Let
q
q
be an odd prime power, denote by
F
q
\mathbb {F}_q
the finite field with
q
q
elements, and set
A
≔
F
q
[
T
]
A ≔\mathbb {F}_q[T]
,
F
≔
F
q
(
T
)
F ≔\mathbb {F}_q(T)
. Let
ψ
:
A
→
F
{
τ
}
\psi : A \to F\{\tau \}
be a Drinfeld
A
A
-module over
F
F
, of rank
r
≥
2
r \geq 2
, with
E
n
d
F
¯
(
ψ
)
=
A
End_{\overline {F}}(\psi ) = A
. For a non-zero ideal
n
\mathfrak {n}
of
A
A
, denote its unique monic generator by
n
n
, denote the degree of
n
n
as a polynomial in
T
T
by
deg
n
\deg n
, and denote the
n
\mathfrak {n}
-division field of
ψ
\psi
by
F
(
ψ
[
n
]
)
F(\psi [\mathfrak {n}])
. A reciprocity law for
ψ
\psi
asserts that, if
gcd
(
c
h
a
r
F
,
r
)
=
1
\gcd (charF, r) = 1
or if
n
\mathfrak {n}
is prime, then a non-zero prime ideal
p
∤
n
\mathfrak {p} \nmid \mathfrak {n}
of
A
A
splits completely in
F
(
ψ
[
n
]
)
F(\psi [\mathfrak {n}])
if and only if the Frobenius trace
a
1
,
p
(
ψ
)
a_{1, \mathfrak {p}}(\psi )
of
ψ
\psi
at
p
\mathfrak {p}
and the first component
b
1
,
p
(
ψ
)
b_{1, \mathfrak {p}}(\psi )
of the Frobenius index of
ψ
\psi
at
p
\mathfrak {p}
satisfy the congruences
a
1
,
p
(
ψ
)
≡
−
r
(
mod
n
)
a_{1, \mathfrak {p}}(\psi ) \equiv -r \pmod n
and
b
1
,
p
(
ψ
)
≡
0
(
mod
n
)
b_{1, \mathfrak {p}}(\psi ) \equiv 0 \pmod n
. We find the Dirichlet density of the set of non-zero prime ideals
p
\mathfrak {p}
for which the latter congruence never holds, that is, for which
b
1
,
p
(
ψ
)
=
1
b_{1, \mathfrak {p}}(\psi ) = 1
. Using similar methods, we prove an asymptotic formula for the function of
x
x
defined by the average
1
#
{
p
:
deg
p
=
x
}
∑
p
:
deg
p
=
x
τ
A
(
b
1
,
p
(
ψ
)
)
\frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi ))
, where
p
=
A
p
\mathfrak {p} = A p
denotes an arbitrary non-zero prime ideal of
A
A
whose monic generator
p
∈
A
p \in A
has degree
x
x
and where
τ
A
(
b
1
,
p
(
ψ
)
)
\tau _A(b_{1, \mathfrak {p}}(\psi ))
denotes the number of monic divisors of
b
1
,
p
(
ψ
)
b_{1, \mathfrak {p}}(\psi )
.