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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.'
address:
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author:
- Ilaria Del Corso$^1$
- Fabio Ferri$^2$
- Davide Lombardo$^1$
bibliography:
- 'biblio.bib'
title: 'How far is an extension of $p$-adic fields from having a normal integral basis?'
---

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Introduction
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Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $G$, ramification index $e_{L/K}$ and inertia degree $f_{L/K}$, and let $\mathcal{O}_K$ (respectively $\mathcal{O}_L$) denote the ring of integers of $K$ (resp. $L$). For any $p$-adic field $F$, let $e_F$ and $f_F$ be the absolute ramification index and inertia degree, $\pi_F$ a uniformiser, and $v_F$ the valuation, normalised by $v_F(\pi_F)=1$. We will write $K[G]$ for the $K$-algebra whose elements are formal linear combinations of elements of $G$ with $K$-coefficients. We similarly define the group ring $\mathcal{O}_K[G]$ and observe that $L$ is in a natural way a $K[G]$-module (resp. $\mathcal{O}_L$ is a $\mathcal{O}_K[G]$-module). The classical normal basis 