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 k2× 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.