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: 鐃緒申鐃緒申文鐃緒申砲弔鐃緒申鐃�...

鐃緒申鐃緒申鐃�II 鐃緒申鐃緒申 No.13

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{平方剰余の相互法則II}

鐃緒申鐃緒申平鐃緒申鐃緒申余鐃緒申鐃緒申鐃祝‖э申両鐃緒申鐃緒申鐃緒申弔辰討鐃緒申泙辰討鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃緒申鐃渋わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃書いてわ申鐃緒申鐃緒申鐃緒申

鐃緒申鐃� 13.1   $ p,\ell$ 鐃緒申鐃緒申曚覆鐃緒申鐃叔随申鐃叔わ申鐃緒申箸鐃緒申鐃� $ {\mathbb{F}}_p$ 鐃塾鰹申鐃緒申鐃塾わ申 $ 1$ 鐃塾醐申鐃緒申 $ \ell$-鐃処根 $ \lambda$ 鐃緒申鐃暑。 鐃緒申鐃緒申 $ a$ 鐃緒申鐃出わ申鐃銃￥申有鐃緒申鐃塾のワ申鐃緒申鐃緒申鐃緒申 $ \tau_a$ 鐃緒申

$\displaystyle \tau_a=\sum_{t=1}^{\ell-1}{\left(\frac{t}{\ell}\right)}\lambda^{at} $

鐃緒申鐃緒申鐃緒申鐃緒申襦� $ \tau_1$ 鐃塾わ申鐃夙わ申単鐃緒申 $ \tau$ 鐃夙わ申鐃緒申鐃緒申

鐃縦わ申鐃叔にわ申鐃緒申鐃緒申鐃緒申琉鐃縮ｏ申砲弔鐃緒申討發�鐃緒申鐃緒申鐃緒申鐃緒申鐃述べてわ申鐃緒申鐃緒申鐃緒申 鐃殉わ申鐃緒申鐃緒申鐃� $ \lambda$ 鐃熟種申鐃塾よう鐃祝わ申鐃銃とるこ鐃夙わ申鐃叔わ申鐃暑。 $ {\mathbb{F}}_{p^{\ell-1}}^\times$ 鐃熟位随申 $ p^{\ell-1}-1$ 鐃塾緒申鷏欧如鐃� 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $ \xi $ 鐃夙わ申鐃緒申鐃緒申 鐃春ワ申鐃緒申沺鐃緒申両鐃緒申鐃緒申鐃緒申砲鐃緒申 $ p^{\ell-1}-1$ 鐃緒申 $ \ell$ 鐃叔鰹申鐃緒申擇鐃暑か鐃初、 $ \xi^{(p^{\ell-1}-1)/\ell}$ 鐃緒申佑鐃緒申襪鰹申箸鐃緒申任鐃緒申襦ｏ申鐃緒申慮鐃緒申鐃� 鐃縮随申鐃熟わ申鐃順う鐃緒申 $ \ell$ 鐃叔わ申鐃暑か鐃初、鐃緒申鐃緒申鐃� $ \lambda$ 鐃夙わ申鐃緒申个鐃緒申鐃緒申鐃� 鐃淑わ申鐃緒申$ \lambda$ 鐃塾種申鐃緒申鐃緒申楼鐃緒申的鐃叔はなわ申鐃緒申 1鐃緒申 $ \ell$ 鐃処根鐃緒申 $ \lambda^a$ $ a=0,1,2,\dots,p-1$ 鐃緒申 $ p$ 鐃縦わ申鐃暑。 鐃緒申鐃緒申鐃緒申鐃出縁申鐃緒申鐃緒申 $ \tau_a$ 鐃緒申鐃叔わ申鐃暑。鐃緒申鐃緒申鐃緒申鐃緒申鐃�(1)鐃熟わ申鐃緒申蕕�鐃緒申鐃塾削申鐃緒申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申劼戮討鐃緒申襦ｏ申鐃緒申鐃緒申鐃緒申鐃�$ a=0$ 鐃塾とわ申鐃緒申鐃緒申鐃緒申 鐃緒申鐃縮で￥申$ \tau_0=0$ 鐃緒申鐃淑りた鐃縦￥申

鐃緒申鐃緒申 13.1   鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申立鐃縦￥申
  1. $ \tau_a={\left(\frac{a}{\ell}\right)}\tau$.
  2. $ \sum_{a=0}^{\ell-1} \tau_a \tau_{-a}=\ell(\ell-1)$.
  3. $ \tau^2=(-1)^{(\ell-1)/2}\ell$ ($ =\ell^*$ 鐃夙緒申).
  4. $ \tau^{p-1}=(\ell^*)^{(p-1)/2}$.
  5. $ \tau^p=\tau_{p}$.

鐃緒申鐃緒申 13.1 (平鐃緒申鐃緒申余鐃緒申鐃緒申鐃祝‖�)   鐃緒申鐃叔随申 $ \ell$ 鐃緒申鐃出わ申鐃銃種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申立鐃縦￥申
  1. $ {\left(\frac{p}{\ell}\right)}={\left(\frac{\ell^*}{p}\right)} $ (但鐃緒申 $ \ell^*=(-1)^{(\ell-1)/2}\ell$)

鐃緒申鐃緒申 13.1   $ p=31$, $ \ell=5$ 鐃祝わ申鐃緒申鐃緒申鐃銃￥申 $ {\mathbb{F}}_p$ 鐃緒申 $ 1$鐃塾醐申鐃緒申 $ \ell$-鐃処根鐃緒申 鐃緒申銚鐃緒申弔鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� $ \tau$ 鐃緒申鐃緒申董鐃�$ \tau^2$ 鐃緒申尊櫃坊彁鐃緒申鐃緒申討澆覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 13.2   鐃緒申鐃叔随申 $ \ell$ 鐃夙￥申鐃緒申 $ K$ 鐃緒申与鐃緒申鐃緒申鐃銃わ申鐃銃￥申 $ 1$ 鐃塾醐申鐃緒申 $ \ell$ 鐃処根 $ \lambda$ 鐃緒申 $ K$ 鐃塾なわ申鐃緒申存鐃淳わ申鐃緒申箸鐃緒申襦� (鐃夙わ申鐃祝￥申$ K$ 鐃緒申標鐃緒申鐃緒申 $ \ell$ 鐃叔はなわ申鐃緒申) $ {\mathbb{F}}_\ell$ 鐃緒申鐃� $ K$-鐃粛関随申鐃緒申鐃緒申鐃緒申 $ V$ 鐃緒申 (鐃緒申鐃緒申鐃緒申鐃夙の駕申法鐃緒申鐃緒申鐃緒申鐃初ー鐃旬にわ申鐃�) $ K$ 鐃緒申離戰鐃緒申肇鐃緒申鐃瞬になり、 $ V$ 鐃緒申鐃緒申鐃緒申僂鐃�

$\displaystyle \langle f,g \rangle=\sum_{\ell=0}^{\ell-1}f(a)g(-a) $

鐃緒申鐃緒申泙襦�(鐃緒申鐃緒申鐃緒申鐃緒申)
  1. $ V$ 鐃緒申 $ K$ 鐃緒申亮鐃緒申鐃緒申鐃緒申鐃処。(鐃緒申鐃塾みでよい鐃緒申)
  2. (鐃緒申標鐃緒申直鐃緒申鐃緒申鐃緒申) $ a=0,1,\dots,\ell-1$ 鐃緒申鐃出わ申鐃銃￥申$ V$ 鐃塾醐申 $ \chi_a$ 鐃緒申

    % latex2html id marker 992 $\displaystyle \chi_a(x)=\lambda^{ax} $

    鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申鐃緒申 $ \langle \chi_a,\chi_b \rangle$ 鐃緒申鐃緒申茵�
  3. $ \{\chi_a\}_{a=0}^{\ell-1}$ 鐃緒申 $ V$ 鐃塾器申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃� 鐃緒申鐃緒申法鐃�$ V$ 鐃塾醐申 $ f$ 鐃緒申

    $\displaystyle f=\sum_{a=0}^{\ell-1} c_a \chi_a $

    鐃夙書くわ申鐃緒申砲蓮鐃� $ c_a \in K$ 鐃緒申匹里茲�鐃祝居申鐃緒申个茲わ申鐃緒申鐃緒申劼戮覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 13.3   鐃緒申鐃緒申鐃銃縁申鐃緒申鐃緒申鐃塾駕申鐃叔￥申 鐃春￥申鐃所エ鐃術器申 $ \mathcal F:V\to V$ 鐃緒申

$\displaystyle \mathcal F[f](a)=\langle f,\chi_a \rangle $

鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃�
  1. $ \langle \mathcal F[f],\chi_a\rangle =\ell f(-a)$ 鐃緒申鐃淑りた鐃縦わ申鐃緒申 鐃薯示わ申鐃淑わ申鐃緒申鐃緒申
  2. $ \mathcal F[\mathcal F[f]]$ 鐃緒申彁鐃緒申鐃緒申鐃� $ \mathcal F$ 鐃塾居申鐃術器申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�
  3. 任鐃春わ申 $ f,g\in V$ 鐃緒申鐃出わ申鐃銃￥申 $ \langle \mathcal F[f],g \rangle=\langle f,\mathcal F [g]\rangle$ 鐃緒申 鐃緒申鐃緒申立鐃縦わ申鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申
  4. $ f_L\in V $ 鐃緒申

    % latex2html id marker 1025 $\displaystyle f_L (a 1_{{\mathbb{F}}_\ell}) ={\left(\frac{a}{\ell}\right)}1_K \qquad (a \in {\mbox{${\mathbb{Z}}$}}) $

    鐃緒申鐃緒申鐃緒申鐃緒申襦�$ f_L$ 鐃緒申 $ \mathcal F$ 鐃塾醐申有鐃駿ワ申鐃夙ワ申任鐃緒申襪鰹申箸鮗┐鐃緒申鐃� 鐃緒申鐃曙が属鐃緒申鐃緒申鐃粛�鐃粛わ申鐃緒申覆鐃緒申鐃緒申鐃�


2002年7鐃緒申15鐃緒申