The genus fields of Artin-Schreier extensions

TL;DR

This paper explicitly describes the genus field and ambiguous ideal classes of Artin-Schreier extensions over rational function fields, and explores their impact on the class group structure, providing analogies to classical formulas.

Contribution

It provides explicit descriptions of genus fields and ambiguous ideal classes for Artin-Schreier extensions, and extends classical class group formulas to this setting.

Findings

Explicit description of genus fields for Artin-Schreier extensions

Analysis of the $p$-part of the class group in these extensions

Analogues of R$ m ilde{e}$dei-Reichardt formulas

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 $K$.