Abhyankar's Affine Arithmetic Conjecture for the Symmetric and Alternating Groups

Abstract

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.