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

Abstract: Let $L/K$ be a Galois extension of $p$-adic fields with Galois group $G$. Denote by $K[G]$ the group ring $\{\sum_{g\in G}a_gg\,\colon\,a_g\in K\}$; the classical normal basis theorem shows that $L$ is a free $K[G]$-module of rank $1$, that is, there exists an element $\alpha\inL$ such that $\{g(\alpha)\}_{g\in G}$ is a basis of $L$ as a $K$-vector space. It is natural to ask whether $\mathcal{O}_L$ is also a free $\mathcal{O}_K[G]$-module of rank $1$, where $\mathcal{O}_L$ and $\mathcal{O}_K$ denote the rings of integers of $L$ and $K$, respectively. A theorem of Noether tells us that this is the case if and only if the extension is (at most) tamely rami ed. When $L/K$ is wildly rami ed, we can still note that there always exists a free $\mathcal{O}_K[G]$-submodule of $\mathcal{O}L$ with finite index. The purpose of this talk is to study the minimal such index, i.e. the quantity $m(L/K) \coloneqq min{\alpha\in\mathcal{O}_L}[\mathcal{O}_L : \mathcal{O}_K[G]\alpha]$. We will provide a general bound that only depends on the invariants of the extension, a complete formula for $m(L/K)$ when $L/\mathbb{Q}_p$ is abelian and a complete formula when $L/K$ is cyclic of degree $p$. This is joint work with Ilaria Del Corso and Davide Lombardo.

Mathematics

Audience: advanced learners

Barcelona Mathematics Informal Seminar (SIMBa)

Series comments: SIMBa is a youth mathematics seminar organized by graduate students of the Barcelona area. It is aimed towards graduate and last course undergraduate students. Our goals are divulging the knowledge from different branches of mathematics for those interested and promote networking between the attendants.

This seminar is backed by the Faculty of Mathematics and Computer Science at Universitat de Barcelona, Faculty of Mathematics and Statistics at Universitat Politècnica de Catalunya, the Department of Mathematics from Universitat Autònoma de Barcelona, CRM, IMUB and BGSMath.