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�U�@��: Algebraic Closure �W�@��: ���� Field ���ʽ� �e�@��: ���� Field ���ʽ�

Algebraic Elements

���] F �O�@�� field, L �O F ���@�� extension. �n���D F �����@�Ӥ��� a �O�_ algebraic over F, �̩w�q�N�������ҬO�_�s�b�@�Ӥ��� 0 �� f (x) $ \in$ F[x] �ϱo f (a) = 0. �@��ӻ��γo�ؤ�k�����Ҥ@�Ӥ����O�_�O algebraic over F, �޳N�W�O�۷��x����. �o�@�`���ڭ̱N�Q�״X�ةM��� algebraic element ���w�q�������ʽ�, �o�˥H��ڭ̭n���Ҥ@�Ӥ����O�_�O algebraic over F �N���h�@�I����k�ӳB�z.

�����`�N�� a $ \in$ L �O algebraic over F ��, �ƹ�W���� f (x) $ \in$ F[x] �B f (a) = 0 ���h�������L�a�h��. ���L�o�䤤���@�Ӭ۷��S�O. �ڭ̭����i�H�Ҽ���� f (a) = 0 �� f (x) $ \in$ F[x] �� degree �̤p�� polynomials. �o�˪� polynomials ���H�U��ӭ��n���ʽ�.

Lemma 10.1.1   ���] F �O�@�� field, L �O F ���@�� extension. �Y a $ \in$ L �O algebraic over F �B f (x) $ \in$ F[x] �O F[x] ������ f (a) = 0 ���D 0 �h������ degree �̤p���@�� polynomial, �h f (x) ���H�U��өʽ�:
  1. �Y g(x) $ \in$ F[x] �B g(a) = 0, �h�s�b h(x) $ \in$ F[x] ���� g(x) = f (x) . h(x).
  2. f (x) �O F[x] ���� irreducible element.

�� ��. (1) �ѩ� F �O�@�� field, �Q�� Euclid's Algorithm (Theorem 7.2.4) ���s�b h(x), r(x) $ \in$ F[x] �ϱo

g(x) = f (x) . h(x) + r(x) (10.1)

�䤤 r(x) = 0 �� deg(r(x)) < deg(f (x)). �N a �N�J���l (10.1) �o

g(a) = f (a) . h(a) + r(a).

�ѩ� f (a) = g(a) = 0, �ڭ̱o r(a) = 0. �p�G r(x)$ \ne$ 0, �h�o�� r(x) $ \in$ F[x] ���� deg(r(x)) < deg(f (x)) �B r(a) = 0. �o�M f (x) ���쪺����ۥ٬�, �G�� r(x) = 0. �]�N�O�� g(x) = f (x) . h(x).

(2) ���] f (x) �b F[x] �����O irreducible, �Y�s�b h(x), l (x) $ \in$ F[x] ���� deg(h(x)) < deg(f (x)), deg(l (x)) < deg(f (x)) �B f (x) = h(x) . l (x). �N a �N�J�W��, �� f (a) = 0 �� h(a) . l (a) = 0. �ѩ� h(x), l (x) $ \in$ F[x] �B a $ \in$ L, �ڭ̪� h(a), l (a) $ \in$ L. �G�� L �O integral domain (Lemma 9.1.1) �o h(a) = 0 �� l (a) = 0. �o�A���M f (x) ������ۥ٬�, �G�� f (x) �O F[x] ���� irreducible element. $ \qedsymbol$

�Y f (x) $ \in$ F[x] �O F[x] ���ŦX f (a) = 0 degree �̤p�� polynomial �B g(x) $ \in$ F[x] ���� g(a) = 0, �h�� Lemma 10.1.1 (1) �� g(x) $ \in$ $ \bigl($f (x)$ \bigr)$. ��b�p�G g(x) �]�O F[x] ���ŦX g(a) = 0 degree �̤p�� polynomial, �h�i�o $ \bigl($f (x)$ \bigr)$ = $ \bigl($g(x)$ \bigr)$. �ѩ� F[x] ���� unit ���O F �����D 0 ���� (Proposition 7.2.3) �Q�� Lemma 8.1.3 ���s�b c $ \in$ F �ϱo f (x) = c . g(x). �ҥH�p�G�ڭ̱N�o�Ǧ��Ƴ̧C�Ӻ��� f (a) = 0 �� polynomial ���H���̪��̰������Y�Ʃұo�� monic polynomial �N�ߤ@�F. �]���ڭ̦��H�U���w�q.

