  We provide an index bound for character sums of polynomials over finite
fields. This improves the Weil bound for high degree polynomials with small
indices, as well as polynomials with large indices that are generated by
cyclotomic mappings of small indices. As an application, we also give some
general bounds for numbers of solutions of some Artin-Schreier equations and
mininum weights of some cyclic codes.
