 
Summary: On the pure Jacobi Sums
Shigeki Akiyama
Let p be an odd prime and F q be the eld of q = p 2 elements. We consider the Jacobi
sum over F q ;
J(; ) =
X
x2Fq
(x) (1 x);
where ; is a non trivial character of F
q
, whose value at 0 is dened to be 0. It is well
known that the absolute value of J(; ) is p
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 d
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:
