---
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 ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, even though it contains $Q$ and is totally decomposable into a tensor product of quaternion algebras.'
address:
- |
    Université Paris 13 (LAGA)\
    CNRS (UMR 7539)\
    Université Paris 12 (IUFM)\
    93430 Villetaneuse, France
- |
    Département de Mathématiques\
    Université catholique de Louvain\
    Chemin du cyclotron, 2\
    B1348 Louvain-la-Neuve\
    Belgique
author:
- 'Anne Quéguiner-Mathieu'
- 'Jean-Pierre Tignol'
title: Algebras with involution that become hyperbolic over the function field of a conic
---

Given two central simple algebras with involution $(A,\sigma)$ and $(B,\tau)$ over a field $F$, we say that $(A,\sigma)$ contains $(B,\tau)$ if $A$ contains a $\sigma$-stable subalgebra isomorphic to $B$ over which the involution induced by $\sigma$ is conjugate to $\tau$. By the double centralizer theorem [@KMRT (1.5)], this is also equivalent to saying that $(A,\sigma)$ is isomorphic to a tensor product $(A,\sigma)\simeq(B,\tau)\otimes(C,\gamma)$ for some central simple algebra with involution $(C,\gamma)$ over $F$.

