鐃緒申鐃緒申惘藹� I 鐃緒申鐃緒申 No.9

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鐃緒申鐃緒申鐃瞬環」とわ申鐃緒申鐃緒申単鐃縮醐申鐃緒申鐃緒申腸調鐃緒申弔任鐃緒申鐃夙わ申鐃緒申鐃瞬環の緒申同鐃緒申鐃竣わ申単鐃縮醐申鐃緒申鐃楯つわ申里鐃緒申鐃緒申鐃粛わ申鐃暑こ鐃夙にわ申鐃殉わ申鐃緒申

鐃緒申鐃� 9.1 鐃緒申鐃緒申

鐃緒申鐃緒申鐃淑�申鐃緒申(unique factorization domain:UFD) 鐃叔わ申鐃緒申箸蓮鐃� $R\setminus \{0\}$ 鐃緒申任鐃春の醐申

鐃緒申鐃叔醐申鐃緒申鐃術わ申

$\displaystyle x=p_1 p_2\dots p_k ($ $p_1,...,p_k$ 鐃緒申 $R$ 鐃緒申鐃叔醐申 $\displaystyle )$

鐃塾よう鐃祝書けわ申箸鐃緒申妨鐃緒申鐃緒申泙鐃緒申鐃� ( $p_1,\dots,p_k$ 鐃緒申鐃緒申砲鐃銃縁申鐃緒申鐃塾わ申鐃緒申鐃獣てわ申茲わ申鐃�)

鐃緒申鐃� 9.2 鐃緒申鐃緒申

鐃緒申単鐃準イ鐃叔ワ申鐃緒申鐃緒申鐃緒申 (principal ideal domain:PID) 鐃叔わ申鐃緒申箸蓮鐃�

鐃塾わ申鐃駿てのワ申鐃叔ワ申鐃暑が単鐃準イ鐃叔ワ申鐃緒申(鐃緒申弔慮鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃暑イ鐃叔ワ申鐃緒申)鐃叔わ申鐃緒申箸鐃緒申妨鐃緒申鐃緒申泙鐃緒申鐃�

鐃緒申鐃� 9.3 鐃緒申鐃緒申

鐃緒申鐃醇ー鐃緒申鐃緒申奪鐃緒申鐃緒申鐃� (Euclidian domain:ED) 鐃叔わ申鐃緒申箸蓮鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�

鐃緒申鐃緒申鐃緒申 $\rho:R\to W$ (鐃瞬重わ申鐃竣わ申調鐃駿わ申鐃緒申鐃�)鐃緒申鐃緒申鐃獣て￥申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃祝醐申鐃緒申鐃殉わ申鐃緒申

鐃塾醐申鐃塾「重わ申鐃緒申 $\rho(a)$ 鐃緒申鐃叔常申 ${\Leftrightarrow}$
鐃塾醐申 $% latex2html id marker 1014 $ (a\neq 0)$$ 鐃緒申鐃出わ申鐃銃￥申

$% latex2html id marker 1016 $\displaystyle b=aq+r,\quad q,r\in R, \quad \rho(r)<\rho(a) $$
鐃夙なわ申鐃緒申存鐃淳わ申鐃暑。

鐃緒申鐃緒申 9.1 鐃緒申

鐃塾醐申

鐃祝つわ申鐃銃￥申鐃緒申鐃緒申鐃緒申弔鐃銃縁申佑任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃叔醐申鐃叔わ申鐃暑。鐃緒申鐃淑わ申鐃� 鐃緒申鐃緒申鐃緒申鐃准わ申弌鐃� 鐃緒申鐃緒申鐃塾どわ申鐃初か鐃緒申鐃緒申鐃旬醐申鐃叔わ申鐃暑。
鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� 鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 9.1 (鐃緒申鐃藷集刻申鐃緒申鐃緒申鐃塾鰹申認)

鐃緒申罅種申鐃緒申鐃獣ワ申鐃緒申鐃緒申箸鐃緒申鐃� $\rho:R\to W$ 鐃緒申鐃緒申鐃� 9.3 鐃祝出てわ申鐃緒申鐃緒申鐃緒申箸鐃緒申泙鐃緒申鐃緒申鐃緒申了鐃緒申鐃叔わ申佞鐃�

鐃緒申鐃緒申分鐃緒申鐃緒申

鐃緒申鐃出わ申鐃銃￥申

鐃緒申鐃緒申能鼎鐃緒申鐃緒申脳鐃緒申里鐃緒申(鐃瞬逸申鐃瞬軽わ申鐃緒申痢鐃�)鐃緒申鐃緒申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 9.2 鐃醇ー鐃緒申鐃緒申奪鐃緒申鐃緒申鐃緒申単鐃準イ鐃叔ワ申鐃緒申鐃緒申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�(鐃述ワ申函鐃�

鐃緒申罅種申鐃緒申鐃獣ワ申鐃緒申鐃緒申箸鐃緒申鐃�

鐃薯そのワ申鐃叔ワ申鐃緒申琉鐃縦とわ申鐃殉わ申鐃緒申

鐃淑わ申単鐃緒申 (0 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�)鐃塾でよい鐃緒申 $% latex2html id marker 1081 $ I \neq 0$$ 鐃淑ら、 $I\setminus\{0\}$ 鐃塾醐申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃瞬軽わ申鐃緒申鐃緒申箸辰討澆覆鐃緒申鐃緒申鐃�)

