I want to show that if $F(T) \in B[T]$, where $B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)$, is monic and has a root $\alpha \in\mathcal K(B)$ then $\alpha$ actually lives in $B$. This will imply that $B$ is an integrally closed domain and hence that $\operatorname{Spec} B$ is normal. But I am stuck at this point.
On the other hand, I know that the geometric interpration is that a quadric of $\operatorname{rk} \ge 3$ doesn't have "bad" singularities. Well, I know that such quadrics are irreducible, but that only helps me to show that $B$ is at least an integral domain.
