Involutions, odd degree extensions and generic splitting

  • Jodi Black

    Bucknell University, Lewisburg, USA
  • Anne Quéguiner-Mathieu

    Université Paris 13, Sorbonne Paris Cité, Villetaneuse, France

Abstract

Let be a quadratic form over a field and let be a field extension of of odd degree. It is a classical result that if is isotropic (resp. hyperbolic) then is isotropic (resp. hyperbolic). In turn, given two quadratic forms over , if then . It is natural to ask whether similar results hold for algebras with involution. We give a general overview of recent and important progress on these three questions, with particular attention to the relevance of hyperbolicity, isotropy and isomorphism over some appropriate function field. In addition, we prove the anisotropy property in some new low degree cases.

Cite this article

Jodi Black, Anne Quéguiner-Mathieu, Involutions, odd degree extensions and generic splitting. Enseign. Math. 60 (2014), no. 3/4, pp. 377–395

DOI 10.4171/LEM/60-3/4-6