Euclidean Domain

�^�U�@�U $\mathbb {Z}$ �� Euclid's Algorithm �i�H��O�� a, b $\in$ $\mathbb {Z}$ , �䤤 b $\ne$ 0, �h�s�b h, r $\in$ $\mathbb {Z}$ , �䤤 r �ŦX r = 0 �� $\left\vert\vphantom{ r}\right.$ r $\left.\vphantom{ r}\right\vert$ < $\left\vert\vphantom{ b}\right.$ b $\left.\vphantom{ b}\right\vert$ �ϱo a = b^.h + r. �Ӧb F[x] �� Euclid's Algorithm �O�� f (x), g(x) $\in$ F[x] �䤤 g(x) $\ne$ 0, �h�s�b h(x), r(x) $\in$ F[x], �䤤 r(x) �ŦX r(x) = 0 �� deg(r(x)) < deg(g(x)) �ϱo f (x) = g(x)^.h(x) + r(x). �o�̭��n��O�b $\mathbb {Z}$ ��@�ӵ��Ȩ�ƱN $\mathbb {Z}$ ��D 0 ��e��D�t��, �Ӧb F[x] ��@�� degree ��ƱN F[x] ��D 0 ��e��D�t��. �ڭ̴N�O�n�^��o�˪��ƪ��S��.

Definition 8.2.1 �] R �O�@�� integral domain. �p�G�s�b�@��

$\displaystyle \Phi$ : R $\displaystyle \setminus$ {0} $\displaystyle \to$ $\displaystyle \mathbb {N}$ $\displaystyle \cup$ {0}

�ϱo��N�� a, b $\in$ R �䤤 b $\ne$ 0 ��i�H�� h, r $\in$ R, �䤤 r �ŦX r = 0 �� $\Phi$ (r) < $\Phi$ (b), �� a = b^.h + r, �h�� R ��@�� Euclidean domain.

��F $\mathbb {Z}$ �M F[x] � �٦��h�� Euclidean domain. �Ҧp $\mathbb {Z}$ [i] = {a + bi | a, b $\in$ $\mathbb {Z}$ } �o�@�� integral domain �Q�� $\Phi$ (a + bi) = a² + b² �o�Ө�ƴN�i�o $\mathbb {Z}$ [i] �O�@�� Euclidean domain (�b��ڭ̲��h�ҩ�, �Y��쪺�P�ǥi�� http://math.ntnu.edu.tw/ $\sim$ li/note �U��q ``Factorization of Commutative Rings'' ��Բ��ҩ�).

�@��Ө��n��Ҥ@�� integral domain �O�_��@�� Euclidean domain �O�ܧx��. �b��ڭ̨ä��Q�׳o��D. �ڭ̶ȦC�X Euclidean domain ��n�ʽ�. �^�U�ڭ̴��Q�� Euclid's Algorithm �ҥX�b $\mathbb {Z}$ �M F[x] ��Ҧ�� ideal ��O principle ideal. �o�@�M�ҩ��i�H��h�� Euclidean domain �W.

�� . �Y I �O R ��@�� ideal. �Ҽ T = { $\Phi$ (a) | a $\in$ I $\setminus$ {0}} �o�@�Ӷ��X. �ѩ� $\Phi$ ��Ȱ�b $\mathbb {N}$ $\cup$ {0} �ҥH T �O $\mathbb {N}$ $\cup$ {0} ��@�Ӥl��X. �]�� T ��s�b�̤p��. ��y�ܻ��s�b d $\in$ I $\setminus$ {0} �ϱo��N�� a $\in$ I $\setminus$ {0} �Ҧ� $\Phi$ (d ) $\le$ $\Phi$ (a). �ڭ̱�� I = $\bigl($ d $\bigr)$ .

�ѩ� d $\in$ I, �۵M�o $\bigl($ d $\bigr)$ $\subseteq$ I. �t� ��N a $\in$ I, �� Euclidean domain ��]��s�b h, r $\in$ R �� a = d^.h + r �B r = 0 �� $\Phi$ (r) < $\phi$ (d ). �p�G r $\ne$ 0, �� r = a - d^.h �B a, d $\in$ I �i�� r $\in$ I. �]�N�O�� r $\in$ I $\setminus$ {0} �B $\Phi$ (r) < $\Phi$ (d ). �o�M $\Phi$ (d ) �O T ��̤p��]�ۥ٬�, �G�� r = 0. �� a = d^.h, �Y a $\in$ $\bigl($ d $\bigr)$ . �G�o�� I $\subseteq$ $\bigl($ d $\bigr)$ . $\qedsymbol$

�ѩ�@�� integral domain �� ideal ��O principle ideal �o�˪� ring �D�`�S�O, �ڭ̤]��@�ӯS�O��W��.

Theorem 8.2.2 �i�D�ڭ̤@�� Euclidean domain �@�w�O�@�� principle ideal domain. �n�`�N, �� principle ideal domain ��|�O�@�� Euclidean domain. ��쪺�P�ǥi�H�Ѧҧڪ��q ``Factorization of Commutative Rings'' �䤤��@�� principle ideal domain ��O Euclidean domain ��Ҥl.