�����ڭ��٬O��@�Ӥ������]�Ƶ��@�ӥ������w�q.
�^�U�@�U�Y R �O integral domain �B d R, �h
d
= {d . r | r
R} �ҥH�ѤW�@�өw�q�ڭ̫ܮe���� d | a �Y�B�߭Y
a
d
. �M�ӭY
a
d
, ��
d
�O�@�� ideal
������N�� r
R �Ҧ�
a . r
d
. �G�o
a
d
. �Ϥ��Y
a
d
, ��
a
a
�o��
a
d
. ���y�ܻ�
a
d
�Y�B�߭Y
a
d
, �]���ڭ̦��H�U������:
Lemma 8.1.2 ���M²����۷����, ���i�D�ڭ̤��������㰣���Y�i�H�ഫ�� ideal �����]�t���Y. �H��ڭ̭n�ͽר⤸�������㰣���Y�ɧڭ̦��ɤ��� divisor ���w�q�B�z, �ڭ̷|�γo�� ideal �����Y�ӱ��Q, �j�a�|�o��o�Ӥ�k�O²��S��K��.
�Y a R �B a
0, �ڭ̫ܧ֪��N���D���N R �����@�� unit
���|�O a ���@�� divisor. �o�O�ѩ�Y u �O R ���� unit, �h
u
= R (Lemma 6.2.4). �G��
a
R =
u
�� u | a. �t�@�譱�� u �O unit ��, a . u �]�O a ��
divisor. �o�]�i��
a . u
=
a
(Lemma 6.5.4) ��
Lemma 8.1.2 ���W�o��. u �M a . u �o�� a �� divisor
�� a �����ѨS���ƻ����U, �ڭ̺٤��� a �� trivial divisor.
�H�U Lemma �O���Q a . u �o�� a �� trivial divisor �M a
��²�����Y.
(2) (3): �i�� Lemma 8.1.2 �������o.
(3) (1): �� a | b ���s�b r
R �ϱo
b = a . r, �A�� b | a ���s�b r'
R �ϱo
a = b . r'. �G��
���F��K�_��, �ڭ̵��� Lemma 8.1.3 �������Y�@�ӯS�����W��.
�Q�� Lemma 8.1.3 ���� (2) �ڭ̪� a b �Y�B�߭Y
a
=
b
, �ҥH���W�o��
�O�@�� equivalence relation.
�^�U�@�U�b
���ڭ̩w a, b �� greatest common divisor �O a, b
�� common divisor ���̤j��, �Ӧb F[x] ���ڭ̩w f (x), g(x) ��
greatest common divisor �O f (x), g(x) �� common divisor �� degree
�̤j��. �b�@�몺 integral domain �O�L�k�w�j�p�� degree ��.
���L�e��ر��p�� greatest common divisor �����@�Ӧ@�P���ʽ� (�Ѩ�
Corollary 7.1.5 (2) �H�� Corollary 7.2.9 (2)),
�ڭ̴N�γo�өʽ�өw integral domain ���� greatest common divisor.
�Y u �O R ���� unit, �h�ѩ�
u
= R (Lemma 6.2.4)
�i������N
a1,..., an �Ҧ�
ai
u
,
i
{1,..., n}. �]�N�O�� u | ai,
i
{1,..., n}. �G�� R ���� unit ���O
a1,..., an �� common divisor. ���L��@�몺 integral domain,
����N��
a1,..., an �� greatest common divisor �����s�b.
�Y�Ϧs�b�� greatest common divisor �]���@�w�ߤ@ (�b F[x]
�����p�N�O�@��). �t� �n�`�N���O�b���w�q���U
���� greatest
common divisor �M Section 7.1 �� Definition 7.1.3 ��
greatest common divisor �ۮt�F�@�ӥ��t��. ���ۧڭ̦C�X greatest
common divisor ���ʽ�.
(2) ���] R ������ӫD 0 ������ greatest common divisor �s�b, �ڭ̧Q�μƾ��k�Ǫk�ҩ����N n �ӫD 0 ���� a1,..., an �� greatest common divisor �]�s�b. ���]���N n - 1 �ӫD 0 ���� a1,..., an - 1 �� greatest common divisor �s�b�B�� d0. �] d0 �M an �ҬO R �����D 0 ����, �Ѱ��]���� greatest common divisor �s�b. �O d �� d0 �M an �� greatest common divisor, �ڭ̭n�ҩ� d �� a1,..., an �� greatest common divisor.
������ d | d0 �B d0 �O
a1,..., an1 �� common divisor
��
d | d0 | ai,
i
{1,..., n - 1}. �A�� d | an �� d �O
a1,..., an ���@�� common divisor.
���ۭY c �O
a1,..., an ���@�� common divisor, �h c ���M�O
a1,..., an - 1 ���@�� common divisor. �G�� d0 �O
a1,..., an - 1 �� greatest common divisor �� c | d0.
������ c �O d0 �M an ���@�� common divisor. �G�� d �O
d0 �M an �� greatest common divisor �� c | d. �]���ѩw�q��
d �O
a1,..., an �� greatest common divisor.
�̫�ڭ̭n�w�q irreducible element �M prime element. Irreducible �O���i���Ѫ��N��, �������N�O���F trivial divisor � �S����L�� divisor.
�ڭ̴��L irreducible element �M prime element ���w�q�W�O���P��, �ҥH���̭�h�W�O��ؤ��P���S��. ���L�H�U�����G�i�D�ڭ̦b integral domain ���U prime element �@�w�O irreducible element.
�e�����g���L�ڭ̳��w�� ideal �����Y�Ӵyø���������㰣���Y. �U���� Lemma �N�O�i�D�ڭ� irreducible element �M prime element �Ҳ��ͪ� principle ideal �ҹ������ʽ�.
: �Ϥ��Y d | a, �h��
a
d
.
�Ѱ��]�S�� nontrivial principle ideal �]�t
a
, �o
d
�O�@�� trivial principle ideal �]�t
a
. ������
d
= R ��
d
=
a
. �Y
d
= R ����
d
=
1
�G�� Lemma
8.1.3 �� d
1, �]�N�O�� d �O�@�� unit. �Y
d
=
a
�P�˥� Lemma 8.1.3 �� d
a. �G�o a
�O�@�� irreducible element.
(2)
: ���] a �O�@�� prime element. �p�G
c . d
a
, ��
a | c . d. �G�� a �O prime �����]�� a | c �� a | d. �o�i�D�ڭ�
c
a
��
d
a
, �G�o��
a
�O�@�� prime ideal.
: ���]
a
�O�@�� prime ideal. ���� c, d
R
����
a | c . d, ��
c . d
a
. �G��
a
�O�@�� prime ideal �����]�o
c
a
��
d
a
. ������
a | c �� a | d, �G�o�� a �O�@�� prime element.