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

TL;DR

This paper investigates the behavior of quasilinear p-forms over certain field extensions, demonstrating invariance of their defect and computing their splitting patterns, thus advancing understanding of their algebraic structure.

Contribution

It proves the invariance of defect over specific field extensions and calculates the full splitting pattern for certain families of quasilinear p-forms, extending prior results.

Findings

Defect remains unchanged over certain radical extensions.

Full splitting patterns are explicitly computed for specific families.

Strengthens Hoffmann's 2004 results on 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.