Character sums over norm groups

In this paper, we give estimates of character sums of Weil type. They are associated with polynomials and rational functions with coefficients in a quadratic extension k2 of a finite field k of q elements, additive and multiplicative characters of k2, and summed over the kernel of the norm map from k× 2 to k×. The bounds are of the magnitude O( √ q) with the implied constant depending on the degree of the polynomial and the total degree of the support of the rational function, similar to Weil’s estimates over k. © 2004 Elsevier Inc. All rights reserved.