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The maximal degree over rational numbers that an n-dimensinonal Kloostermansum 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 alwaysachieved 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 theKloosterman sums are distinct except Frobenius conjugation. For these fields wecompletely determine the degrees of all the Kloosterman sums. Even if thefinite field does not satisfy this condition we can still often find points inwhich the Kloosterman sum has smaller degree than (p-1)/d.

On the degree of a Kloosterman sum as an algebraic integer