�� II �� No.5

��ϡ�¿�༰�Ĥ򥤥ǥ��ǳ�뤳�Ȥˤ�ä��Τ��뤳�Ȥ��Ǥ��뤳�Ȥ�Ҥ٤�� ϡ��¸��ΤȤ��Τ褦��ΤȤ��Ӥ��ˤ��롣��ˤʤ�Τϼ��Ǥ��롣

�� 5.1 ��

�γ��

�ȡ� $K\setminus k$ �θ�

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1.

$\begin{displaymath}I=\{f\in k[X]; f(a)=0\} \end{displaymath}$

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2.

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3.

ñ�ʹĽ�Ʊ��

$\begin{displaymath}\phi: k[X]/(f_0) \to K: \overline{f(X)}\mapsto f(a) \end{displaymath}$

��¸�ߤ��롣

4.

$\phi$ ��

�Ǥ��롣

5.

$[k[a]:k]=\deg(f_0)$

�� 5.1 ��

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��κǾ�¿�༰�Ȥ�� (�ä˻��ꤷ�ʤ��С��Ǿ�¿�༰�Ȥ��Х�˥å��ʤ�Τ򤨤�֤Τ� ��̤Ǥ��롣)

�� 5.1

$\begin{displaymath}\mbox{${\Bbb Q}$}[X]/(X^2-2)\cong\mbox{${\Bbb Q}$}[\sqrt{2}] \end{displaymath}$

�� 5.2

$\begin{displaymath}\mbox{${\Bbb R}$}[X]/(X^2+1)\cong{\Bbb C} \end{displaymath}$

�� 5.3
$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^4-16X^2+4)&\cong\mbox{${\Bbb Q}$ }[\sqrt{5}+\sqrt{3}]\\ \overline{f(X)}&\mapsto f(\sqrt{5}+\sqrt{3}) \end{align*}$
Ʊ�ͤˡ�
$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^4-16X^2+4)&\cong\mbox{${\Bbb Q}$ }[\sqrt{5}-\sqrt{3}]\\ \overline{f(X)}&\mapsto f(\sqrt{5}-\sqrt{3}) \end{align*}$
�Ǥ⤢�롣��ϡ�
$\begin{align*}& \mbox{${\Bbb Q}$ }[\sqrt{5}+\sqrt{3}]\cong\mbox{${\Bbb Q}$ }[\sqrt{5}-\sqrt{3}]\\ & f(\sqrt{5}+\sqrt{3})\mapsto f(\sqrt{5}-\sqrt{3}) \end{align*}$
�Ȥ��Ʊ��¸�ߤ򼨺��Ƥ��롣

�� 5.4
$\begin{align*}\mbox{${\Bbb Q}$ }[X]/(X^2-2)&\cong\mbox{${\Bbb Q}$ }[X]/(X^2-2X-1)\\ \overline{f(X)}&\mapsto \overline{f(X-1)} \end{align*}$

�� 5.5 $\omega=\frac{-1+\sqrt{-3}}{2}$ �Ȥ��ȡ�

$\begin{displaymath}\mbox{${\Bbb Q}$}[\sqrt[3]{5}]\cong\mbox{${\Bbb Q}$}[\sqrt[3]{5}\omega] \end{displaymath}$

��Ω�ġ�

�� 5.1 $\mbox{${\Bbb Q}$ }[X]/(X^2+X+1)$ �� $\mbox{${\Bbb Q}$ }[X]/(X^2+3)$ �Ȥδ֤�Ʊ��ĺ��ʤ�� $\sqrt{2}+3\sqrt{5}+4\sqrt{10}$ �� $\mbox{${\Bbb Q}$ }[\sqrt{5}]$ ��κǾ�¿�༰�� ʤ��

�� 5.2 �� $\mbox{${\Bbb Q}$ }[\sqrt[3]{2}+\sqrt{3}]$ �� $\mbox{${\Bbb Q}$ }[X]/(f(X))$ �Ȥ�Ʊ��ˤʤ�褦�� $f\in \mbox{${\Bbb Q}$ }[X]$ ��ᡢ�ºݤˤ��Ʊ��ʤ��

����� II ���� No.5

�� II �� No.5