Definition 10.1.2   ���] F �O�@�� field, L �O F ���@�� extension field �B a $ \in$ L �O algebraic over F. �Y p(x) $ \in$ F[x] �O F[x] ���D 0 polynomial ������ p(a) = 0 degree �̤p�� monic polynomial, �h�� p(x) �O a over F �� minimal polynomial. �S�p�G deg(p(x)) = n, �h�� a �O algebraic over F of degree n.

�ڭ̪��D�� [L : F] �O�������ɭ�, L �����������O algebraic over F. �Y [L : F] = n �B a $ \in$ L, �h�ѩ� 1, a,..., an �@�w linearly independent over F, �G���s�b f (x) $ \in$ F[x] �B deg(f (x))$ \le$n �ϱo f (a) = 0 (�Ԩ� Theorem 9.3.7 ������) �G�� minimal polynomial ���w�q��: �Y p(x) �O a �� minimal polynomial, �h deg(p(x))$ \le$deg(f (x))$ \le$n. �������ڭ̱o a �� degree �p��ε��� [L : F]. �ڭ̱N�o�ӵ��G�g���H�U�� Lemma.

Lemma 10.1.3   ���] F �O�@�� field, L �O F ���@�� finite extension, �h L �����N���������O algebraic over F �B�� degree �p��ε��� [L : F].

�� L ���O finite extension over F ��, L �����M���i��s�b�����O algebraic over F. �p�G a $ \in$ L �O algebraic over F, �ڭ̷Q���D F �M L �����O�_�i�H���@�� field K �O F ���@�� finite extension ���� a $ \in$ K?

Definition 10.1.4   ���] F �O�@�� field, L �O F ���@�� extension field. �Y K �O L ���@�� extension field �B F $ \subseteq$ K $ \subseteq$ L, �h�� K �O L over F ���@�� subextension �άO intermediate field.

Proposition 10.1.5   ���] F �O�@�� field, L �O F ���@�� extension field. �Y a $ \in$ L �O algebraic over F �B�� degree �� n, �h�s�b L over F ���@�� subextension K ���� a $ \in$ K �B [K : F] = n.

�� ��. �Ҽ $ \phi$ : F[x]$ \to$L �䤤����N�� f (x) $ \in$ F[x], $ \phi$(f (x)) = f (a). �ѩ� a $ \in$ L, �ҥH�۵M�� $ \phi$(f (x)) = f (a) $ \in$ L, �]�� $ \phi$ �T��O�@�ӱq F[x] �M�g�� L �����. �ܮe������ $ \phi$ �O�@�� ring homomorphism.

����O ker($ \phi$) �O? �ѩ� F[x] �O�@�� principle ideal domain �B ker($ \phi$) �O F[x] ���@�� ideal, �ڭ̪��s�b p(x) $ \in$ f[x] �ϱo ker($ \phi$) = $ \bigl($p(x)$ \bigr)$. �ƹ�W �ڭ̥i�H�� ker($ \phi$) = $ \bigl($p(x)$ \bigr)$ �䤤 p(x) �O a �� minimal polynomial. �o�O�]���Y f (x) $ \in$ ker($ \phi$), �h�� $ \phi$(f (x)) = f (a) = 0. �G�� Lemma 10.1.1 �� f (x) $ \in$ $ \bigl($p(x)$ \bigr)$. �Ϥ�, ����N f (x) $ \in$ $ \bigl($p(x)$ \bigr)$, �s�b h(x) $ \in$ F[x] �ϱo f (x) = p(x) . h(x), �]���� p(a) = 0 �o f (a) = p(a) . h(a) = 0. �G�o�� ker($ \phi$) = $ \bigl($p(x)$ \bigr)$, �䤤 p(x) �O a �� minimal polynomial.

��� First Isomorphism Theorem (6.4.2) ��

F[x]/$\displaystyle \bigl($p(x)$\displaystyle \bigr)$ $\displaystyle \simeq$ im($\displaystyle \phi$).

�M�� p(x) �O F[x] ���@�� irreducible element (Lemma 10.1.1), �G�� $ \bigl($p(x)$ \bigr)$ �O F[x] ���@�� maximal ideal (Lemma 8.3.2), �o�� F[x]/$ \bigl($p(x)$ \bigr)$ �O�@�� field (Theorem 6.5.11). ������ im($ \phi$) �O�@�� field.

�ܩ󤰻�O im($ \phi$) �O? �ѩw�q��

im($\displaystyle \phi$) = {f (a) | f (x) $\displaystyle \in$ F[x]}.

