How far is an extension of $p$-adic fields from having a normal integral basis?

TL;DR

This paper investigates the minimal index of free modules within the ring of integers of Galois extensions of p-adic fields, providing exact results for specific cases such as cyclic extensions of degree p.

Contribution

It precisely determines the minimal index of free modules in the ring of integers for certain Galois p-adic extensions, including cyclic degree p extensions.

Findings

Exact minimal index values for specific Galois extensions

Results for cyclic extensions of degree p

Enhanced understanding of normal integral bases in p-adic fields

Abstract

Let $L/K$ be a finite Galois extension of $p$-adic fields with group $G$. It is well-known that $\mathcal{O}_L$ contains a free $\mathcal{O}_K[G]$-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree $p$ of $p$-adic fields.