We study the zeros
(
mod
p
)
\pmod p
of the polynomial
β
p
(
X
)
=
Σ
k
(
B
k
/
k
)
(
X
p
−
1
−
k
−
1
)
{\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1)
for
p
p
an odd prime, where
B
k
{B_k}
denotes the
k
k
th Bernoulli number and the summation extends over
1
⩽
k
⩽
p
−
2
1 \leqslant k \leqslant p - 2
. We establish a reciprocity law which relates the congruence
β
p
(
r
)
≡
0
(
mod
p
)
{\beta _p}(r) \equiv 0\;\pmod p
to a congruence
f
p
(
n
)
≡
0
(
mod
r
)
{f_p}(n) \equiv 0\,\pmod r
for
r
r
a prime less than
p
p
and
n
∈
Z
n \in {\mathbf {Z}}
. The polynomial
f
p
(
x
)
{f_p}(x)
is the irreducible polynomial over
Q
{\mathbf {Q}}
of the number
Tr
L
Q
(
ζ
)
ζ
\operatorname {Tr}_L^{{\mathbf {Q}}(\zeta )}\zeta
, where
ζ
\zeta
is a primitive
p
2
{p^2}
th root of unity and
L
⊂
Q
(
ζ
)
L \subset {\mathbf {Q}}(\zeta )
is the extension of degree
p
p
over
Q
{\mathbf {Q}}
. These congruences are closely related to the prime divisors of the indices
I
(
α
)
=
(
O
:
Z
[
α
]
)
I(\alpha ) = (\mathcal {O}:{\mathbf {Z}}[\alpha ])
, where
O
\mathcal {O}
is the integral closure in
L
L
and
α
∈
O
\alpha \in \mathcal {O}
is of degree
p
p
over
Q
{\mathbf {Q}}
. We establish congruences
(
mod
p
)
\pmod p
involving the numbers
I
(
α
)
I(\alpha )
and show that their prime divisors
r
≠
p
r \ne p
are closely related to the congruence
r
p
−
1
≡
1
(
mod
p
2
)
{r^{p - 1}} \equiv 1\,\pmod {p^2}
.