Anne Quéguiner-Mathieu and Jean-Pierre Tignol: Algebras with involution that become hyperbolic over the fonction field of a conic,

Submission: 2008, May 26

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.

2000 Mathematics Subject Classification: 16W10 ; 11E04

Keywords and Phrases: Algebras with involution ; fonction field of a conic ; minimal quadratic forms.

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