On zeros of Kloosterman sums

A Kloosterman zero is a non-zero element of ${{\mathbb F}_q}$ for which the Kloosterman sum on ${{\mathbb F}_q}$ attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in ${{\mathbb F}_q}$ belongs to a proper subfield of ${{\mathbb F}_q}$ with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.