Notes on a problem of H. Cohn

We prove some partial results concerning the following problem: Assume that F is a finite field, a(i) is a complex number for each i is an element of F such that a(0) = 0, a(1) = 1, /a(i)/ = 1 for all i is an element of F/{0} , and Sigma(i is an element of F)a(i + j)(a) over bar(i) = = 1 for all i is an element of F/{0}. Does it follow that the function i --> a(i) is a mult iplicative character of F? We prove (in the case /F/ = p, p is a prime) on the one hand that there is only a finite number of complex solutions: on th e other hand we solve completely a mod p version of the problem. The proofs are mainly elementary. except for applying a theorem of Chevalley from alg ebraic geometry. (C) 1999 Academic Press.