  We prove that for any prime $p>2$, $q=p^\nu$ a power of $p$, $n\ge p$ and
$G=S_n$ or $G=A_n$ (symmetric or alternating group) there exists a Galois
extension $K/\mathbb F_q(T)$ ramified only over $\infty$ with
$\mathrm{Gal}(K/\mathbb F_q(T))=G$. This confirms a conjecture of Abhyankar for
the case of symmetric and alternating groups over finite fields of odd
characteristic.
