The genus fields of Artin-Schreier extensions

Details The genus fields of Artin-Schreier extensions The genus fields of Artin-Schreier extensions

2009-06-25

arxiv.org/abs/0906.4626v2

Mathematics

Number Theory

9 pages, Corrected typos

Scientific paper

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 $K$.

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