������, im($ \phi$) �̪��������O�ѬY�� F[x] �̪� polynomial �N�J a �ұo. �ҥH�Y c $ \in$ F, �ڭ̦۵M�� $ \phi$(c) = c $ \in$ im($ \phi$), �G�o F $ \subseteq$ im($ \phi$) $ \subseteq$ L. �t�@�譱�N a �N�J x �o�@�� polynomial �o�� a: �]�N�O�� $ \phi$ �N x �e�� a (�Y $ \phi$(x) = a), �G�� a $ \in$ im($ \phi$). �ҥH�Y�O K = im($ \phi$), �h�� K �O L over F ���@�� subextension �B a $ \in$ K. �̫�Ѱ��] a over F �� degree �O n, �]�N�O�� a �� minimal polynomial p(x) �� degree �O n, �]���� Lemma 9.3.6 �� dimF(F[x]/$ \bigl($p(x)$ \bigr)$) = deg(p(x)) = n. �G�� K $ \simeq$ F[x]/$ \bigl($p(x)$ \bigr)$ �� [K : F] = n. $ \qedsymbol$

�Y�ȥѩw�q�Ӭ� Proposition 10.1.5 ���� im($ \phi$) = {f (a) | f (x) $ \in$ F[x]} �u�O�@�� ring, �����󥦷|�O field �O? �Y�A�O�o Theorem 9.3.7 �o�N�@�I�����_�ǤF. �]�� im($ \phi$) $ \subseteq$ L �۵M�O integral domain, �ӥ� Proposition 10.1.5 ���ҩ��]�� dimF(im($ \phi$)) = n.

�ڭ̤]�ܮe���ˬd {f (a) | f (x) $ \in$ F[x]} �|�O L ���]�t F �H�� a �̤p�� ring, �o�O�]���Y R �O�@�� ring �B�]�t F �H�� a, �h����N�� f (x) $ \in$ F[x], �ѩ� f (a) �Ȳo�A�� a �M F �������������[�k�H�έ��k, �O�ѤF�o�dz��O R ���������B��ҥH���M�o f (a) $ \in$ R. �������ڭ̱o {f (a) | f (x) $ \in$ F[x]} $ \subseteq$ R, �A�[�W {f (a) | f (x) $ \in$ F[x]} �����O�@�� ring �ҥH���۵M�O�]�t F �H�� a �̤p�� ring �F!

���F��K�ڭ̩w�H�U���Ÿ�, �b�@�몺�N�ƮѤW�o�өw�q�O�зǪ��B�`�Q�ϥΪ��w�q.

Definition 10.1.6   ���] F �O�@�� field, L �O F ���@�� extension field �B a $ \in$ L. �ڭ̥O F[a] ���� L ���]�t F �H�� a �̤p�� ring; �ڭ̤]�O F(a) ���� L ���]�t F �H�� a �̤p�� field.

�e���w�� F[a] �N�O im($ \phi$) = {f (a) | f (x) $ \in$ F[x]}. ���� F(a) ���������S�O��˩O? �Q�� quotient field ���ʽ� (Proposition 7.4.2) �ܮe������

F(a) = {f (a)/g(a) | f (x), g(x) $\displaystyle \in$ F[x] �B g(a)$\displaystyle \ne$0}.

�ѳo�̥i�ݥX: �@��ӻ� F[a] �M F(a) �O���ۦP��; ���L�e�����L�Y a �O algebraic over F, �h F[a] �|�O�@�� field, �ҥH F[a] �۵M�O�]�t F �H�� a �̤p�� field. ���y�ܻ��� a �O algebraic over F ��, �ڭ̦� F[a] = F(a). �]�� F(a) �N�O Proposition 10.1.5 ���ҭn�䪺 K, �ҥH�ڭ̱N Proposition 10.1.5 ����H��i�H�o:

Corollary 10.1.7   ���] F �O�@�� field, L �O F ���@�� extension field. �Y a $ \in$ L �O algebraic over F �B p(x) $ \in$ F[x] �� a over F �� minimal polynomial, �h

F(a) $\displaystyle \simeq$ F[x]/$\displaystyle \bigl($p(x)$\displaystyle \bigr)$    and    [F(a) : F] = deg(p(x)).

Remark 10.1.8   �P�ǩγ|�_�� F[a] �̪����������O f (a) �䤤 f (x) $ \in$ F[x] �o�ؼˤl, �� F(a) �̪����������O f (a)/g(a) �䤤 f (x), g(x) $ \in$ F[x] �o�ؼˤl: ��Ӽˤl�t�o��h, ���i��| F[a] = F(a) �O? �o�O�]���� a �O algebraic over F ��, F[a] (�� F(a)) �̪���������ܪk�O���ߤ@��. �Ҧp�Y p(x) $ \in$ F[x] �O a �� minimal polynomial, �p�G�O g(x) = f (x) + p(x), �h g(a) = f (a). �ҥH���M���i��Τ��P���Φ��g�U�Ӫ��������̪��ȬO�ۦP��.

