Bounds of Gauss sums in finite fields

We consider Gauss sums of the form [GRAPHICS] with a nontrivial additive character chi not equal chi(0) of a finite field F(p)m of pm elements of characteristic p. The classical bound \G(n)(a)\ less than or equal to (n - 1) p(m/2) becomes trivial for n greater than or equal to p(m/2) + 1. We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on \G(n)(a)\ which is nontrivial for the values of n of order up to p(m/2+1/6). We also show that for almost all primes one can obtain a bound which is nontrivial for the values of n of order up to p(m/2+1/2).