Descent and Decomposability of Algebras of Exponent 2 in characteristic 2

arXiv:2610.40673v1 Announce Type: new Abstract: We define an invariant of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2. It is built from the discriminant Pfister forms of symplectic involutions, takes values in a quotient of the third part of the graded group associated to the fundamental filtration of the Witt group of quadratic forms, and vanishes on tensor products of three quaternion algebras, thus giving a necessary condition for decomposability. Variants of the consctuction yield necessary and sufficient conditions for a biquaternion algebra to descend from a quadratic extension of the base field, and for a quaternion algebra to descent from a biquadratic extension that is not purely inseparable. Our results extend known results both in characteristic 2 and in characteristic different from 2, see the introduction. Using residue maps for quadratic forms, we show that these invariants do not vanish in general, and we give an explicit central simple algebra of degree 8 and exponent 2 with nonzero invariant, which is therefore indecomposable. As explained in the appendix, the purely inseparable case is quite different; in particular, a quaternion algebra descents if and only if its Brauer class also does.

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