  For a prime power $q$, we show that the discriminants of monic polynomials in
$\mathbb{F}_q[x]$ of a fixed degree $m$ are equally distributed if
$\gcd(q-1,m(m-1))=2$ when $q$ is odd and $\gcd(q-1,m(m-1))=1$ if $q$ is even. A
theorem in the converse direction is proved when $q-1$ is squarefree.
