The descent of biquaternion algebras in characteristic two

TL;DR

This paper introduces an invariant for biquaternion algebras over fields of characteristic two, which detects when such algebras descend from a subfield, and explores its properties under various field extensions.

Contribution

It defines a new invariant for biquaternion algebras in characteristic two and characterizes when these algebras descend from a subfield, advancing understanding of their structure.

Findings

Invariant is trivial iff the algebra descends from the subfield

Behavior of the invariant under field extensions analyzed

Several examples illustrating the invariant's properties

Abstract

In this paper we associate an invariant to a biquaternion algebra $B$ over a field $K$ with a subfield $F$ such that $K/F$ is a quadratic separable extension and $\operatorname{char}(F)=2$. We show that this invariant is trivial exactly when $B \cong B_0 \otimes K$ for some biquaternion algebra $B_0$ over $F$. We also study the behavior of this invariant under certain field extensions and provide several interesting examples.