Abstract
We study trigonometric sums in finite fields \(F_Q \). The Weil estimate of such sums is well known: \(|S(f)| \leqslant ({\text{deg }}f - 1)\sqrt Q \), where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, \((Q,2) = 2\), for which \(|S(f)|\) attains the largest possible value and, in particular, \(|S(f)| = ({\text{deg }}f - 1)\sqrt Q \).
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