Involutions, odd-degree extensions and generic splitting

Abstract

Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two quadratic forms $q,q^{\prime }$ over $F$, if $q_L\cong q_L^{\prime }$ then $q\cong q^{\prime }$. It is natural to ask whether similar results hold for algebras with involution. We give a survey of the progress on these three questions with particular attention to the relevance of hyperbolicity, isotropy and isomorphism over some {appropriate} function field. Incidentally, we prove the anisotropy property in some {new} low degree cases.