Galois scaffolds for extraspecial p-extensions in characteristic 0

TL;DR

This paper establishes conditions under which certain ramified extensions of local fields with extraspecial p-groups admit Galois scaffolds, leading to insights on the structure of their rings of integers and associated orders.

Contribution

It provides new sufficient conditions for the existence of Galois scaffolds in extraspecial p-extensions over characteristic 0 local fields, and explores their implications for ring of integers and Hopf orders.

Findings

Conditions for Galois scaffold existence in extraspecial p-extensions

Criteria for the ring of integers to be free over its associated order

Stricter conditions ensuring the associated order is a Hopf order

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 $K[G]$.