ ��ߤ����ǤǤ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 8.20   ����������ơ� $\sqrt{6}$ ��̵�����Ǥ��뤳�Ȥ�������ʤ������������� �ǰ���ʬ��ΰ�������μ����Ѥ��ʤ��Ǿ������뤳�ȡ�

���� 8.21   �����Ʊ��������ǡ� ���̤ˡ�ʿ�����Ǥʤ�(�Ĥޤꡢ $\{n^2; n\in {\mbox{${\mathbb{Z}}$}}\}$ �θ��ǤϤʤ�)���� $m$ �� �������ơ� $\sqrt{m}$ ��̵�����Ǥ��뤳�Ȥ�������ʤ�����

���� 8.22   ����������¿�༰

$$p(X)=\sum_{j=0}^n a_j X^j \qquad(a_j\in {\mbox{${\mathbb{Z}}$}} (j=0,1,2,\dots,n),\quad a_n \neq 0)$$

�ˤĤ��ơ� $p$ �Ρ� 0 �Ǥʤ�ͭ���� $r$ �����ä��Ȥ��롣 ���ʤ����$p(r)=0$ �Ȥ��롣 $r$ �����ʬ�� $\frac{l}{m}$ �Ƚ񤤤��Ȥ���

1. $r$ ��ʬ�� $l$ �� $p$ ������� $a_0$ ������Ǥ��뤳�Ȥ򼨤��ʤ�����
2. $r$ ��ʬ�� $m$ �� $p$ �κǹ⼡�η��� $a_n$ ������Ǥ��뤳�Ȥ򼨤��ʤ�����