Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

TL;DR

This paper extends Albert-Draxl's theorem by exploring the transfer of quadratic forms and quaternion algebras over quadratic extensions, providing new insights into their structure and embedding properties.

Contribution

It establishes a modified version of the Albert-Draxl theorem involving corestriction of quaternion algebras over quadratic extensions.

Findings

Corestriction of quaternion algebra can be embedded over quadratic extensions.

Transfer of quadratic forms can reveal subforms over base fields.

New criteria for isotropy of transferred quadratic forms.

Abstract

A theorem of Albert-Draxl states that if a tensor product of two quaternion division algebras $Q_1$, $Q_2$ over a field $F$ is not a division algebra, then there exists a separable quadratic extension of $F$ that embeds as a subfield in $Q_1$ and in $Q_2$. We establish a modified version of this result where the tensor product of quaternion algebras is replaced by the corestriction of a single quaternion algebra over a separable field extension. As a tool in the proof, we show that if the transfer of a nonsingular quadratic form $\varphi$ over a quadratic extension is isotropic for a linear functional $s$ such that $s(1)=0$, then $\varphi$ contains a nondegenerate subform defined over the base field.