On the Values of Kloosterman Sums

Given a prime p and a positive integer n, we show that the shifted Kloosterman sums Sigma_{xisinF p n}Psi(x + alphax^pn-2)=Sigma_{xisinF* p n}Psi(x+alphax^-1)+1, alphaisinF*_pn where Psi is a nontrivial additive character of a finite field F_pn of pⁿ elements, do not vanish if alpha belongs to a small subfield F_pm sube F_pn. This complements recent results of P. Charpin and G. Gong which in turn were motivated by some applications to bent functions.