�����`�N�� a L �O algebraic over F ��, �ƹ�W����
f (x)
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)
F[x] �� degree �̤p��
polynomials. �o�˪� polynomials ���H�U��ӭ��n���ʽ�.
(2) ���] f (x) �b F[x] �����O irreducible, �Y�s�b
h(x), l (x) 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)
F[x] �B a
L, �ڭ̪�
h(a), l (a)
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.
�Y
f (x) F[x] �O F[x] ���ŦX f (a) = 0 degree �̤p�� polynomial
�B
g(x)
F[x] ���� g(a) = 0, �h�� Lemma 10.1.1 (1) ��
g(x)
f (x)
. ��b�p�G g(x) �]�O F[x] ���ŦX g(a) = 0
degree �̤p�� polynomial, �h�i�o
f (x)
=
g(x)
. �ѩ�
F[x] ���� unit ���O F �����D 0 ���� (Proposition 7.2.3)
�Q�� Lemma 8.1.3 ���s�b c
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.
�ڭ̪��D�� [L : F] �O�������ɭ�, L �����������O algebraic over
F. �Y [L : F] = n �B a L, �h�ѩ�
1, a,..., an �@�w linearly
independent over F, �G���s�b
f (x)
F[x] �B
deg(f (x))
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))
deg(f (x))
n. �������ڭ̱o a �� degree
�p��ε��� [L : F]. �ڭ̱N�o�ӵ��G�g���H�U�� Lemma.
�� L ���O finite extension over F ��, L �����M���i��s�b�����O
algebraic over F. �p�G a L �O algebraic over F, �ڭ̷Q���D
F �M L �����O�_�i�H���@�� field K �O F ���@�� finite
extension ���� a
K?
����O
ker() �O? �ѩ� F[x] �O�@�� principle ideal domain �B
ker(
) �O F[x] ���@�� ideal, �ڭ̪��s�b
p(x)
f[x] �ϱo
ker(
) =
p(x)
. �ƹ�W �ڭ̥i�H��
ker(
) =
p(x)
�䤤 p(x) �O a �� minimal polynomial. �o�O�]���Y
f (x)
ker(
), �h��
(f (x)) = f (a) = 0. �G�� Lemma 10.1.1
��
f (x)
p(x)
. �Ϥ�, ����N
f (x)
p(x)
, �s�b
h(x)
F[x] �ϱo
f (x) = p(x) . h(x), �]���� p(a) = 0 �o
f (a) = p(a) . h(a) = 0. �G�o��
ker(
) =
p(x)
, �䤤 p(x)
�O a �� minimal polynomial.
��� First Isomorphism Theorem (6.4.2) ��
�ܩ�O
im() �O? �ѩw�q��
�Y�ȥѩw�q�Ӭ� Proposition 10.1.5 ����
im() = {f (a) | f (x)
F[x]} �u�O�@�� ring, �����|�O field �O? �Y�A�O�o
Theorem 9.3.7 �o�N�@�I�����_�ǤF. �]��
im(
)
L �۵M�O integral domain, �ӥ� Proposition
10.1.5 ���ҩ��]��
dimF(im(
)) = n.
�ڭ̤]�ܮe���ˬd
{f (a) | f (x) 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)
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)
R. �������ڭ̱o
{f (a) | f (x)
F[x]}
R,
�A�[�W
{f (a) | f (x)
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.
�e���w�� F[a] �N�O
im() = {f (a) | f (x)
F[x]}. ����
F(a) ���������S�O��˩O? �Q�� quotient field ���ʽ� (Proposition
7.4.2) �ܮe������
���U�ӧڭ̴N�ӬݩM a �O algebraic over F ����������O����?
(2) (3): �Y K �O L over F �� subextension (�Y
F
K
L), �h�Ѱ��] a
K ��
F[a]
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) (1): ���] F[a] = F(a), �]�N�O�� F[a] �O�@��
field. �p�G a = 0
F, �����M a �O algebraic over F (�`�N F
�����������M�O algebraic over F). �p�G a
0, �h�� a
F[a]
�B F[a] �O�@�� field ��
a-1
F[a]. �O�ѤF F[a]
�̪��������O f (a), �䤤
f (x)
F[x] �o�اΦ�, �ҥH�ڭ̦�
a-1 = f (a), �䤤
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) F[x] �ϱo f (a) = 0.
���M�F�n�Τ��̤�k�|�]���D�Ӧ��Үt�O. ��軡�Y a2
L �B�ڭ̪�
a2 �O algebraic over F, �p�G
f (x)
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
K. �M�� K �O�@�� field �B a
K, �ҥH���M a2
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.