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

TL;DR

This paper constructs specific totally ramified Galois extensions of local fields in characteristic p with Galois scaffolds, advancing understanding of their algebraic structure and associated orders.

Contribution

It introduces a method to produce G-extensions with Galois scaffolds and explores their properties related to the ring of integers and Hopf orders.

Findings

Existence of G-extensions with Galois scaffolds

Construction of extensions with free ring of integers over the associated order

Extensions where the associated order is a Hopf order in the group ring

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 $K[G]$.