 
Summary: On the pure Jacobi Sums
Shigeki Akiyama
Let p be an odd prime and Fq be the field of q = p2
elements. We consider the Jacobi
sum over Fq;
J(, ) =
xFq
(x)(1  x),
where , is a non trivial character of F×
q , whose value at 0 is defined to be 0. It is well
known that the absolute value of J(, ) is
q = p, when is not principal. According
to [11], [9], call the Jacobi sum J(, ) pure if J(, )/p is a root of unity.
Let ord() be the order of in F×
q . From now on in this paper, we assume that ord() = 2
and ord() = n 3. This special type of Jacobi sums play an important role in evaluating
the argument of Gauss sum:
G() =
xFq
