Estimates of character sums in finite fields

We prove analogs of the Burgess estimates for character sums over n-dimensional segments in the field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{F}_{p^n } $$\end{document}.