  In this paper we study products of quadratic residues modulo odd primes and
prove some identities involving quadratic residues. For instance, let $p$ be an
odd prime. We prove that if $p\equiv5\pmod8$, then
$$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where
$(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th
power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class
number formula, quartic Gauss sums, Stickelberger's congruence and values of
Dirichlet L-series at negative integers.
