dimension of variety is independent of the choice of algebraically closed field?

Question

Suppose I have an affine algebraic set $X$ defined by some equations $f_i(x_1, \ldots, x_n) = 0$ for $i \in I$, where each $f_i$ is a polynomial with integer coefficients. I suspect that the dimensions of $X(\mathbb{C})$ and $X(\overline{\mathbb{Q}})$, considering $X$ as algebraic set inside affine $n$ space over $\mathbb{C}$ and $\overline{\mathbb{Q}}$, respectively, have the same dimensions, both being algebraically closed fields. How can one show this? Thank you!