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

Anne Quéguiner-Mathieu (1) , Jean-Pierre Tignol (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Département de mathématique [Louvain]

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.

Mots clés: Algebras with involution; Fonction field of a conic

Domaines: Anneaux et algèbres [math.RA]

Anne Quéguiner-Mathieu, Jean-Pierre Tignol. Algebras with involution that become hyperbolic over the fonction field of a conic. 2008. ⟨hal-00282238⟩