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SUMMARY:Fabio Ferri (University of Exeter)
DTSTART:20210310T110000Z
DTEND:20210310T120000Z
DTSTAMP:20260423T004547Z
UID:SIMBa/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SIMBa/6/">Ho
 w far is an extension of $p$-adic fields from having a normal integral bas
 is?</a>\nby Fabio Ferri (University of Exeter) as part of Barcelona Mathem
 atics Informal Seminar (SIMBa)\n\n\nAbstract\nLet $L/K$ be a Galois extens
 ion of $p$-adic \nfields 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\n$\\{g(\\alpha)
 \\}_{g\\in G}$ is a basis of $L$ as a $K$-vector space. It is natural to a
 sk whether $\\mathcal{O}_L$ is also a free $\\mathcal{O}_K[G]$-module of r
 ank $1$\, 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\ne
 d. When $L/K$ is wildly rami\ned\, we can still note that there always exi
 sts a free $\\mathcal{O}_K[G]$-submodule of $\\mathcal{O}L$ with \nfinite 
 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 onl
 y 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 $L/K$ i
 s cyclic of degree $p$. This is joint work with Ilaria Del Corso and David
 e Lombardo.\n
LOCATION:https://researchseminars.org/talk/SIMBa/6/
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