Let
A
=
F
q
[
T
]
A=\mathbb {F}_q[T]
be the ring of polynomials over the finite field
F
q
\mathbb {F}_q
and
0
≠
a
∈
A
0 \neq a \in A
. Let
C
C
be the
A
A
-Carlitz module. For a monic polynomial
m
∈
A
m\in A
, let
C
(
A
/
m
A
)
C(A/mA)
and
a
¯
\bar {a}
be the reductions of
C
C
and
a
a
modulo
m
A
mA
respectively. Let
f
a
(
m
)
f_a(m)
be the monic generator of the ideal
{
f
∈
A
,
C
f
(
a
¯
)
=
0
¯
}
\{f\in A, C_f(\bar {a}) =\bar {0}\}
on
C
(
A
/
m
A
)
C(A/mA)
. We denote by
ω
(
f
a
(
m
)
)
\omega (f_a(m))
the number of distinct monic irreducible factors of
f
a
(
m
)
f_a(m)
. If
q
≠
2
q\neq 2
or
q
=
2
q=2
and
a
≠
1
,
T
a\neq 1, T
, or
(
1
+
T
)
(1+T)
, we prove that there exists a normal distribution for the quantity
\[
ω
(
f
a
(
m
)
)
−
1
2
(
log
deg
m
)
2
1
3
(
log
deg
m
)
3
/
2
.
\frac {\omega (f_a(m))-\frac {1}{2}(\log \deg m)^2}{\frac {1}{\sqrt {3}}{(\log \deg m)^{3/2}}}.
\]
This result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of
b
b
modulo
n
n
, where
b
b
is an integer with
|
b
|
>
1
|b|>1
, and
n
n
a positive integer.