Zariski closure of an infinite cyclic group of diagonal matrices

Question

Suppose that $\Gamma=\{$exp $kX\mid k\in\mathbb{Z}\}$ where $X\in\mathfrak{gl}(n,\mathbb{R})$ is a diagonal matrix. How do we prove that the Zariski closure of $\Gamma$ must contain exp $tX$ for all real number $t$?

Answer 1

Hint: The group $\mathbb{Z}$ is Zariski dense in $\mathbb{R}$, since any nonzero polynomial $f(x)$ has only finitely many roots, so any polynomial $f(x)$ such that $f(k) = 0$ for all $k\in \mathbb{Z}$ must vanish identically. Similarly $\mathbb{Z}^n$ is Zariski dense in $\mathbb{R}^n$.