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