  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.
