  Let $f$ be a complex valued function on a finite field $F$ such that $f(0) =
0$, $f(1) = 1$, and $|f(x)| = 1$ for $x \neq 0$. Cohn asked if it follows that
$f$ is a nontrivial multiplicative character provided that $\sum_{x \in F} f(x)
\bar{f(x+h)} = -1$ for $h \neq 0$. We prove that this is the case for finite
fields of prime cardinality under the assumption that the nonzero values of $f$
are roots of unity.
