Twisted Artin-Schreier Theory I: $(C_p \rtimes C_{p-1})$-extensions of local function fields

Over any field $K$ of characteristic $p$, we parametrize the $(C_p\rtimes C_{p-1})$-extensions of $K$ using pairs of elements $(g,a) \in K\times K^\times$, via a "twisted" variant of Artin-Schreier theory. Focusing on the case of the local function field $K := \mathbb F_q(\!(π)\!)$ over a finite fie