  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 $K$.
