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

2020/12/15 by Hoffmann, Detlev W.
#11E04 11E81 12F15 16H05 16K20 17A35 17A75 #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2012.09644

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.

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