ï¿½ï¿½ï¿½ï¿½ï¿½III ï¿½ï¿½ï¿½ï¿½ No.10

ï¿½ï¿½

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

ï¿½Ä¤ï¿½ï¿½Î¤ï¿½ï¿½È¤ï¿½ï¿½ï¿½ï¿½Åªï¿½Ç¤ï¿½ï¿½ë¡£

ï¿½ï¿½ï¿½ï¿½Ê¬ï¿½ï¿½ ï¿½ï¿½ ï¿½ï¿½Þ¤ï¿½ï¿½ï¿½( ï¿½ï¿½ ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½) ï¿½ï¿½Í¿ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½È¡ï¿½ $\operatorname{Gal}(L/M)$ ï¿½ï¿½ $\operatorname{Gal}(L/K)$ ï¿½ï¿½ï¿½ï¿½Ê¬ï¿½ï¿½ï¿½È¤ß¤Ê¤ï¿½ï¿½ï¿½ï¿½È¤ï¿½ï¿½Ç¤ï¿½ï¿½ë¡£
ï¿½ï¿½ ï¿½Î¤ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Ä¤ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ ï¿½ï¿½ $M_1 \subset M_2$ ï¿½ï¿½ß¤ï¿½ï¿½ï¿½ï¿½Ê¤ï¿½Ð¡ï¿½

$\displaystyle \operatorname{Gal}(L/M_1)\supset \operatorname{Gal}(L/M_2)$

ï¿½ï¿½ß¤ï¿½ï¿½ï¿½ï¿½ï¿½

ï¿½ï¿½ 10.1 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 889 $ (\sqrt{2})/$$ $\mbox{${\mathbb{Q}}$}$

ï¿½Î¸ï¿½ $\sigma$ ï¿½Ï¡ï¿½ $% latex2html id marker 895 $ \sigma(\sqrt{2})$$ ï¿½ï¿½ $% latex2html id marker 897 $ \sqrt{2}$$ ï¿½ï¿½ $% latex2html id marker 899 $ -\sqrt{2}$$ ï¿½Ç¤ï¿½ï¿½ë¤«ï¿½Ë¤ï¿½Ã¤ï¿½ï¿½ï¿½Þ¤ê¡¢

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 903 $\displaystyle (\sqrt{2})/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 905 $\displaystyle ) \cong C_2 \quad($$ ï¿½Ì¿ï¿½ 2 ï¿½Î½ï¿½ï¿½ $\displaystyle )$

ï¿½ï¿½ 10.2 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 915 $ (\sqrt{2},\sqrt{5})/$$ $\mbox{${\mathbb{Q}}$}$

ï¿½Î¸ï¿½ $\sigma$ ï¿½Ï¡ï¿½ $% latex2html id marker 921 $ \sigma(\sqrt{2})$$ ï¿½ï¿½ $% latex2html id marker 923 $ \sigma(\sqrt{5})$$ ï¿½ï¿½ï¿½ï¿½Þ¤ê¡¢

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 927 $\displaystyle (\sqrt{2},\sqrt{5})/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle ) \cong C_2 \times C_2$

ï¿½ï¿½ 10.3 $% latex2html id marker 936 $ \omega=(-1+\sqrt{-3})/2$$ ï¿½È¤ï¿½ï¿½ë¡£ $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 940 $ (\sqrt[3]{2},\omega)/$$ $\mbox{${\mathbb{Q}}$}$

ï¿½Î¸ï¿½ $\sigma$ ï¿½Ï¡ï¿½ $% latex2html id marker 946 $ \sigma(\sqrt[3]{2})$$ ï¿½ï¿½ $\sigma(\omega)$ ï¿½ï¿½ï¿½ï¿½Þ¤ê¡¢

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 952 $\displaystyle (\sqrt[3]{2},\omega)/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle ) \cong \mathfrak{S}_3$

ï¿½ï¿½ï¿½ï¿½ 10.1 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 963 $ (\sqrt{2}+\sqrt{7})/$$ $\mbox{${\mathbb{Q}}$}$