Abhyankar's affine arithmetic conjecture for the symmetric and alternating groups

Abstract

We prove that for any prime p>2, q=pν a power of p, n≥p and G=Sn or G=An (symmetric or alternating group), there exists a Galois extension K/Fq(T) ramified only over ∞ with Gal(K/Fq(T))=G. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.