Chain Lemma for Biquaternion Algebras in Characteristic 2

TL;DR

This paper establishes bounds on the number of steps needed to transform one symbol presentation of a biquaternion algebra in characteristic two into another, with specific constraints on unchanged entries per step.

Contribution

It introduces explicit step bounds for transforming symbol presentations of biquaternion algebras in characteristic two, improving understanding of their algebraic structure.

Findings

At most three steps to change one symbol presentation to another with one entry fixed.

Up to fifteen steps if two entries are fixed in each step.

Up to forty-five steps using more basic transformations.

Abstract

In this paper, we prove that for a given biquaternion algebra over a field of characteristic two, one can move from one symbol presentation to another by at most three steps, such that in each step at least one entry remains unchanged. If one requires that in each step two entries remain the same then their number increases to fifteen. We provide even more basic steps that in order to move from one symbol presentation to another one needs to use up to forty-five of them.