Rationally isotropic quadratic spaces are locally isotropic. III

Abstract:

Let $R$ be a regular semilocal domain containing a field such that all the residue fields are infinite. Let $K$ be the fraction field of $R$. Let $(R^n,q\colon R^n \to R)$ be a quadratic space over $R$ such that the quadric $\{q=0\}$ is smooth over $R$. If the quadratic space $(R^n,q\colon R^n \to R)$ over $R$ is isotropic over $K$, then there is a unimodular vector $v \in R^n$ such that $q(v)=0$. If $\mathrm {char}(R)=2$, then in the case of even $n$ the assumption on $q$ is equivalent to the fact that $q$ is a nonsingular quadratic space and in the case of odd $n > 2$ this assumption on $q$ is equivalent to the fact that $q$ is a semiregular quadratic space.