The ring of integers of degree p extensions of p-adic fields – Daniel Gil-Munoz (Charles University - Univerzita Karlova)
mercoledì 23 ott 2024 3:00 PM — 4:00 PM Aula Seminari (Department of Mathematics) Seminars
Galois module theory aims to describe the structure of the ring of integers of Galois extensions of local and global fields as a module over a suitable ring depending on the Galois group of the extension. The best choice is the associated order in the Galois group algebra, defined as the set of elements in the Galois group algebra whose action on the extension leaves the ring of integers invariant. This situation generalizes naturally to Hopf-Galois extensions, which are those extensions for which there are Hopf algebras acting on the top field in such a way that it mimics the Galois action (the so called Hopf-Galois structures). Namely, we can define the notion of associated order at any Hopf-Galois structure, and it endows the ring of integers with module structure. Whether such a module is free or not is a problem of long-standing interest. In this talk we shall study this question for the case of ramified degree p extensions of p-adic fields, where p is an odd prime. We shall review the case of cyclic degree p extensions, which was solved in the 1972 work by F. Bertrandias, J.P. Bertrandias and M.-J. Ferton, and afterwards, we shall see a solution for the general case.