On the degree of a Kloosterman sum as an algebraic integer

TL;DR

This paper investigates the algebraic integer degrees of n-dimensional Kloosterman sums over finite fields, establishing conditions for maximal degrees and analyzing their distribution across different fields.

Contribution

It provides a complete determination of Kloosterman sum degrees for certain finite fields and explores degree variations in other fields.

Findings

Maximal degree of Kloosterman sums is (p-1)/gcd(p-1,n+1).

Maximal degree points have nonzero absolute trace.

Degrees are often smaller in fields not meeting specific conditions.

Abstract

The maximal degree over rational numbers that an n-dimensinonal Kloosterman sum defined over a finite field of characteristic p can achieve is known to be (p-1)/d where d=gcd(p-1,n+1). Wan has shown that this maximal degree is always achieved in points whose absolute trace is nonzero. By the works of Fischer, Wan we know that there exist many finite fields for which the values of the Kloosterman sums are distinct except Frobenius conjugation. For these fields we completely determine the degrees of all the Kloosterman sums. Even if the finite field does not satisfy this condition we can still often find points in which the Kloosterman sum has smaller degree than (p-1)/d.