Minimal Polynomials and Distinctness of Kloosterman Sums

Abstract

In this paper, we use the Stickelberger theorem on Gauss sums to study the minimal polynomial and distinctness of n-dimensional Kloosterman sums over a finite field Fq of q elements. In particular, we improve some results of Fisher [Contemp. Math. 133 (1992), 81–102; Kloosterman sums as algebraic integers, Math. Ann., to appear.] obtained by ℓ-adic methods. We also show that the minimal polynomials of certain exponential sums (including many Kloosterman sums) are p-Eisensteinian in a broader sense.