Galois scaffolds for $p$-extensions in characteristic $p$

Abstract

Let $K$ be a local field of characteristic $p>0$ with perfect residue field and let $G$ be a finite $p$-group. In this paper we use Saltman's construction of a generic $G$-extension of rings of characteristic $p$ to construct totally ramified $G$-extensions $L/K$ that have Galois scaffolds. We specialize this construction to produce $G$-extensions $L/K$ such that the ring of integers $O_L$ is free of rank 1 over its associated order $A_0$, and extensions such that $A_0$ is a Hopf order in the group ring $K[G]$.