@misc{ent-pir-22-symgrp-junk, author = {Entin, Alexei and Pirani, Noam}, title = {Abhyankar's Affine Arithmetic Conjecture for the Symmetric and Alternating Groups}, howpublished = {Online document at ArXiv Math.}, url = (https://arxiv.org/abs/2205.03879} number = {2205.03879}, pages = {13}, year = 2022, month = may, doi = {10.48550/arXiv.2205.03879}, comment = {Not related to AA at all, in spite of the title}, abstract = {We prove that for any prime $p > 2$, $q = p^\nu$ a power of $p$, $n \geq p$ 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 $\infinity$ with $\mathrm{Gal}(K/\mathbb{F}_q(T)) = G$, with the possible exception of $G = S_{p+1}$ if $F_q \supset F_{p^2}$. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic (with the possible exception of $S_{p+1}$ with $F_q \supset F_{p^2}$).} }