  We generalize Burgess' results on partial Gaussian sums to arbitrary finite
fields. The main ingredients are the classical method of amplification, two
deep results on multiplicative energy for subsets in finite fields which are
obtained respectively by the tools from additive combinatorics and geometry of
numbers, and a technique of Chamizo for treating the difficulty caused by
additive character. Our results include the recent works on character sums in
finite fields by M.-C. Chang and S. V. Konyagin.
