On the degree of a Kloosterman sum as an algebraic integer

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.