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�� IB No.12��

$\fbox{��Υơ��}$ ��Ǹ�ʬ��ġ�(2)

�� 12.1 ��

�� $a,b\in R$ �Ȥˤ��ơ�

$a \in b R$ �ΤȤ�� ܸ��Ǥ��Ȥ�� $b\vert a$ �ǽ�ɽ�� Ȥ��ơ� �� 󸵤Ǥ��Ȥ⤤��
�� $u \in R^\times$ ��äơ� ��ߤ��Ȥ�� Ȥ� Ʊȼ�Ǥ��Ȥ��

̿�� 12.1 �� θ� �ˤ��ơ�

$(a) \subset (b) {\Leftrightarrow} b\vert a$ .
�� Ʊȼ ${\Leftrightarrow}$ .

�� 12.2 ��

��Ϳ��Ƥ��Ȥ��롣 $d_0\in R$ �� $a,b\in R$ �� ��( $\gcd$ ) �Ǥ��Ȥ�

$% latex2html id marker 868 $\displaystyle \forall d \in R \left ( \ d \vert d_0 \quad {\Leftrightarrow}\quad ( d \vert a \text{ �� }d\vert b) \ \right) $$

��Ω�ĤȤ��˸��

�� 12.1 ñ�।�ǥ�� Υ��ǥ��

$\displaystyle I_1 \subset I_2 \subset I_3 \subset I_4 \subset \dots$

��ɬ��ɤ��ǻߤޤ롣��ʤ�� äơ�

$\displaystyle I_N=I_{N+1}=I_{N+2}=\dots$

��ʤꤿ�ġ�

��ϥ͡��Ĥΰ��ü�ʾ��Ǥ��롣 ��ǡ�

���͡���Ǥ��Ȥϡ�

��Ǥ�դΥ��ǥ��뤬ͭ�¸Ĥθ�� ˸��͡��ĤΥ��ǥ��⡢ɬ��ɤ��ǻߤޤ뤳�Ȥ��Ǥ��롣 ��ϤۤȤ��Ʊ��ʤΤǿʤ��ٶ��ͤϤ�äƤߤ��ɤ�� (;�̤ʤ��͡��Ĥϴ��ˤ��ѽ��פ��оݤǤ��롣 �ξ�ͭ��ʴĤ��ƥ͡��ĤǤ��롣(�ҥ�٥�Ȥδ��))

̿�� 12.2 ��Ǹ�ʬ��Ĥʤ�С� $R\setminus \{0\}$ �γƸ��

$% latex2html id marker 896 $\displaystyle u p_1 p_2 \dots p_l \qquad(l \in \mathbb{N}, u\in R^\times , p_1,\dots,p_l$$ �� $R$ ��Ǹ� $\displaystyle )$

�Ƚ񤯤��Ȥ��Ǥ��뤬��ν��Ʊȼ��ư��Ū�Ǥ��롣 ��ʤ��

	$% latex2html id marker 898 $\displaystyle u p_1 p_2 \dots p_l =v q_1q_2 \dots q_m$$
	$% latex2html id marker 899 $\displaystyle (l,m \in \mathbb{N}, u,v\in R^\times , p_1,\dots,p_l,q_1,\dots,q_m$$ �� Ǹ� $\displaystyle )$

�ʤ�С� �Ǥ��äơ��ʤ��Ĥ�� $\sigma\in \mathfrak{S}_l$ ��ä� �� ˤ�� $% latex2html id marker 912 $ q_{\sigma(j)}$$ �Ϥ��줾��Ʊȼ�ˤʤ롣

�� 12.1 ��

�θ�

�κ��󸵤�2�Ĥ��ä��Ȥ��С� ��ϸߤ��Ʊȼ�Ǥ��뤳�Ȥ��衣

ED: Euclidean domain
PID: principal ideal domain
UFD: unique factorization domain ľ��ϡְ��ʬ��ġס�

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2016-12-19

����� IB No.12����

�� IB No.12��