Galois scaffolds for cyclic $p^n$-extensions in characteristic $p$

TL;DR

This paper establishes conditions under which certain cyclic p^n-extensions in characteristic p fields admit Galois scaffolds, impacting the structure of their rings of integers and associated orders.

Contribution

It provides new sufficient conditions for the existence of Galois scaffolds in cyclic p^n-extensions in characteristic p, linking to the structure of rings of integers and Hopf orders.

Findings

Conditions for Galois scaffold existence in cyclic p^n-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 $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. 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 $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.