鐃緒申鐃緒申 9.5 $% latex2html id marker 1096 $ \omega=\frac{-1+\sqrt{-3}}{2}$$ (鐃緒申鐃塾誌申鐃処根鐃塾逸申鐃�)鐃夙わ申鐃緒申箸鐃緒申鐃� $\mbox{${\mathbb{Q}}$}$ $[\omega]$ 鐃緒申鐃塾でわ申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 9.6 $% latex2html id marker 1102 $ {\mbox{${\mathbb{Z}}$}}[\sqrt{-1}]$$ 鐃熟「重み関随申鐃緒申 $\rho$ 鐃緒申

$% latex2html id marker 1105 $\displaystyle \rho:{\mbox{${\mathbb{Z}}$}}[\sqrt{-1}]$$	$\displaystyle \to$	$% latex2html id marker 1107 $\displaystyle {\mbox{${\mathbb{Z}}$}}_{\geq 0}$$
$\displaystyle \cup \!\! \!\! \shortmid$		$\displaystyle \cup \!\! \!\! \shortmid$
$% latex2html id marker 1110 $\displaystyle m+n\sqrt{-1}$$	$\displaystyle \mapsto$	$\displaystyle m^2+n^2$

鐃緒申鐃緒申鐃緒申鐃緒申襪鰹申箸砲鐃所、鐃醇ー鐃緒申鐃緒申奪百弔箸覆襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�(鐃述ワ申函鐃� $% latex2html id marker 1114 $ \alpha,\beta \in {\mbox{${\mathbb{Z}}$}}[\sqrt{-1}], \beta \neq 0$$ 鐃夙わ申鐃緒申箸鐃緒申鐃� $\alpha$ 鐃緒申 $\beta$ 鐃叔鰹申辰鐃緒申鐃緒申鐃緒申鐃緒申砲蓮鐃� $\alpha/\beta$ 鐃緒申 $% latex2html id marker 1122 $ {\mbox{${\mathbb{Z}}$}}[\sqrt{-1}]$$ 鐃塾醐申鐃叔駈申鐃緒申鐃緒申討澆覆鐃緒申鐃緒申鐃�)

鐃緒申鐃緒申 9.7

任鐃春わ申複鐃叔随申鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃緒申 $% latex2html id marker 1127 $ \gamma\in {\mbox{${\mathbb{Z}}$}}[\sqrt{-2}]$$ 鐃緒申鐃緒申鐃獣て￥申鐃緒申 $\gamma$ 鐃夙の居申離 $\vert\gamma-z\vert$ 鐃緒申鐃淑駕申鐃叔わ申鐃暑こ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申(鐃述ワ申函鐃淑ｏ申鐃淑随申名鐃緒申 $% latex2html id marker 1137 $ {\mbox{${\mathbb{Z}}$}}[\sqrt{-2}]$$ 鐃塾醐申鐃緒申廛鐃緒申奪箸鐃緒申討澆覆鐃緒申鐃緒申鐃�)
$% latex2html id marker 1139 $ {\mbox{${\mathbb{Z}}$}}[\sqrt{-2}]$$ 鐃熟「重み関随申鐃緒申 $\rho$ 鐃緒申

$% latex2html id marker 1142 $\displaystyle \rho:{\mbox{${\mathbb{Z}}$}}[\sqrt{-2}]$$ $\displaystyle \to$ $% latex2html id marker 1144 $\displaystyle {\mbox{${\mathbb{Z}}$}}_{\geq 0}$$

$\displaystyle \cup \!\! \!\! \shortmid$ $\displaystyle \cup \!\! \!\! \shortmid$

$% latex2html id marker 1147 $\displaystyle m+n\sqrt{-2}$$ $\displaystyle \mapsto$ $\displaystyle m^2+2n^2$

鐃緒申鐃緒申鐃緒申鐃緒申襪鰹申箸砲鐃所、鐃醇ー鐃緒申鐃緒申奪百弔箸覆襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 9.8 $% latex2html id marker 1152 $ \omega=\frac{-1+\sqrt{-3}}{2}$$ (

鐃塾誌申鐃処根鐃塾逸申鐃�) 鐃夙わ申鐃殉わ申鐃緒申鐃緒申鐃塾とわ申鐃緒申 ${\mbox{${\mathbb{Z}}$}}[\omega]$ 鐃熟ユー鐃緒申鐃緒申奪百弔任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 9.9 鐃緒申鐃緒申 $% latex2html id marker 1159 $ {\mbox{${\mathbb{Z}}$}}[\sqrt{-1}]$$ 鐃塾ワ申鐃叔ワ申鐃緒申鐃緒申単鐃淑件申鐃祝なわ申鐃緒申鐃淑わ申鐃緒申鐃緒申

$% latex2html id marker 1161 $ (3+4\sqrt{-1},5)$$
$% latex2html id marker 1163 $ (5+12\sqrt{-1},13)$$
$% latex2html id marker 1165 $ (7+8\sqrt{-1},9)$$

$% latex2html id marker 1142 $\displaystyle \rho:{\mbox{${\mathbb{Z}}$}}[\sqrt{-2}]$$	$\displaystyle \to$	$% latex2html id marker 1144 $\displaystyle {\mbox{${\mathbb{Z}}$}}_{\geq 0}$$
$\displaystyle \cup \!\! \!\! \shortmid$		$\displaystyle \cup \!\! \!\! \shortmid$
$% latex2html id marker 1147 $\displaystyle m+n\sqrt{-2}$$	$\displaystyle \mapsto$	$\displaystyle m^2+2n^2$