�� II �� No.6

�� 6.1 $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ �� $\mbox{${\Bbb Q}$ }[\sqrt{2},\sqrt{3}]$ �� $\mbox{${\Bbb Q}$ }$ �Ȥ� �֤��ΤǤ��롣

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�� 6.2

�� $\mbox{${\Bbb Q}$ }$ ��Ǥ��롣

1.

�ޤ�� $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ ��

$\begin{displaymath}X^4-10X^2+1= ((X-\sqrt{2})^2-3) ((X+\sqrt{2})^2-3) \end{displaymath}$

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2.

�� $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ ��δ��ʬ��Ǥ��롣

3.

$\mbox{${\Bbb Q}$ }[X]$ �Ǥ�

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��ܤΰ㤤�� $1, ((X-\sqrt{2})^2-3), ((X+\sqrt{2})^2-3),f$ �Τ��줫�ΤϤ��Ǥ��롣 ��Τ��

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4.

�� $\mbox{${\Bbb Q}$ }$ ��Ǥ��뤳�Ȥ��顢

�� 6.2 ��

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�� 6.5 $\sqrt[3]{5}$ �� $\sqrt[3]{5}\omega$ (â�� $\omega=\frac{-1+\sqrt{-3}}{2})$ ) �Ȥ� $\mbox{${\Bbb Q}$ }$ ��ߤ��˶��Ǥ��뤬�� $\mbox{${\Bbb Q}$ }[\sqrt[3]{5}]$ ��Ǥ��Ĥ϶��ǤϤʤ��

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�ؤ�Ʊ�� $\phi$ �Ǥ��äơ� Ǥ�դ� $x\in K$ ��Ф�� $\phi(x)=x$ ��Ω�Ĥ�Τ��

�� 6.6 $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ ��餽�켫�Ȥؤν�Ʊ�� $\phi(a+b\sqrt{2})=a-b\sqrt{2}$ �� $\mbox{${\Bbb Q}$ }[\sqrt{2}]$ �� $\mbox{${\Bbb Q}$ }$ �� Ʊ��Ǥ��롣

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�ؤ�Ʊ�� $\phi:k\to k'$ ��Ϳ��Ƥ��Ȥ�� $\phi$ ��Ĥ��Τ褦��

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�ؤ�Ʊ��˿��Ф��Ȥ��Ǥ��롣 ��򤳤ιֵ��Ǥϲ�� $\hat\phi$ �Ȥ��Ȥˤ��롣(��Ū�ʵ��ǤϤʤ��)
$\begin{align*}&\hat\phi( a_nX^n+a_{n-1}X^{n-1}+\dots+a_1X+a_0)\\ =& \phi(a_n)X^n+\phi(a_{n-1})X^{n-1}+\dots+\phi(a_1)X+\phi(a_0) \end{align*}$

�� 6.2 ��

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