Galois scaffolds for extraspecial p-extensions in characteristic 0

Abstract

Let $K$ be a local field of characteristic 0 with residue characteristic $p$. Let $G$ be an extraspecial $p$-group and let $L/K$ be a totally ramified $G$-extension. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers ${\mathfrak{O}}_L$ to be free of rank 1 over its associated order ${\mathfrak{A}}_{L/K}$, and to stricter conditions which imply that ${\mathfrak{A}}_{L/K}$ is a Hopf order in the group ring $K[G]$.