  We generalize Erd\H{o}s's conjecture on integer covering systems to
polynomial rings on finite field, for example, $\mathbb F_2[x]$, and proved the
theorem: Every system of $n$ cosets of ideals in $\mathbb F_q[x]$ covering all
polynomials whose degree is less than $n$ covers $\mathbb F_q[x]$. When $q=2$,
our result is sharp.
