Point counts of a variety (awaiting review)

If $X$ is a variety defined over a finite field $\mathbb{F}_q$, one can show that it will have finitely many points that are defined over that field. Upon extending the base field to $\mathbb{F}_{q^2}$, $\mathbb{F}_{q^3}$, $\mathbb{F}_{q^4}$, etc. the number of points of $X$ defined over those fields remains finite, as long as we consider only field extensions of finite degree over $\mathbb{F}_q$.