In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for
Z
\mathbb {Z}
, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed
10
16
10^{16}
. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough’s bound to
616
,
000
616,000
. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for
F
q
[
x
]
\mathbb {F}_q[x]
of multiplicity
s
s
is bounded by a constant depending only on
s
s
and
q
q
. Consequently, we successfully resolve the minimum modulus problem for
F
q
[
x
]
\mathbb {F}_q[x]
and disprove a conjecture by Azlin [Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011].