Galois Scaffolds and Galois Module Structure for Totally Ramified $C_p^2$-Extensions in Characteristic 0

TL;DR

This paper extends the concept of Galois scaffolds from characteristic p to characteristic 0 for totally ramified cyclic extensions of degree p^2, providing tools to analyze Galois module structures.

Contribution

It adapts existing Galois scaffold conditions from characteristic p to characteristic 0 for degree p^2 extensions, advancing the understanding of Galois module structures in this setting.

Findings

Established conditions for Galois scaffold existence in characteristic 0

Provided a basis for analyzing valuation behavior in these extensions

Extended prior characteristic p results to characteristic 0

Abstract

Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified $p$-extension of local fields $L/K$, a Galois Scaffold gives us a $K$-basis for $K[G]$ whose effect on the valuation of elements of $L$ is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree $p^2$ in characteristic $p$. We take their work and adapt it to cyclic extensions of degree $p^2$ in characteristic $0$.