A Kloosterman zero is a non-zero element of F_q for which the Kloosterman sum on 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 F_q belongs to a proper subfield of 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.

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