Suppose $(F) \subset k[x_1, \cdots, x_n] $ defines an affine irreducible variety $V(F)$ with a non-singular point $p=(a_n, \ldots, a_n)\in V$. Define the linear approximation $L=\displaystyle\sum_i \frac{\partial F}{\partial x_i}(p) (x_i-a_i) $ around $p$. The field $k$ is arbitrary.
Can we determine the Krull dimension of the ideal $(F, L)$? What about $\operatorname{rad} (F, L)=\sqrt{(F, L)}$?