Splitting quaternion algebras defined over a finite field extension

TL;DR

This paper investigates the splitting of quaternion algebras over finite field extensions, providing bounds on the degree of 2-extensions needed to split central simple algebras of degree 16.

Contribution

It introduces new methods to determine splitting fields for quaternion algebras over finite extensions, establishing explicit degree bounds for such extensions.

Findings

Every central simple algebra of degree 16 is split by a 2-extension of degree at most 2^{16}

Systems of quadratic forms over fields can be used to analyze algebra splitting

New bounds on splitting fields for quaternion algebras over finite extensions

Abstract

We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every central simple algebra of degree 16 is split by a 2-extension of degree at most 2^{16}.