Some Congruences of Kloosterman Sums and their Minimal Polynomials

TL;DR

This paper investigates properties of Kloosterman sums over finite fields, establishing minimal polynomials and characterizations modulo 27 using advanced number theory tools like Stickelberger's theorem and the Gross-Koblitz formula.

Contribution

It provides new results on the minimal polynomial of Kloosterman sums over Q and characterizes ternary Kloosterman sums modulo 27, advancing understanding of their algebraic structure.

Findings

Determined the minimal polynomial over Q of Kloosterman sums.

Characterized ternary Kloosterman sums modulo 27.

Applied Stickelberger's theorem and Gross-Koblitz formula to these problems.

Abstract

We prove two results on Kloosterman sums over finite fields, using Stickelberger's theorem and the Gross-Koblitz formula. The first result concerns the minimal polynomial over Q of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27.