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

Abstract

Over any field $K$ of characteristic $p$, we parametrize the $(C_p⋊C_{p-1})$-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 $K:={\mathbb{F}}_q((\pi ))$ over a finite field ${\mathbb{F}}_q$ of characteristic $p$, we use this parametrization to obtain an exact formula for the count of all $(C_p⋊C_{p-1})$-extensions of $K$ with a given discriminant.

arXiv comment22 pages, 3 figures