I am confused with some computation in Galois theory (this is not homework, just my weird curiosity).
Let $k$ be a field of positive characteristic $p\neq 2$ that contains all roots of unity (e.g. algebraically closed), let $K$ be the Artin-Schreier extension $k(x)[\mu]/(\mu^p-\mu-x)$ and let $L$ be the extension of $K$ obtained by adjoining the square root of $\mu$, call it $\alpha$. Now $K$ is Galois over $k(x)$ with Galois group cyclic of order $p$, as it is an Artin-Schreir extension. $L$ is Galois over $K$ with Galois group cyclic of order 2, as a Kummer extension. The Galois group $G$ of $L$ over $k(x)$ must be an extension of $C_p$ by $C_2$ $$ 1 \to C_2 \to G \to C_p \to 1 $$ By Schur-Zassenhaus (most likely there is a simpler way to see it) since the orders of $C_p$ and $C_2$ are coprime, $G$ is a semiderect product, and since there is no non-trivial $C_p$-module structure on $C_2$, it is a direct product.
I wonder what is the action of the generator of $C_p$ on $L$ and what is the subextension of $L$ of degree 2 fixed by the action of $C_p$.