���U�ӧڭ̴N�ӬݩM a �O algebraic over F ����������O����?

Theorem 10.1.9   ���] F �O�@�� field, L �O F ���@�� extension field �B a $ \in$ L, �h�U�������� a ���ԭz�O������.
  1. a �O algebraic over F.
  2. �s�b K �O L over F �� subextension ���� a $ \in$ K �B [K : F] �O������.
  3. F[a] = F(a).

�� ��. �ѫe�� Proposition 10.1.5 �i�� (1) $ \Rightarrow$ (2), �ҥH�ڭ̶ȭn���� (2) $ \Rightarrow$ (3) �H�� (3) $ \Rightarrow$ (1).

(2) $ \Rightarrow$ (3): �Y K �O L over F �� subextension (�Y F $ \subseteq$ K $ \subseteq$ L), �h�Ѱ��] a $ \in$ K �� F[a] $ \subseteq$ K. �A�Ѱ��] K �O F ���@�� finite extension, �M�� Proposition 9.4.3 �i�o F[a] �O�@�� field. �G�� F[a] = F(a).

(3) $ \Rightarrow$ (1): ���] F[a] = F(a), �]�N�O�� F[a] �O�@�� field. �p�G a = 0 $ \in$ F, �����M a �O algebraic over F (�`�N F �����������M�O algebraic over F). �p�G a$ \ne$ 0, �h�� a $ \in$ F[a] �B F[a] �O�@�� field �� a-1 $ \in$ F[a]. �O�ѤF F[a] �̪��������O f (a), �䤤 f (x) $ \in$ F[x] �o�اΦ�, �ҥH�ڭ̦� a-1 = f (a), �䤤

f (x) = cnxn + ... + c1x + c0,    ci $\displaystyle \in$ F.

�G��

a-1 = cn . an + ... + c1 . a + c0

�o

1 = cn . an + 1 + ... + c1 . a2 + c0 . a.

�]���Y�O

g(x) = cnxn + 1 + ... + c1x2 + c0x - 1,

�h g(a) = 0. �ѩ� g(x) $ \in$ F[x] �B g(x)$ \ne$ 0, �G�� a �O algebraic over F. $ \qedsymbol$

Theorem 10.1.9 ���F�ڭ̤@�ӫܦn����k������ a �O�_�O algebraic over F. �]�N�O������n�ˬd a �O algebraic over F �ڭ̥i�H�����u���h��@�� f (x) $ \in$ F[x] �ϱo f (a) = 0. ���M�F�n�Τ��̤�k�|�]���D�Ӧ��Үt�O. ��軡�Y a2 $ \in$ L �B�ڭ̪� a2 �O algebraic over F, �p�G f (x) $ \in$ F[x] ���� f (a2) = 0, �h�O g(x) = f (x2), �ڭ̥i�o g(a) = f (a2) = 0. �]���� a �]�O algebraic over F. �]�N�O�� a2 �O algebraic over F ��, a �]�|�O algebraic over F. ���O�ϹL��, �p�G�w�� a �O algebraic over F, �ڭ̴N�L�k�Q�κ��� a �� polynomial �ӻs�y�@�Ӻ��� a2 �� polynomial �F. �P�ǩγ|�Q�Y f (a) = 0, �ڭ̥i�H�O g(x) = f (x1/2), �h g(a2) = f (a) = 0 �r! �o�O���諸, �]�� f (x) �Y���_�Ʀ���, �h g(x) = f (x1/2) �N���A�O�@�� polynomial �F. �ҥH�b�o�ت��p�U�N���i��Q�Χ� polynomial ����k���ҩ� a2 �O algebraic over F. ���� a �O algebraic over F �ɧQ�� Theorem 10.1.9 ���s�b�@�� field K �O F �� finite extension �B a $ \in$ K. �M�� K �O�@�� field �B a $ \in$ K, �ҥH���M a2 $ \in$ K, �ҥH�A�Τ@�� Theorem 10.1.9 (�άO�Q�� Lemma 10.1.3) �ڭ̱o�� a2 �]�O algebraic over F. �H��ڭ̱`�|����������k�ӳB�z���������D.


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�U�@��: Algebraic Closure �W�@��: ���� Field ���ʽ� �e�@��: ���� Field ���ʽ�
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