For the minimal polynomial
f
(
x
)
=
x
k
+
b
1
x
k
−
1
+
⋯
+
b
k
f(x) = {x^k} + {b_1}{x^{k - 1}} + \cdots + {b_k}
of
∑
n
=
0
p
−
1
exp
(
2
π
i
n
k
/
p
)
\sum \nolimits _{n = 0}^{p - 1} {\exp (2\pi i{n^k}/p)}
over
Q
Q
, where
p
p
is a
prime
≡
1
(
mod
k
)
\operatorname {prime} \equiv 1(\bmod k)
, we evaluate
b
1
,
b
2
{b_1},{b_2}
and prove
p
|
b
i
(
i
=
1
,
…
,
k
)
\left . p \right |{b_i}(i = 1, \ldots ,k)
but
p
2
∤
b
j
(
j
=
2
,
k
)
{p^2}\nmid {b_j}(j = 2,k)
. Also, we raise the interesting conjecture that
p
2
∤
b
j
{p^2}\nmid {b_j}
for
k
≥
j
≥
2
k \geq j \geq 2
.