On a Character Sum Problem of Cohn Original Research Article

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≠0. H. Cohn asked if it follows that f is a nontrivial multiplicative character provided that ∑xset membership, variantF f(x) image=−1 for h≠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.