The genus fields of Artin-Schreier extensions

Abstract

Let $q$ be a power of a prime number $p$. Let $k={\mathbb{F}}_q(t)$ be the rational function field with constant field ${\mathbb{F}}_q$. Let $K=k(\alpha )$ 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 ${\mathbb{F}}_q[t]$ in $K$. And we also give an analogy of R$\acute{e}$dei-Reichardt's formulae for $K$.

9 pages, Corrected typos