Isotropy and full splitting pattern of quasilinear \(p\)-forms (Q6588185)

Suppose $F$ is a field and $\varphi$ a quadratic form over it. The splitting pattern $\mathrm{SP}(\varphi)$ of $\varphi$ will be defined as the set of dimensions realized by the anisotropic part of $\varphi$ over all possible field extensions of \textit{M. Knebusch} [Proc. Lond. Math. Soc. (3) 33, 65--93 (1976; Zbl 0351.15016)] studied the splitting pattern when characteristic of the field is not 2. Knebusch presented an explicit tower of fields $F_0\subseteq F_1\subseteq\cdots\subseteq F_{h}$, called the standard splitting tower. The $\mathrm{SP}(\varphi)$ has a property: $m\in\mathrm{SP}(\varphi)$ if and only if m is the dimension of the anisotropic part of $\varphi$ over one of the fields $F_{i}$ in the standard splitting tower. It can also consider an analogous standard splitting tower over fields of characteristic 2. But, in this case several different types of quadratic forms must be considered. For nonsingular quadratic forms, the standard splitting tower has been defined by \textit{M. Knebusch} [Proc. Lond. Math. Soc. (3) 34, 1--31 (1977; Zbl 0359.15013)], and it has been extended to the general case by \textit{A. Laghribi} [Math. Z. 240, No. 4, 711--730 (2002; Zbl 1007.11017)]. The author of the paper under review, extends the knowledge on isotropy and full splitting patterns of totally singular quadratic forms to prove her results in a more general setting of quasilinear $p$-forms, as they are a generalization of the concept of totally singular quadratic forms over fields of characteristic 2.