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

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 ≥ 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 Mühlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms.