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 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.