鐃緒申鐃緒申鐃�III 鐃緒申鐃緒申 No.10

$\fbox{ガロア群には十分たくさんの元があること}$ 鐃淑駕申鐃緒申鐃緒申鐃塾講居申鐃叔は「ワ申鐃緒申鐃緒申鐃緒申鐃緒申廚噺鐃緒申鐃緒申鐃粛㍼申村鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申味鐃緒申鐃暑こ鐃夙にわ申鐃暑。鐃緒申鐃緒申法鐃緒申鐃緒申痢廚噺鐃緒申鐃緒申鐃粛㍼申鐃緒申鐃緒申鐃� $\mbox{${\mathbb{Q}}$}$ 鐃緒申鐃緒申分鐃塾とわ申鐃銃含わ申茲�申覆鐃塾をさわ申鐃緒申鐃夙にわ申鐃暑。

鐃緒申

鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申鐃夙わ申鐃緒申 $\operatorname{Gal}(L/K)$ 鐃塾醐申鐃熟どのわ申鐃初い鐃緒申鐃緒申里鐃緒申鐃緒申鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 10.1

鐃緒申

鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申里箸鐃緒申鐃�

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

鐃緒申 10.1

鐃緒申

鐃塾ワ申鐃緒申鐃緒申鐃緒申鐃緒申里箸鐃緒申鐃�

鐃緒申鐃緒申分鐃緒申鐃緒申鐃緒申泙鐃緒申鐃�( 鐃緒申鐃緒申鐃緒申鐃緒申鐃�) 鐃緒申与鐃緒申鐃緒申鐃緒申函鐃� $\operatorname{Gal}(L/M)$ 鐃緒申 $\operatorname{Gal}(L/K)$ 鐃緒申鐃緒申分鐃緒申鐃夙みなわ申鐃緒申鐃夙わ申鐃叔わ申鐃暑。
鐃緒申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申弔鐃緒申鐃緒申鐃緒申鐃緒申 $M_1 \subset M_2$ 鐃緒申澆鐃緒申鐃緒申覆鐃出￥申

$\displaystyle \operatorname{Gal}(L/M_1)\supset \operatorname{Gal}(L/M_2)$
鐃緒申澆鐃緒申鐃緒申鐃�

鐃緒申単鐃祝醐申鐃緒申鐃出￥申鐃緒申鐃緒申

鐃緒申鐃獣わ申 $G=\operatorname{Gal}(G/K)$ 鐃緒申鐃獣わ申鐃瞬わ申鐃緒申鐃緒申鐃緒申鐃緒申里任鐃緒申襦ｏ申鐃緒申鐃緒申

鐃緒申

鐃緒申鐃緒申分鐃緒申 $H=\operatorname{Gal}(M/K)$ 鐃緒申鐃出縁申鐃緒申鐃緒申鐃暑。

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃縦わ申鐃藷げよう鐃緒申

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

鐃塾醐申 $\sigma$ 鐃熟￥申 $% latex2html id marker 901 $ \sigma(\sqrt{2})$$ 鐃塾行わ申鐃処が $% latex2html id marker 903 $ \sqrt{2}$$ 鐃緒申 $% latex2html id marker 905 $ -\sqrt{2}$$ 鐃叔わ申鐃暑か鐃祝わ申辰鐃緒申鐃殉り、

$\displaystyle \operatorname{Gal}($ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 909 $\displaystyle (\sqrt{2})/$$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $% latex2html id marker 911 $\displaystyle ) \cong C_2 \quad($$ 鐃縮随申 2 鐃塾緒申鐃� $\displaystyle )$

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

鐃塾醐申 $\sigma$ 鐃熟￥申 $% latex2html id marker 923 $ \sigma(\sqrt{2})$$ 鐃緒申 $% latex2html id marker 925 $ \sqrt{5}$$ 鐃塾行わ申鐃緒申鐃� 鐃緒申泙蝓�

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

鐃緒申 10.3 $% latex2html id marker 934 $ \omega=(-1+\sqrt{-3})/2$$ 鐃夙わ申鐃暑。 $\operatorname{Gal}($ $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 938 $ (\sqrt[3]{2},\omega)/$$ $\mbox{${\mathbb{Q}}$}$

鐃塾醐申 $\sigma$ 鐃熟￥申 $% latex2html id marker 944 $ \sigma(\sqrt[3]{2})$$ 鐃緒申 $\omega$ 鐃塾行わ申鐃緒申鐃� 鐃緒申泙蝓�

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

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

鐃熟どのよう鐃淑件申鐃祝なわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃�