quadratic reciprocity ruleQuadraticReciprocityRule2001-08-13 10:23:162007-06-19 20:52:32Theorem\usepackage{amssymb}
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\newtheorem*{thm}{Theorem}\begin{thm}[Law of Quadratic Reciprocity]
Let $p$ and $q$ be two distinct odd primes. Then:
$$ \left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{(p-1)(q-1)/4} $$
where $\left(\frac{\cdot}{\cdot}\right)$ is the \PMlinkname{Jacobi}{JacobiSymbol} symbol (or Legendre symbol).
\end{thm}
The following is an equivalent formulation of the Law of Quadratic Reciprocity:
\begin{thm}[Quadratic Reciprocity (second form)]
Let $p,q$ be distinct odd primes. Then:
\begin{enumerate}
\item $\displaystyle \left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$ if one of $p$ or $q$ is congruent to $1$ modulo $4$;
\item $\displaystyle \left(\frac{p}{q}\right) = - \left(\frac{q}{p}\right)$ if both $p$ and $q$ are congruent to $3$ modulo $4$.
\end{enumerate}
\end{thm}