Isotropy and full splitting pattern of quasilinear $p$-forms

Abstract

For a quasilinear $p$-form defined over a field $F$ of characteristic $p>0$, we prove that its defect over the field $F(\sqrt[p^{n_1}]{a_1}, \dots, \sqrt[p^{n_r}]{a_r})$ equals to its defect over the field $F(\sqrt[p]{a_1}, \dots, \sqrt[p]{a_r})$, strengthening a result of Hoffmann from 2004. We also compute the full splitting pattern of some families of quasilinear $p$-forms.