The genus fields of Artin–Schreier extensions

Abstract

Let q be a power of a prime number p. Let k = F q ( t ) be the rational function field with constant field F q . Let K = k ( α ) be an Artin–Schreier extension of k. In this paper, we explicitly describe the ambiguous ideal classes and the genus field of K. Using these results, we study the p-part of the ideal class group of the integral closure of F q [ t ] in K. We also give an analogue of the Rédei–Reichardt formula for K.