Splitting of quadratic Pfister forms over purely inseparable extensions in characteristic 2

Abstract

We give examples of anisotropic n-fold quadratic Pfister forms (n≥2) over a suitably chosen field F of characteristic 2 that split over a purely inseparable extension E of F of exponent ≥2 but that do not split over any subextension of F inside E of lower exponent. For n≥3, we also give examples of such Pfister forms that split over a purely inseparable extension E of F of degree ≥4 but that do not split over any simple subextension of F inside E. As a consequence, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss who asked whether an octonion division algebra over a field of characteristic 2 that splits over a purely inseparable extension will already split over some quadratic subextension. We use the fact that the splitting of such an algebra is controlled by the splitting of the associated norm form which in turn is given by a 3-fold Pfister form.