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Legendre symbol

DEFINITION 6.1   Let $ p$ be an odd prime. Let $ a$ be an integer which is not divisible by $ p$ . Then we define the Legendre symbol $ \left(\frac{a}{p}\right)$ by the following formula.

\begin{displaymath}
{\left(\frac{a}{p}\right)}=
\begin{cases}1 & \text{if }(X^2...
...ucible over }\mathbb{F}_p\\
-1 & \text{otherwise}
\end{cases}\end{displaymath}

We further define

$\displaystyle {\left(\frac{a}{p}\right)}= 0$$\displaystyle \text { if } a \in p\mathbb{Z}.
$

LEMMA 6.2   Let $ p$ be an odd prime. Then:
  1. $ {\left(\frac{a}{p}\right)}= a^{(p-1)/2} \mod p $
  2. $ {\left(\frac{ab}{p}\right)}= {\left(\frac{a}{p}\right)}
{\left(\frac{b}{p}\right)} $

We note in particular that $ {\left(\frac{-1}{p}\right)}=(-1)^{(p-1)/2} $ .

DEFINITION 6.3   Let $ p,\ell$ be distinct odd primes. Let $ \lambda$ be a primitive $ \ell$ -th root of unity in an extension field of $ {\mathbb{F}}_p$ . Then for any integer $ a$ , we define a Gauss sum $ \tau_a$ as follows.

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

$ \tau_1$ is simply denoted as $ \tau$ .

LEMMA 6.4  
  1. $ \tau_a={\left(\frac{a}{\ell}\right)}\tau$ .
  2. $ \sum_{a=0}^{l-1} \tau_a \tau_{-a}=\ell(\ell-1)$ .
  3. $ \tau^2=(-1)^{(\ell-1)/2}\ell$ ( $ =\ell^*$ (say)).
  4. $ \tau^{p-1}=(\ell^*)^{(p-1)/2}$ .
  5. $ \tau^p=\tau_{p}$ .

THEOREM 6.5   $ {\left(\frac{p}{\ell}\right)}={\left(\frac{\ell^*}{p}\right)} $ ( where $ \ell^*=(-1)^{(\ell-1)/2}\ell$ ) $ {\left(\frac{-1}{\ell}\right)}=(-1)^{(\ell-1)/2} $ $ {\left(\frac{2}{\ell}\right)}=(-1)^{(\ell^2-1)/8} $


next up previous
Next: About this document ... Up: Congruent zeta functions. No.6 Previous: Congruent zeta functions. No.6
2007-06-01