While most proofs of the Weil bound on one-variable Kloosterman sums over finite fields are carried out in all characteristics, the original proof of this bound, by Weil, assumes the characteristic is odd. We show how to make Weil's argument work in even characteristic, for both ordinary Kloosterman sums and sums twisted by a multiplicative character. © 2002 Elsevier Science (USA). All rights reserved.
Conrad, K. (2002). On Weil’s proof of the bound for Kloosterman sums. Journal of Number Theory, 97(2), 439–446. https://doi.org/10.1016/S0022-314X(02)00011-2