On algebraically maximal valued fields that are not defectless

Abstract

An example originally given by F.~Delon shows the existence of an algebraically maximal discretely valued field of characteristic $p>0$ which admits purely inseparable extensions of degree $p^2$ with defect $p$. These extensions are not generated by a single element. Using a trick introduced in an earlier paper of the author, we construct algebraically maximal valued fields, of characteristic $p$ as well as of characteristic $0$, which admit separable extensions of degree $p^2$ with defect $p$. They are of rank 2 and it is an open question whether such examples having rank 1 exist.

To appear in the Proceedings of the ddg40 conference (Banyuls-sur-mer, 2025), to be published in the "Séminaires et Congrès" of the Société mathématique de France