Algebras with involution that become hyperbolic over the fonction field of a conic

TL;DR

This paper classifies degree-4 central simple algebras with involution that become hyperbolic over the function field of a conic associated with a quaternion algebra, and provides an example of a degree-8 algebra with specific properties.

Contribution

It offers a classification of such algebras and constructs a novel example of a degree-8 algebra with particular involution properties.

Findings

Classified degree-4 algebras with involution becoming hyperbolic over the conic function field.

Constructed a degree-8 division algebra with orthogonal involution that contains but does not contain a specific quaternion algebra.

Provided explicit examples illustrating the theoretical classification.

Abstract

We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra $Q$. We classify these algebras in degree~4 and give an example of such a division algebra with orthogonal involution of degree~8 that does not contain $Q$ with its canonical involution, even though it contains $Q$ and is totally decomposable into a tensor product of quaternion algebras with involution.