��III �� No.10

$\fbox{��ˤϽ�ʬ��θ��뤳��}$ �ʲ��ιֵ��Ǥϡ֥��פȸ��ͭ�¼�� ̣��뤳�Ȥˤ��롣��ˡ��Ρפȸ��ͭ�� $\mbox{${\mathbb{Q}}$}$ �� ʬ�ΤȤ��ƴޤ�褦�ʤ�Τ򤵤��Ȥˤ��롣

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�Υ��Ǥ��Ȥ�� $\operatorname{Gal}(L/K)$ �θ��ϤɤΤ��餤��Τ��

�� 10.1

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�Υ��ΤȤ��

$\displaystyle \vert\operatorname{Gal}(L/K)\vert=[L:K]$

�� 10.1

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�Υ��ΤȤ��

��ʬ�� ޤ��( �� ) ��Ϳ��ȡ� $\operatorname{Gal}(L/M)$ �� $\operatorname{Gal}(L/K)$ ��ʬ��Ȥߤʤ��Ȥ��Ǥ��롣
�� Τ��Ĥ�� $M_1 \subset M_2$ ��ߤ��ʤ�С�

$\displaystyle \operatorname{Gal}(L/M_1)\supset \operatorname{Gal}(L/M_2)$
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��ñ�˸��С��

��ä� $G=\operatorname{Gal}(G/K)$ ��ä� �֤��ΤǤ��롣��

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��ʬ�� $H=\operatorname{Gal}(M/K)$ ��б��롣

�� 10.1 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 895 $ (\sqrt{2})/$$ $\mbox{${\mathbb{Q}}$}$

�θ� $\sigma$ �ϡ� $% latex2html id marker 901 $ \sigma(\sqrt{2})$$ �ιԤ��褬 $% latex2html id marker 903 $ \sqrt{2}$$ �� $% latex2html id marker 905 $ -\sqrt{2}$$ �Ǥ��뤫�ˤ�ä��ޤꡢ

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 909 $\displaystyle (\sqrt{2})/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 911 $\displaystyle ) \cong C_2 \quad($$ �̿� 2 �ν�� $\displaystyle )$

�� 10.2 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 917 $ (\sqrt{2},\sqrt{5})/$$ $\mbox{${\mathbb{Q}}$}$

�θ� $\sigma$ �ϡ� $% latex2html id marker 923 $ \sigma(\sqrt{2})$$ �� $% latex2html id marker 925 $ \sqrt{5}$$ �ιԤ�� ޤꡢ

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 929 $\displaystyle (\sqrt{2},\sqrt{5})/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle ) \cong C_2 \times C_2$

�� 10.3 $% latex2html id marker 934 $ \omega=(-1+\sqrt{-3})/2$$ �Ȥ��롣 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 938 $ (\sqrt[3]{2},\omega)/$$ $\mbox{${\mathbb{Q}}$}$

�θ� $\sigma$ �ϡ� $% latex2html id marker 944 $ \sigma(\sqrt[3]{2})$$ �� $\omega$ �ιԤ�� ޤꡢ

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 950 $\displaystyle (\sqrt[3]{2},\omega)/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle ) \cong \mathfrak{S}_3$

�� 10.1 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 957 $ (\sqrt{2}+\sqrt{7})/$$ $\mbox{${\mathbb{Q}}$}$

�ϤɤΤ褦�ʷ��ˤʤ��

�����III ���� No.10

��III �� No.10