Solution
= Solution
The extension $K/F$ is a <vector space> over $F$. Since $K$ is finite, a maximal linearly independent subset is finite and spans it; let its size be $d$. Relative to this <basis>, every element of $K$ has a unique list of $d$ coefficients in $F$. There are $q$ choices for each coefficient, so
$$
\boxed{|K|=q^d,\qquad d=[K:F].}
$$
The inclusion $F\subseteq K$ makes $d\geq1$.