On the Weil descent of Artin-Schreier algebraic function fields over finite fields

Let us consider a generalized Artin-Schreier algebraic function field extension $F$ of the rational function field $\F_{p^n}(x)$ defined over the finite field extension $K=\F_{p^n}$ of the prime field $\F_p$. We assume that $K$ is algebraically closed in $F$. We give general results on the descent over the fields $k= \F_{p^t}$ for $t$ dividing $n$. Then, we completely handle the bi-cyclic case of the descent over the fields $k_1=\F_{p}$ and $k_2= \F_{p^2}$ of all the sub-extensions of $F$ defined over $\F_{p^4}$. We give explicit examples with small prime numbers $p$.