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

El proper dimecres, 10 de març, se celebrarà una nova xerrada —en format virtual— del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Fabio Ferri
Universitat: University of Exeter

Data: Wednesday, March 10th, 2021.
Hora: 12:00,virtual coffee break; 12:20, talk.
Lloc: Zoom (the link will be posted on our website).
Idioma: English.

Títol: ¿How far is an extension of $p$-adic fields from having a normal integral basis?
Resum: Let $L / K$ be a Galois extension of $p$ -adic fields with Galois group $G$. Denote by $K[G]$ the group ring $\left\{\sum_{g \in G} a_{g} g: a_{g} \in K\right\} ;$ the classical normal basis theorem shows that $L$ is a free $K[G]$ -module of rank 1 . that is, there exists an element $\alpha \in L$ 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 ramified. When $L / K$ is wildly ramified, 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 $m(L / K):=$ $\min _{\alpha \in \mathcal{O}_{L}}\left[\mathcal{O}_{L}: \mathcal{O}_{K}[G] \alpha\right] .$ 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.

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