Abstract
We prove some partial results concerning the following problem:Assume that F is a finite field, aiis a complex number for each i∈F such that a0=0,a1=1, |ai|=1for all i∈F\\{0},and∑i∈Fai+jāi=−1for all i∈F\\{0}.Does it follow that the function i→aiis a multiplicative character of F? We prove (in the case |F|=p,pis a prime) on the one hand that there is only a finite number of complex solutions; on the other hand we solve completely a modpversion of the problem. The proofs are mainly elementary, except for applying a theorem of Chevalley from algebraic geometry
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