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 -adic fields from having a normal integral basis?
Resum: Let be a Galois extension of
-adic fields with Galois group
. Denote by
the group ring
the classical normal basis theorem shows that
is a free
-module of rank 1 . that is, there exists an element
such that
is a basis of
as a
-vector space. It is natural to ask whether
is also a free
-module of rank 1 , where
and
denote the rings of integers of
and
, respectively. A theorem of Noether tells us that this is the case if and only if the extension is (at most) tamely ramified. When
is wildly ramified, we can still note that there always exists a free
-submodule of
with finite index. The purpose of this talk is to study the minimal such index, i.e. the quantity
We will provide a general bound that only depends on the invariants of the extension, a complete formula for
when
is abelian and a complete formula when
is cyclic of degree
. This is joint work with Ilaria Del Corso and Davide Lombardo.
Si voleu estar al cas de les xerrades previstes, podeu consultar el calendari. Si voleu proposar una xerrada, ompliu el formulari. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu
.