Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2

TL;DR

This paper constructs examples of quaternion and octonion division algebras over fields of characteristic 2 that split over certain purely inseparable extensions but not over any smaller subextensions, answering a specific open question.

Contribution

It provides explicit examples of splitting behavior of quaternions and octonions over purely inseparable extensions in characteristic 2, addressing an open question in the field.

Findings

Examples of quaternion and octonion algebras splitting over degree ≥4 purely inseparable extensions.

Counterexamples showing they do not split over any smaller subextensions.

Analysis of isotropy behavior of associated norm forms.

Abstract

We give examples of quaternion and octonion division algebras over a field $F$ of characteristic $2$ that split over a purely inseparable extension $E$ of $F$ of degree $\geq 4$ but that do not split over any subextension of $F$ inside $E$ of lower exponent, or, in the case of octonions, over any simple subextension of $F$ inside $E$. Thus, we get a negative answer to a question posed by Bernhard M\"uhlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms.